recursive program synthesis€¦ · can apply length (the recursive call) on the input values...

Recursive Program Synthesis

Aws Albarghouthi (UToronto), Sumit Gulwani (MSR), and Zachary Kincaid (UToronto)

CAV 2013Saint Petersburg, Russia

Program Synthesis

2

MagicSpecification Program

Program Synthesis

2

MagicSpecification Program

What are your Dreams?

3

MagicSpecification Program

What are your Dreams?

3

MagicSpecification ProgramFirst-order Logic

What are your Dreams?

3

MagicSpecification ProgramFirst-order Logic

Temporal Logic

What are your Dreams?

3

Magic ProgramFirst-order Logic

Temporal LogicI/O Examples

Why Synthesis from I/O?

I/O examples are easy to specify

4

Why Synthesis from I/O?

I/O examples are easy to specify

Non-expert users can specify I/O behaviour

• See, e.g., FlashFill for Excel, Smartphone scripts, etc.

• Desired programs are usually simple

4

Contribution

5

EscherI/O Examples RecursiveProgram

Contribution

5

EscherI/O Examples RecursiveProgram

User

Contribution

5

EscherI/O Examples RecursiveProgram

Parameterized by a set of building blocks (operations)integers, lists, trees, etc.

User

Contribution

5

EscherI/O Examples RecursiveProgram

Parameterized by a set of building blocks (operations)

Novel search-based synthesis techniqueintegers, lists, trees, etc.

User

Contribution

5

EscherI/O Examples RecursiveProgram

Parameterized by a set of building blocks (operations)

Novel search-based synthesis techniqueintegers, lists, trees, etc.

User

No templates required

High Level View

6

Forward Search

High Level View

6

Forward Search

I1 In. . .

High Level View

6

Forward Search

I1 In. . .

O1. . .On

High Level View

6

Forward Search

I1 In. . .

O1. . .On

P1 Pn. . .

High Level View

6

Forward Search Conditional Inference

I1 In. . .

O1. . .On

P1 Pn. . .

I1

O1

+

I2

O2

P1 P2

High Level View

6

Forward Search Conditional Inference

I1 In. . .

O1. . .On

P1 Pn. . .

I1

O1

+

I2

O2

P1 P2 =

if ( )

else

CP1

P2

High Level View

6

Forward Search Conditional Inference

I1 In. . .

O1. . .On

P1 Pn. . .

I1

O1

+

I2

O2

P1 P2 =

if ( )

else

CP1

P2

High Level View

6

Forward Search Conditional Inference

I1 In. . .

O1. . .On

P1 Pn. . .

I1

O1

+

I2

O2

P1 P2 =

if ( )

else

CP1

P2

Recursive call synthesisReuse I/O as recursive call specification or query user

Example

Synthesize list length from examples:

7

Example

Synthesize list length from examples:

7

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Inputs:

Example

Synthesize list length from examples:

7

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Inputs:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Forward Search

8

Components: inc, isEmpty, tail, zero

Ex: Forward Search

8

Components: inc, isEmpty, tail, zero

Programs of size 1:

Ex: Forward Search

8

Components: inc, isEmpty, tail, zero

Programs of size 1:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Forward Search

8

Components: inc, isEmpty, tail, zero

Programs of size 1:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Forward Search

8

Components: inc, isEmpty, tail, zero

Programs of size 1:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

matches output for first inputRecall outputs

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Goal Graph (GG):

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Goal Graph (GG):

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

AND node

Goal Graph (GG):

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

Ex: Conditional Inference

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

AND node

Goal Graph (GG):

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Outputs:

“Backward” search

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Ex: Forward Search 2

10

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Programs of size 1:

Programs of size 2:

Satisfies one of our goals

Ex: Conditional Inference 2

11

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Conditional Inference 2

11

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Conditional Inference 2

11

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

if ( )

else

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Only allow calls satisfying well-founded relation

Ex: Forward Search 3

12

Programs of size 2

Programs of size 3

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Use I/O values to simulate recursive call

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

...

Only allow calls satisfying well-founded relation

But what if there isn’t such an I/O pair?

Query the user for new examples!

(too long for 15 min talk)

Ex: Conditional Inference

13

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

One iteration later:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Conditional Inference

13

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

One iteration later:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Ex: Conditional Inference

13

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

One iteration later:

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Recursive Program Synthesis 937

h Program Pi

1P1 = i ! "[],[2],[1,2]#P2 = zero ! "0,0,0#

2P3 = tail(P1) ! "err,[],[2]#P4 = inc(P2) ! "1,1,1#P5 = isEmpty(P1) ! "T,F,F#

3P6 = length(P3) ! "err,0,1#P7 = tail(P3) ! "err,err,[]#

4 P8 = inc(P7) ! "err,1,2#

Fig. 1. Synthesized programs Pi organized by heuristic value h

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5 P8

Pr

!0,1,2"

!T,F,F" !?,1,2"!0,?,?"cond then else

goal

P2P5

if isEmpty(i)then 0else inc(length(tail(i)))

(a) (b) (c)

P5

P8P2

Fig. 2. Goal graph after (a) first, (b) second and third, and (c) final alternations

employed by Escher that specifies how to reach the goal !0,1,2" by synthe-sizing a program Pr. Specifically, we need a program Pcond that returns !T,F,F"for inputs ![],[2],[1,2]", and a program Pelse that returns !?,1,2" for inputs![],[2],[1,2]", where ‘?’ denotes a “don’t care” value, since the first input []will not enter the else branch. Note that the then branch is already solvableusing P2, and is therefore annotated with it.

Second Alternation. The forward search now synthesizes programs by apply-ing components to programs found for the heuristic value 1. Note that Eschercan apply length (the recursive call) on the input values (program P1), butthis obviously results in a non-terminating program (namely, the program letlength(i) = length(i)). Escher employs a termination argument that de-tects and discards non-terminating programs. Next, by applying tail to P1, weget the program P3 that computes !err,[],[2]", where err is a special symbolrepresenting an error value, since tail is undefined on the empty list. Program P4results from applying inc to P2 and generates the value vector !1,1,1". ProgramP5 is generated by applying isEmpty to P1, resulting in the value vector !T,F,F".Now, the conditional inference phase discovers that P5 solves the !T,F,F" subgoalin the goal graph, and annotates it with P5 (see Figure 2(b)).

In the third alternation, new programs are generated, but none of our goalsare solved.

Synthesized program:

Escher: RecapForward Search

• Apply component to all synthesized programs

• Heuristic-based search

14

Escher: RecapForward Search

• Apply component to all synthesized programs

• Heuristic-based search

Conditional Inference• Uses goal graph to synthesize conditionals

• Conditionals synthesized on demand

14

Escher: RecapForward Search

• Apply component to all synthesized programs

• Heuristic-based search

Conditional Inference• Uses goal graph to synthesize conditionals

• Conditionals synthesized on demand

Recursive call synthesis• Use I/O specification

• Query user for more output values if needed14

Implementation

Prototype implementation of Escher

• Pluggable components

Experimented with:

• integer, list, and tree manipulating programs

15

Experiments

16

Escher w/o ObsEquiv

w/o GoalGraph

w/o OE+GG

collect_leaves 0.04 0.09 68.9 81.8

count_leaves 0.06 0.2 9.3 12.3

hbal_tree 1.5 MEM TIME MEM

nodes_at_level 10.74 MEM TIME MEM

(tree manipulating programs)

Time in seconds

Experiments

17

(#components sensitivity: mult)

Escher: 0.7swith all components

Experiments

17

(#components sensitivity: mult)

0

10

20

30

40

0 1 2 3 4 5

Sketch sensitivtySAT-based synthesis [ASPLOS’06]

NUMBER OF EXTRA COMPONENTS

TIM

E (S

)

Escher: 0.7swith all components

Experiments

17

(#components sensitivity: mult)

0

10

20

30

40

0 1 2 3 4 5

Sketch sensitivtySAT-based synthesis [ASPLOS’06]

NUMBER OF EXTRA COMPONENTS

TIM

E (S

)

Had to supply Sketch with high-level conditional

Escher: 0.7swith all components

Conclusion

18

EscherI/O Examples RecursiveProgram

User

Conclusion

18

EscherI/O Examples RecursiveProgram

User

Interactive

Conclusion

18

EscherI/O Examples RecursiveProgram

User

InteractiveHighly-customizable

Conclusion

18

EscherI/O Examples RecursiveProgram

User

InteractiveHighly-customizable

Novel synthesis techniques

Questions?

How can we synthesize loops?

19

Questions?

How can we synthesize loops?

Integration with SMT-based search

19

Questions?

How can we synthesize loops?

Integration with SMT-based search

Applications of Escher in eduction

• e.g., for interacting with students

19

Questions?

How can we synthesize loops?

Integration with SMT-based search

Applications of Escher in eduction

• e.g., for interacting with students

Theoretical under-pinnings

• e.g., see Madhusudan [CSL’11]

19

Thank You