DECIDABILITY OF RMLVIA GAME SEMANTICS

Andrzej Murawski University of Warwick

reporting on joint work with C. Cotton-Barratt, D. Hopkins, L. Ong and N. Tzevelekos

ALGORITHMIC GAME SEMANTICS

M1 ≅M2

12

IDEALIZED ALGOL

Contextual approximation and higher-order procedures 3

2 Idealized Algol

We consider a finitary version IA

f

of Idealized Algol with active expressions [1]. Itstypes are generated by the following grammar.

✓ ::= � | ✓ ! ✓ � ::= com | exp | var

IA

f

can be viewed as a simply-typed �-calculus over the base types com, exp, var (ofcommands, expressions and variables respectively) augmented with the constants listedbelow

skip : com i : exp (0 i max ) succ : exp ! exp pred : exp ! exp

ifzero� : exp ! � ! � ! � seq� : com ! � ! � deref : var ! exp

assign : var ! exp ! com cell� : (var ! �) ! �while : exp ! com ! com mkvar : (exp ! com) ! exp ! var

where � ranges over base types and exp = { 0, · · · ,max }. Other IAf

-terms are formedusing �-abstraction and application

� ` M : ✓ ! ✓0 � ` N : ✓

� ` MN : ✓0�, x : ✓ ` M : ✓0

� ` �x✓.M : ✓ ! ✓0

using the obvious rules for constants and free identifiers. Each of the constants corre-sponds to a different programming feature. For instance, the sequential composition ofM and N (typically denoted by M ;N ) is expressed as seq�MN , assignment of N toM (M :

=N ) is represented by assignMN and cell�(�x.M) amounts to creating alocal variable x visible in M (new x in M ). mkvar is the so-called bad-variable con-structor that makes it possible to construct terms of type var with prescribed read- andwrite-methods. whileMN corresponds to while M do N . We shall write ⌦� for thedivergent constant that can be defined using while 1 do skip.

The operational semantics of IAf

, based on call-by-name evaluation, can be foundin [1]; we will write M + if M reduces to skip. We study the induced contextualapproximation.

Definition 1. We say that � ` M1

: ✓ contextually approximates � ` M2

: ✓ if, for anycontext C[�] such that C[M

1

], C[M2

] are closed terms of type com, we have C[M1

]+implies C[M

2

]+. We then write � ` M1

@⇠ M2

.

Even though the base types are finite, IAf

contextual approximation is not decidable [11].To obtain decidability one has to restrict the order of types, defined by:

ord(�) = 0 ord(✓ ! ✓0) = max(ord(✓) + 1, ord(✓0)).

Definition 2. Let i � 0. The fragment IAi of IAf

consists of IAf

-terms x1

: ✓1

, · · · , xn : ✓n `M : ✓ such that ord(✓j) < i for any j = 1, · · · , n and ord(✓) i.

Contextual approximation is known to be decidable for IA1

, IA2

and IA

3

[13], but it isundecidable for IA

4

[11].

Definition 3. – The level of a �-redex (�x✓.M)N is the order of the type of �x✓.M .

Contextual approximation and higher-order procedures 3

2 Idealized Algol

We consider a finitary version IA

f

of Idealized Algol with active expressions [1]. Itstypes are generated by the following grammar.

✓ ::= � | ✓ ! ✓ � ::= com | exp | var

IA

f

can be viewed as a simply-typed �-calculus over the base types com, exp, var (ofcommands, expressions and variables respectively) augmented with the constants listedbelow

skip : com i : exp (0 i max ) succ : exp ! exp pred : exp ! exp

ifzero� : exp ! � ! � ! � seq� : com ! � ! � deref : var ! exp

assign : var ! exp ! com cell� : (var ! �) ! �while : exp ! com ! com mkvar : (exp ! com) ! exp ! var

where � ranges over base types and exp = { 0, · · · ,max }. Other IAf

-terms are formedusing �-abstraction and application

� ` M : ✓ ! ✓0 � ` N : ✓

� ` MN : ✓0�, x : ✓ ` M : ✓0

� ` �x✓.M : ✓ ! ✓0

using the obvious rules for constants and free identifiers. Each of the constants corre-sponds to a different programming feature. For instance, the sequential composition ofM and N (typically denoted by M ;N ) is expressed as seq�MN , assignment of N toM (M :

=N ) is represented by assignMN and cell�(�x.M) amounts to creating alocal variable x visible in M (new x in M ). mkvar is the so-called bad-variable con-structor that makes it possible to construct terms of type var with prescribed read- andwrite-methods. whileMN corresponds to while M do N . We shall write ⌦� for thedivergent constant that can be defined using while 1 do skip.

The operational semantics of IAf

, based on call-by-name evaluation, can be foundin [1]; we will write M + if M reduces to skip. We study the induced contextualapproximation.

Definition 1. We say that � ` M1

: ✓ contextually approximates � ` M2

: ✓ if, for anycontext C[�] such that C[M

1

], C[M2

] are closed terms of type com, we have C[M1

]+implies C[M

2

]+. We then write � ` M1

@⇠ M2

.

Even though the base types are finite, IAf

contextual approximation is not decidable [11].To obtain decidability one has to restrict the order of types, defined by:

ord(�) = 0 ord(✓ ! ✓0) = max(ord(✓) + 1, ord(✓0)).

Definition 2. Let i � 0. The fragment IAi of IAf

consists of IAf

-terms x1

: ✓1

, · · · , xn : ✓n `M : ✓ such that ord(✓j) < i for any j = 1, · · · , n and ord(✓) i.

Contextual approximation is known to be decidable for IA1

, IA2

and IA

3

[13], but it isundecidable for IA

4

[11].

Definition 3. – The level of a �-redex (�x✓.M)N is the order of the type of �x✓.M .

Contextual approximation and higher-order procedures 3

2 Idealized Algol

We consider a finitary version IA

f

of Idealized Algol with active expressions [1]. Itstypes are generated by the following grammar.

✓ ::= � | ✓ ! ✓ � ::= com | exp | var

IA

f

can be viewed as a simply-typed �-calculus over the base types com, exp, var (ofcommands, expressions and variables respectively) augmented with the constants listedbelow

skip : com i : exp (0 i max ) succ : exp ! exp pred : exp ! exp

ifzero� : exp ! � ! � ! � seq� : com ! � ! � deref : var ! exp

assign : var ! exp ! com cell� : (var ! �) ! �while : exp ! com ! com mkvar : (exp ! com) ! exp ! var

where � ranges over base types and exp = { 0, · · · ,max }. Other IAf

-terms are formedusing �-abstraction and application

� ` M : ✓ ! ✓0 � ` N : ✓

� ` MN : ✓0�, x : ✓ ` M : ✓0

� ` �x✓.M : ✓ ! ✓0

using the obvious rules for constants and free identifiers. Each of the constants corre-sponds to a different programming feature. For instance, the sequential composition ofM and N (typically denoted by M ;N ) is expressed as seq�MN , assignment of N toM (M :

=N ) is represented by assignMN and cell�(�x.M) amounts to creating alocal variable x visible in M (new x in M ). mkvar is the so-called bad-variable con-structor that makes it possible to construct terms of type var with prescribed read- andwrite-methods. whileMN corresponds to while M do N . We shall write ⌦� for thedivergent constant that can be defined using while 1 do skip.

The operational semantics of IAf

, based on call-by-name evaluation, can be foundin [1]; we will write M + if M reduces to skip. We study the induced contextualapproximation.

Definition 1. We say that � ` M1

: ✓ contextually approximates � ` M2

: ✓ if, for anycontext C[�] such that C[M

1

], C[M2

] are closed terms of type com, we have C[M1

]+implies C[M

2

]+. We then write � ` M1

@⇠ M2

.

Even though the base types are finite, IAf

contextual approximation is not decidable [11].To obtain decidability one has to restrict the order of types, defined by:

ord(�) = 0 ord(✓ ! ✓0) = max(ord(✓) + 1, ord(✓0)).

Definition 2. Let i � 0. The fragment IAi of IAf

consists of IAf

-terms x1

: ✓1

, · · · , xn : ✓n `M : ✓ such that ord(✓j) < i for any j = 1, · · · , n and ord(✓) i.

Contextual approximation is known to be decidable for IA1

, IA2

and IA

3

[13], but it isundecidable for IA

4

[11].

Definition 3. – The level of a �-redex (�x✓.M)N is the order of the type of �x✓.M .

TYPE ORDER

Let T be a TM using Exp(k − 1, n) space for any input of size n.

Lemma. Consider the languages L5,L6 ⊆ (Σ#T )∗ defined by

L5 = {cinit # cR1 #d1#⋯ c

RN #dN #f

R ∣ cj ∈ ConfigT , f ∈ AcceptT , ∀inext(ci) = di}

and

L6 = {c1#dR1 #⋯ cN #d

RN ∣ cj ∈ ConfigT , ∃inext(ci) ≠ di}.

Then T accepts w if and only if L5 /⊆ L6.

Simulation of (k − 1)-EXPSPACE in IAk1 is afforded by an IAk

2 -term nestn,k.

Lemma.

Let k ≥ 2. nestn,k belongs to IAk2 , has polynomial size in n and is β-reducible to fExp(k−1,n)x.

ord(β) = 0

ord(θ → θ′) = max(ord(θ) + 1,ord(θ′))

10

M1@⇠ M2

IAi

Let i � 0. The fragment IAi consists of terms

x1 : ✓1, · · · , xn : ✓n ` M : ✓

such that ord(✓j) < i for any j = 1, · · · , n and ord(✓) i.

1

M1!∼M2

IAi

Let i ≥ 0. The fragment IAi consists ofterms

x1 ∶ θ1,⋯, xn ∶ θn ⊢M ∶ θ

such that ord(θj) < i for any j = 1,⋯, nand ord(θ) ≤ i.

fragment status

IA0 "

IA1 "

IA2 "

IA3 "

IA4 #

Ghica, McCusker, M., Ong, Walukiewicz

fragment status

IA0 PSPACE-completeIA1 PSPACE-completeIA2 PSPACE-completeIA3 EXPTIME-completeIA4 #

M., Walukiewicz

1

DECIDABILITYM1

<∼ M2

IAi

Let i ≥ 0. The fragment IAi consists of terms

x1 ∶ ✓1,�, xn ∶ ✓n �M ∶ ✓such that ord(✓j) < i for any j = 1,�, n and ord(✓) ≤ i.

fragment statusIA0 ,IA1 ,IA2 ,IA3 ,IA4 /

Ghica, McCusker, M., Ong, Walukiewicz

fragment statusIA0 PSPACE-completeIA1 PSPACE-completeIA2 PSPACE-completeIA3 EXPTIME-completeIA4 /

M., Walukiewicz

1

M1!∼M2

IAi

Let i ≥ 0. The fragment IAi consists ofterms

x1 ∶ θ1,⋯, xn ∶ θn ⊢M ∶ θ

such that ord(θj) < i for any j = 1,⋯, nand ord(θ) ≤ i.

fragment status

IA0 "

IA1 "

IA2 "

IA3 "

IA4 #

Ghica, McCusker, M., Ong, Walukiewicz

fragment status

IA0 PSPACE-completeIA1 PSPACE-completeIA2 PSPACE-completeIA3 EXPTIME-completeIA4 #

M., Walukiewicz

1

O-POINTERS

M1 ≅M2

q0

q1

q2

q3

q4

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

M1 ≅M2

q0

q1

q2

q3

q4

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

M1 ≅M2

q0

q1

q2

q3

q4

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

P-POINTERS

• IA3q0 q1 q2 q1 q2 q3O P O P O P

13

RML (CBV)

θ ∶∶= β ∣ θ → θ β ∶∶= unit ∣ int ∣ ref int

ref ∶ int → ref int

14

RML (O-STRICTNESS)

• O-strictunit→ unit !(unit→ unit)→ unit !(unit→ ⋯→ unit)→ unit !

• not O-strictunit→ unit→ unit ?

unit→ (unit→ unit)→ unit ?

((unit→ unit)→ unit) → unit "

[Hopkins, M., Ong; ICALP’11]

15

unit→ unit→ unit

q0 a0 q1 a1 q1 a1 q1 a1 q2 a2

O P O P O P O P O P

((com → com) → com)→ com

q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4 q3O P O P O P O P O P

16

M1 ≅M2

q0

q1

q2

q3

q4

q0

a0

q1

a1

q2

a2

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

M1 ≅M2

q0

q1

q2

q3

q4

q0

a0

q1

a1

q2

a2

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

POINTERS VIA INFINITE ALPHABET

q0 a0 q1 a1 q1 a1 q1 a1 q2 a2

O P O P O P O P O P

q0 a0 qn1

1 an1

1 qn2

1 an2

1 qn3

1 an3

1 qn2

2 an2

2

O P O P O P O P O P

17

RML (O-STRICTNESS)new results

• O-strictunit→ unit !(unit→ unit)→ unit !(unit→ ⋯→ unit)→ unit !

• not O-strictunit→ unit→ unit !unit→ unit→ unit→ unit !unit→ (unit→ unit)→ unit ?

((unit→ unit)→ unit) → unit "

[Cotton-Barratt, Hopkins, M., Ong; FOSSACS’15]

18

new results

• O-strictunit→ unit !(unit→ unit)→ unit !(unit→ ⋯→ unit)→ unit !

• not O-strictunit→ unit→ unit !unit→ unit→ unit→ unit !unit→ (unit→ unit)→ unit ?

((unit→ unit)→ unit) → unit "

[Cotton-Barratt, Hopkins, M., Ong; FOSSACS’15]

18

GROUND ML (CBV)Ground ML

θ ∶∶= β ∣ θ → θ β ∶∶= unit ∣ int ∣ ref(β)

19

Ground ML

θ ∶∶= β ∣ θ → θ β ∶∶= unit ∣ int ∣ ref(β)

ref (ref (int))

19

GROUND ML (O-STRICTNESS)Ground ML

• O-strictunit→ unit !(unit→ unit)→ unit !(unit→ ⋯→ unit)→ unit !

• not O-strictunit→ unit→ unit "unit→ (unit→ unit)→ unit "((unit→ unit)→ unit) → unit "

[M., Tzevelekos; ICALP’12]

20

unit→ unit→ unit

q0 a0 q1 a1 q1 a1 q1 a1 q2 a2

O P O P O P O P O P

((com → com) → com)→ com

q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4 q3O P O P O P O P O P

16

M1 ≅M2

q0

q1

q2

q3

q4

q0

a0

q1

a1

q2

a2

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12

M1 ≅M2

q0

q1

q2

q3

q4

q0

a0

q1

a1

q2

a2

Undecidability creeps in as soon as it is necessary

to represent pointers from O-moves.

• IA3q0 ⋯ q1 ⋯ q2O P O

• IA4q0 q1 q2 q3 q2 q3 q2 q3 ⋯ q4O P O P O P O P O

12