a modified presentation of the decidability proof for ... · a modiﬁed presentation of the...

A modified presentation of the decidability prooffor language equivalence

on deterministic pushdown automata

Petr Jancar

Dept of Computer ScienceTechnical University Ostrava (FEI VSB-TU)

Czech Republicwww.cs.vsb.cz/jancar

ABCDays on List Automata, Forgetting Automata, and RestartingAutomata

Prague, March 27-29, 2009

Petr Jancar (TU Ostrava) DPDA - decidability ABCDays’09, March 2009 1 / 31

DPDA results

Senizergues G.:L(A)=L(B)? Decidability results from complete formal systems.Theoretical Computer Science 251(1-2): 1-166 (2001)(a preliminary version appeared at ICALP’97; Godel prize 2002)

Stirling C.: Decidability of DPDA equivalence.Theoretical Computer Science 255, 1-31, 2001

Senizergues G.: L(A)=L(B)? A simplified decidability proof.Theoretical Computer Science 281(1-2): 555-608 (2002)

Stirling C.: Deciding DPDA equivalence is primitive recursive.ICALP 2002, Lecture Notes in Computer Science 2380, 821-832,Springer 2002 (longer draft paper, 38 pages)


Example of a DPDA M

q0Z0a

−→ q0A

q0Z0b

−→ q0B

q0Aa

−→ q0AA

q0Ab

−→ q0BA

q0Ba

−→ q0AB

q0Bb

−→ q0BB

q0Ac

−→ q1A

q0Bc

−→ q1B

q1Aa

−→ q1

q1Bb

−→ q1

q0Ad

−→ q2

q0Bd

−→ q2

q2Aε

−→ q2

q2Bε

−→ q2

Q = {q0, q1, q2}, A = {a, b, c , d}, V = {Z0, A, B}

q0Z0

⊥

q0A

⊥

a

q0B

A

⊥

b

q1B

A

⊥

c

C0a

−→ C1b

−→ C2c

−→ C3

C0 = q0Z0⊥,C1 = q0A⊥,

C2 = q0BA⊥,C3 = q1BA⊥


Example of a DPDA M

q0Z0a

−→ q0A

q0Z0b

−→ q0B

q0Aa

−→ q0AA

q0Ab

−→ q0BA

q0Ba

−→ q0AB

q0Bb

−→ q0BB

q0Ac

−→ q1A

q0Bc

−→ q1B

q1Aa

−→ q1

q1Bb

−→ q1

q0Ad

−→ q2

q0Bd

−→ q2

q2Aε

−→ q2

q2Bε

−→ q2

Q = {q0, q1, q2}, A = {a, b, c , d}, V = {Z0, A, B}

q0Z0

⊥

q1B

A

⊥

abcq1A

⊥

b

q1⊥

a

L(M) = L(C0) where C0 = q0Z0⊥, i.e.

{w ∈ A∗ | q0Z0⊥w

−→ q⊥ for some q}

(M accepts by the empty stack)We have shown abcba ∈ L(M).


Example of a DPDA M

q0Z0a

−→ q0A

q0Z0b

−→ q0B

q0Aa

−→ q0AA

q0Ab

−→ q0BA

q0Ba

−→ q0AB

q0Bb

−→ q0BB

q0Ac

−→ q1A

q0Bc

−→ q1B

q1Aa

−→ q1

q1Bb

−→ q1

q0Ad

−→ q2

q0Bd

−→ q2

q2Aε

−→ q2

q2Bε

−→ q2

Q = {q0, q1, q2}, A = {a, b, c , d}, V = {Z0, A, B}

q0Z0

⊥

q0B

A

⊥

abq2A

⊥

d

q2⊥

ε

So q0Z0⊥abd−→ q2⊥, which means abd ∈ L(M).

(We could find out that L(M) ={wcwR | w ∈ {a, b}+} ∪ {wd | w ∈ {a, b}+})


Problem DPDA-Lang-Eq

Instance : Q, A, V, a finite set ∆ of rules pXa

−→ qα where a ∈ A∪ {ε},p.q ∈ Q, X ∈ V, α ∈ V∗, satisfying the determinism conditions:

if pXa

−→ qα and pXa

−→ q′α′ are in ∆ then qα = q′α′;

if pXε

−→..,i.e., if pX is unstable, then ∀a ∈ A : pX 6a

−→,

and two configurations, say C0 = q0Z0⊥, C′0 = q′

0Z′0⊥.

Question : Is L(C0) = L(C′0) ?

——————————————————————————————–

Note:

Disjoint union of two DPDAs is a DPDA.

L1 = L2 iff L1 · {$} = L2 · {$} for a fresh endmarker $.(Thus restricting to prefix-free DCFL [recognized by the empty stack]does not mean losing generality.)


Normalized dpda

pX

α

⊥

We say that a dpda M is normalized if

all ε-moves are popping, i.e., of the type pXε

−→ r ,

pXa

−→ r or pXa

−→ rZα for a stable rZ (a ∈ A)

each stable pX (other than p⊥) has a rulepX

a−→ . . . for each a ∈ A.

Proposition. Each dpda M can be transformed (by a quick algorithm) to anormalized dpda M ′ so that each configuration C of M is (easily) mappedto C′ of M ′ such that L(C) = L(C′).


Configurations as trees

Configuration C has a corresponding c-tree TC ;

we assume Q = {q1, q2, q3}, a rule q3Aε

−→ q2, and q1B, q1A, q2A arestable.

q1B

A

⊥ q1⊥ q2⊥ q3⊥ q1⊥ q2⊥ q3⊥

q1A q2A q2⊥

q1Bq1

q2q3

A c-tree is a [finite or infinite] ordered |Q|-branching tree, with the innernodes labelled from Vnew = {qX | qX is stable}, and the leaves from{⊥, ℓ1, ℓ2, . . . }.


Transitions T1a

−→ T2

A rule pXa

−→ qα (e.g., pXa

−→ q1BA), for a stable pX , can be presentedas TpX

a−→ Tqα

ℓ1 ℓ2 ℓ3

pX a−→

ℓ1 ℓ2 ℓ3 ℓ1 ℓ2 ℓ3

q1A q2A ℓ2

q1B

and it entails that for all T1, . . . ,Tb, where b = |Q|, we have

TpX (T1, . . . ,Tb)a

−→ Tqα(T1, . . . ,Tb)

TpX

T1 . . .Tba

−→Tqα

T1 . . .Tb


Stratified equivalence (induced by words upto a given length)

Given a normalized dpda M = (Q,A,V, ∆),and c-trees T1, T2, we put

T1 ∼m T2 (m ∈ N)

iff T1w

−→ ⇔ T2w−→ for all w ∈ A∗, |w | ≤ m

(Thus T1 6∼m T2 iff T1u

−→ ⊥, T2u

−→ T 6= ⊥, or vice versa,for some u, |u| < m . )

Note:

L(T1) = L(T2) iff ∀m : T1 ∼m T2, denoted T1 ∼ T2.

(Relations ∼m and ∼ are equivalences.)

We define eq-lev(T1, T2) as the maximal m such that T1 ∼m T2;

for T1 ∼ T2 we put eq-lev(T1, T2) = ω.Petr Jancar (TU Ostrava) DPDA - decidability ABCDays’09, March 2009 10 / 31

A first idea to try to decide if T0 ∼ T ′0 (for A = {a, b})

Breadth-first construction of TreeA∗ = {ε, a, b, aa, ab, ba, bb, aaa, . . . }where lab(ε) = (T0, T

′0) and (only) the following instruction is used:

Basic labelling instruction:

if lab(u) = (T1, T2) then lab(ua) = (T ′1, T

′2) where T1

a−→ T ′

1, T2a

−→ T ′2.

T0, T′0

Ta, T′a Tb, T

′b

Taa, T′aa Tab, T

′ab Tba, T

′ba Tbb, T

′bb

a b

a b a b

a b a b a b a b

If lab(u) = (⊥,⊥), the branch u is terminated succesfully.If lab(u) = (⊥, T ) or lab(u) = (T ,⊥) for T 6= ⊥, algorithm returns NO.


Semidecidability of the negative case T0 6∼ T ′0

If T0 6∼ T ′0 then there is a word w = a1a2 . . . am such that

TreeA∗

(T0, T′0)

w

(⊥, T ′w )

w is anoffending branch :eq-lev decreasesat each step on w

eq-lev(T0, T′0)

eq-lev(Ta1 , T′a1

)eq-lev(Ta1a2 , T

′a1a2

)· · ·eq-lev(Ta1a2...am , T ′

a1a2...am−1)

eq-lev(Ta1a2...am , T ′a1a2...am

)

= m

= m−1= m−2· · ·= 1= 0

∼m

∼m−1

∼m−2

· · ·∼1

∼0

6∼m+1

6∼m

6∼m−1

· · ·6∼2

6∼1


Positive case T0 ∼ T ′0 needs a sound branch termination

In the positive case T0 ∼ T ′0, our construction of TreeA∗ labels

all nodes w by pairs of equivalent trees (Tw ∼ T ′w ), and can go forever.

Butwe can terminate branch u whenever we find (somehow effectively)that u is not (a prefix of) an offending branch.E.g.:

u is labelled by (T , T ) (the pair (⊥,⊥) is a special case)a pair appears on branch u for the second time.

(E , F )

(E , F )


Two steps to get the decidability of T0 ∼ T ′0

A safe (modification of the basic) labelling of TreeA∗ :

we keep equivalence in the positive case (when T0 ∼ T ′0):

if lab(w) = (T , T ′) then T ∼ T ′;

we do not ‘correct’ an offending branch (when T0 6∼ T ′0):

each (originally) offending w remains offending(eq-lev can even decrease by more than 1 in a step)

A sound and complete (and effective) branch termination condition:

each terminated branch w is guaranteed to be not offending(eq-lev does not decrease in some step).

termination of any branch will be guaranteed (when T0 ∼ T ′0).


A safe relabelling option (a subtree replacement)

TreeA∗

C B

T1

T

T ′

v

C B

T2

T

T ′

v

C

T1

u

B

T2

u

|u| < |v |

eq-lev(T1, T2) ≥ eq-lev(C , B) − |u|)(congruence:) T1 ∼k T2 implies T (T1) ∼k T (T2).T1 ∼k T2 and eq-lev(T (T1), T

′) < k

implies eq-lev(T (T1), T′) = eq-lev(T (T2), T

′).


An (infinite) sequence of substitutions

Given tree D = D(ℓ1, ℓ2, . . . , ℓn), where D 6= ℓn, by

Dn-lim = Dn-lim(ℓ1, ℓ2, . . . , ℓn−1)

we mean D(ℓ1, ℓ2, . . . , ℓn−1, D(ℓ1, ℓ2, . . . , ℓn−1, D(ℓ1, ℓ2, . . . , ℓn−1, . . . )))

D

ℓ1. . . ℓn−1

D

ℓ1. . . ℓn−1

D

ℓ1. . . ℓn−1 ·

··Petr Jancar (TU Ostrava) DPDA - decidability ABCDays’09, March 2009 16 / 31

A useful solution of a recursive equation

Tn ∼k

D

T1 . . . Tn−1 Tnimplies Tn ∼k

Dn-lim

T1 . . . Tn−1

IfE

T1 . . .Tn ∼m

F

T1 . . .Tn andE

T ′1 . . .T ′

n 6∼m

F

T ′1 . . .T ′

n

then for some j , 1 ≤ j ≤ n, some k ≤ m and some D 6= ℓj , not dependingon T1, . . . ,Tn, we have

Tj ∼k

D

T1 . . .Tn and T ′j 6∼k

D

T ′1 . . .T ′

n

By reordering Q we achieve j = n and thus Tn ∼k

Dn-lim

T1 . . . Tn−1


Aim of balancing (to terminate each branch of TreeA∗)

E0

G1 . . .Gn

F0

G1 . . .Gn

. . .

E1

e11 . . . e1

n

G1 . . .Gn

F1

e11 . . . e1

n

G1 . . .Gn

. . .

. . .

Er+1

er+11 . . . er+1

n

. . .

e21 . . . e2

n

e11 . . . e1

n

G1 . . .Gn

Fr+1

er+11 . . . er+1

n

. . .

e21 . . . e2

n

e11 . . . e1

n

G1 . . .Gn

Branch-termination rule:if the pairs of heads are thesame (E0 = E1 = . . .Er+1,F0 = F1 = . . .Fr+1) and theword er+1er . . . e1 of extensionsis of type n then terminate thebranch.(for n = 0 it is just a repeat)


Soundness for n = 1 (1 tail, 4 pairs)

E

G1

F

G1 ,

E

e11

G1

F

e11

G1 ,

E

e21

e11

G1

F

e21

e11

G1 ,

E

e11

e21

e11

G1

F

e11

e21

e11

G1

If pair0, pair1, pair2 are in ∼m while pair3 is in 6∼m then

G1

D

G1 ,e11

G1

D

e11

G1 ,

e21

e11

G1

D

e21

e11

G1 ,

e11

e21

e11

G1

D

e11

e21

e11

G1

pair ′0, pair ′1, pair ′2 are in ∼k while pair ′3 is in 6∼k . Thus

e11 ∼k

D

e11 and e1

1 6∼k

D

e11

where e11 = e1

1(Dn-lim) – a contradiction (reduced to n = 0).


n = 2 (2 tails, 8 pairs)

E

G1G2

F

G1G2 ,. . . ,

E

e11e1

2

e21e2

2

e11e1

2

G1G2

F

e11e1

2

e21e2

2

e11e1

2

G1G2 ,. . . ,

E

e11e1

2

e21e2

2

e11e1

2

e31e3

2

e11e1

2

e21e2

2

e11e1

2

G1G2

F

e11e1

2

e21e2

2

e11e1

2

e31e3

2

e11e1

2

e21e2

2

e11e1

2

G1G2

Using G2 ∼k Dn-lim(G1), denoting e ij = e i

j (ℓ1, Dn-lim(ℓ1)), etc., we reduce:

e12

G1

D

e11 e1

2G1 ,

e12

e21

G1

D

e11 e1

2e21

G1 ,

e12

e31

e21

G1

D

e11 e1

2e31

e21

G1 ,

e12

e21

e31

e21

G1

D

e11 e1

2e21

e31

e21

G1


Patterns for extensions

E

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e31e3

2e33e3

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e41e4

2e43e4

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e31e3

2e33e3

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

G1G2G3G4


Patterns for extensions - shrinking

E

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e31e3

2e33e3

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e41e4

2e43e4

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

e31e3

2e33e3

4

e11e1

2e13e1

4

e21e2

2e23e2

4

e11e1

2e13e1

4

G1G2G3G4

E

e11 e1

2 e13

e21 e2

2 e32

e11 e1

2 e13

e31 e3

2 e33

e11 e1

2 e13

e21 e2

2 e32

e11 e1

2 e13

G1G2G3


Word patterns

Given a (maybe infinite) alphabet Σ, we say that a word w ∈ Σ∗ is

of type 0 if |w | ≥ 1,

of type n + 1 if w = vuv for some v of type n and some u, |u| ≥ 1.

Proposition. For a finite Σ, |Σ| = h,every word w ∈ Σ∗ with |w | ≥ fh(n) contains a subword of type n.

It suffices if in the word of length fh(n + 1) we surely findtwo equal separated subwords of length fh(n):

· · · · · · · · · · · · · · · · · · · · ·

Given h ∈ N, we define

fh(0) = 1, fh(n + 1) = (1 + fh(n)) · hfh(n) (fh(n) ≤ cc·· n-tower)


Successive bounded extensions guarantee termination

E0

G1 . . .Gn

F0

G1 . . .Gn

. . .

E1

e11 . . . e1

n

G1 . . .Gn

F1

e11 . . . e1

n

G1 . . .Gn

. . .

. . .

Er+1

er+11 . . . er+1

n

. . .

e21 . . . e2

n

e11 . . . e1

n

G1 . . .Gn

Fr+1

er+11 . . . er+1

n

. . .

e21 . . . e2

n

e11 . . . e1

n

G1 . . .Gn

It is sufficient that(r is large enough and)the word

E1

e11 . . . e1

n

F1

e11 . . . e1

n

E2

e21 . . . e2

n

F2

e21 . . . e2

n

E3

e31 . . . e3

n

F3

e31 . . . e3

n

. . .

is from a fixed alphabet.


Recall the safe relabelling option (a subtree replacement)

TreeA∗

C B

T1

T

T ′

v

C B

T2

T

T ′

v

C

T1

u

B

T2

u

|u| < |v |

eq-lev(T1, T2) ≥ eq-lev(C , B) − |u|)(congruence:) T1 ∼k T2 implies T (T1) ∼k T (T2).T1 ∼k T2 and eq-lev(T (T1), T

′) < k

implies eq-lev(T (T1), T′) = eq-lev(T (T2), T

′).


d -prefix form of a tree (with prefix Pd of depth d)

The (d + 1)-th node of each branch (if the branch is not shorter)is replaced by a variable-leaf ...

Pd

T1 T2. . . Tn

d levels

at most bd var-leaves(b ... branching degree)

IfPd

T1 · · ·Tn

u−→ T

where |u| < d

then T =E

T1 · · ·Tn where 1 ≤ height(E ) < 2d

(Assuming (wlog) that the tree-height can only increase by 1 in a step)


Words w(X , j), number M

Given the set of rules TX (ℓ1, . . . , ℓb)a

−→ T (ℓ1, . . . , ℓb),we can attach to each X ∈ Vnew = {qY | qY is stable}and each j , 1 ≤ j ≤ b

a shortest word w(X , j) such that

X

ℓ1 . . . ℓbw(X ,j)−→ ℓj

if there is one.

Let

M = 1 + max{ length(w(X , j)) | X ∈ V, 1 ≤ j ≤ b }

(M is (can be) exponential in the size of the DPDA-Lang-Eq instance)


Balancing

TreeA∗

C B

T T ′

v

XT1 . . .Tb

PM

T ′1 . . .T ′

n

ET1 . . .Tb

FT ′

1 . . .T ′n

v

suppose the trees are in the depicted form,

Cv

−→ T avoiding T1 . . .Tb , and |v | = M .

Then replace each Tj by(

B =PM

T ′1 . . .T ′

n

w(X ,j)−→

)

Fj

T ′1 . . .T ′

n.

(or by B when there is no w(X , j)). We thus get:Petr Jancar (TU Ostrava) DPDA - decidability ABCDays’09, March 2009 28 / 31

Balancing

XT1 . . .Tb

PM

T ′1 . . .T ′

n

ET1 . . .Tb

FT ′

1 . . .T ′n

v

XT1 . . .Tb

PM

T ′1 . . .T ′

n=B

E

F1 . . .Fb

T ′1 . . .T ′

n

F

T ′1 . . .T ′

n

v

The height of E , F1, . . . ,Fb, F is bounded (by 2M).

Balancing strategy: do rhs or lhs balancing when possible but you canswitch the sides only if some Fj(T

′1 . . .T ′

n) has been ‘exposed’.Note: ‘shortly’ after a balancing step

a new balancing happens or both-side pivots are allowed.Petr Jancar (TU Ostrava) DPDA - decidability ABCDays’09, March 2009 29 / 31

Balancing pivots from an (infinite) branch are on a ‘path’

B1u1−→ B2

u2−→ B3u3−→ · · ·

Let D1 =X1

T1 . . .Tb =

X1

P11M . . .P1b

M

G1 . . .Gn

be the subtree of B1 which is used in B1u1u2u3...−→ but its proper subtrees are

avoided (it is possible that D1 = B1).

We take Bi1 as the first pivot behind D1, so Bi1 =

H1

P11M . . .P1b

M

G1 . . .Gn ;

height(H1) is bounded (by M3).Let D2 be the (first) subtree of Bi1 which

is used by Bi1

ui1ui1+1ui1+2−→ . . . but its proper

subtrees are avoided. We take Bi2 as thefirst pivot behind D2, etc. D2 =

X2

P21M . . .P2b

M

e11 . . . e1

n

G1 . . .Gn


Successive bounded extensions

The pairs resulting from balancing with

Bi1 =

H1

P11M . . .P1b

M

G1 . . .Gn , Bi2 =

H2

P21M . . .P2b

M

e11 . . . e1

n

G1 . . .Gn , Bi3 =

H3

P31M . . .P3b

M

e21 . . . e2

n

e11 . . . e1

n

G1 . . .Gn , . . .

create the required sequence of ‘boundedly extended pairs’

E0

G1 . . .Gn

F0

G1 . . .Gn ,

E1

e11 . . . e1

n

G1 . . .Gn

F1

e11 . . . e1

n

G1 . . .Gn ,. . .


a modified presentation of the decidability proof for ... · a modiﬁed presentation of the...

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