Equivalences of pushdown systems are hard Petr Jan car Dept of - - PowerPoint PPT Presentation

Equivalences of pushdown systems are hard

Petr Janˇ car

Dept of Computer Science Technical University Ostrava (FEI Vˇ SB-TUO), Czech Republic www.cs.vsb.cz/jancar

FoSSaCS’14, part of ETAPS 2014 Grenoble, 11 Apr 2014

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 1 / 73

Deterministic pushdown automata; language equivalence

3 ∗ (5 + 7) q YES/NO (empty stack acceptance) ⊥ B A B finite control unit M = (Q, Σ, Γ, δ, q0, Z0) stack (LIFO) Decidability of L(M1) ? = L(M2) was open since 1960s (Ginsburg, Greibach). First-order schemes (1970s, 1980s, ..., B. Courcelle, ....).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 2 / 73

Solution

S´ enizergues 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; G¨

• del prize 2002)

Stirling C.: Decidability of DPDA equivalence. Theoretical Computer Science 255, 1-31, 2001 S´ enizergues 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 on the author’s web page) S´ enizergues G.: The Bisimulation Problem for Equational Graphs of Finite Out-Degree. SIAM J.Comput., 34(5), 1025–1106 (2005) (a preliminary version appeared at FOCS’98)

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 3 / 73

Outline

Part 1

Deterministic case is in TOWER. Equivalence of first-order schemes (or det-FO-grammars, or deterministic pushdown automata (DPDA)) is in TOWER, i.e. “close” to elementary. (The known lower bound is P-hardness.)

Part 2

Nondeterministic case is Ackermann-hard. Bisimulation equivalence of first-order grammars (or PDA with deterministic popping ε-moves) is Ackermann-hard, and thus not primitive recursive (but decidable).

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Part 1

Equivalence of det-FO-grammars (or of DPDA) is in TOWER.

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(Det-)labelled transition systems (LTSs); trace equivalence

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(Det-)labelled transition systems (LTSs); trace equivalence

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(Det-)labelled transition systems (LTSs); trace equivalence

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 8 / 73

(Det-)labelled transition systems (LTSs); trace equivalence

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(Det-)labelled transition systems (LTSs); trace equivalence

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FO-grammar G = (N, A, R) ... rules A(x1, . . . , xm)

a

− → E

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 11 / 73

FO-grammar G = (N, A, R) ... rules A(x1, . . . , xm)

a

− → E

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 12 / 73

FO-grammar G = (N, A, R) ... rules A(x1, . . . , xm)

a

− → E

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 13 / 73

FO-grammar G = (N, A, R) ... rules A(x1, . . . , xm)

a

− → E

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 14 / 73

(D)pda from a first-order term perspective

Q = {q1, q2, q3} (pushing) rule q2A

a

− → q1BC configuration q2ABA (popping) rule q2A

b

− → q2 q2C

ε

− → q3

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 15 / 73

Bounding lengths of witnesses (where EL keeps dropping)

Theorem. There is an elementary function g such that for any det-FO grammar G = (N, A, R) and T ∼ U

• f size n we have

EL(T, U) ≤ tower(g(n)). tower(0) = 1 tower(n+1) = 2tower(n)

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 16 / 73

Bounding lengths of witnesses (where EL keeps dropping)

Theorem. There is an elementary function g such that for any det-FO grammar G = (N, A, R) and T ∼ U

• f size n we have

EL(T, U) ≤ tower(g(n)). tower(0) = 1 tower(n+1) = 2tower(n) Proof is based on two ideas:

1 “Synchronize” the growth of lhs-terms

and rhs-terms while not changing the respective eq-levels. (Hence no repeat.)

2 Derive a tower-bound on the size of

terms in the (modified) sequence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 16 / 73

Congruence properties of ∼k and ∼

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Congruence properties of ∼k and ∼

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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Balancing (the crucial tool for “synchronizing”)

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“Stair subsequence” of pairs (on balanced witness path)

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Stair subsequence of pairs (written horizontally)

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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(ℓ, n)-(sub)sequences, with 2ℓ pairs

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Final (conditional) step of the “TOWER-proof”

Recall: There is no EL-decreasing (1, 0)-sequence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 40 / 73

Final (conditional) step of the “TOWER-proof”

Recall: There is no EL-decreasing (1, 0)-sequence.

• Claim. Any EL-decreasing (ℓ+1, n+1)-sequence gives rise to

an EL-decreasing (ℓ, n)-sequence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 40 / 73

Final (conditional) step of the “TOWER-proof”

Recall: There is no EL-decreasing (1, 0)-sequence.

• Claim. Any EL-decreasing (ℓ+1, n+1)-sequence gives rise to

an EL-decreasing (ℓ, n)-sequence.

• Corollary. There is no EL-decreasing (n+1, n)-sequence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 40 / 73

Final (conditional) step of the “TOWER-proof”

Recall: There is no EL-decreasing (1, 0)-sequence.

• Claim. Any EL-decreasing (ℓ+1, n+1)-sequence gives rise to

an EL-decreasing (ℓ, n)-sequence.

• Corollary. There is no EL-decreasing (n+1, n)-sequence.

Recall that h(1) = 1 + q, h(j+1) = h(j) · (1 + qh(j)) and that h(j) “stairs” gives rise to (j, n)-sequence (n being the “small” thickness).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 40 / 73

Final (conditional) step of the “TOWER-proof”

Recall: There is no EL-decreasing (1, 0)-sequence.

• Claim. Any EL-decreasing (ℓ+1, n+1)-sequence gives rise to

an EL-decreasing (ℓ, n)-sequence.

• Corollary. There is no EL-decreasing (n+1, n)-sequence.

Recall that h(1) = 1 + q, h(j+1) = h(j) · (1 + qh(j)) and that h(j) “stairs” gives rise to (j, n)-sequence (n being the “small” thickness).

• Corollary. There are less than h(n+1) stairs, and h(n+1) ≤ tower(g(n)).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 40 / 73

Repeating heads yield an “equation”

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Repeating heads yield an “equation”

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Repeating heads yield an “equation”

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Repeating heads yield an “equation”

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Repeating heads yield an “equation”

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From (ℓ, n) to (ℓ−1, n−1) ... decreasing thickness

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From (ℓ, n) to (ℓ−1, n−1) ... decreasing thickness

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From (ℓ, n) to (ℓ−1, n−1) ... decreasing thickness

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From (ℓ, n) to (ℓ−1, n−1) ... decreasing thickness

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Bounding lengths of witnesses (End of Part 1)

Theorem. There is an elementary function g such that for any det-FO grammar G = (N, A, R) and T ∼ U

• f size n we have

EL(T, U) ≤ tower(g(n)). Proof is based on two ideas:

1 “Synchronize” the growth of lhs-terms

and rhs-terms while not changing the respective eq-levels. (Hence no repeat.)

2 Derive a tower-bound on the size of

terms in the (modified) sequence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 50 / 73

Part 2

Bisimulation equivalence for FO-grammars is Ackermann-hard.

Note: Benedikt M., G¨

• ller S., Kiefer S., Murawski A.S.:

Bisimilarity of Pushdown Automata is Nonelementary. LICS 2013 (no ε-transitions)

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Ackermann function, class ACK, ACK-completeness

Family f0, f1, f2, . . . of functions: f0(n) = n+1 fk+1(n) = fk(fk(. . . fk(n) . . . )) = f (n+1)

k

(n) Ackermann function fA: fA(n) = fn(n).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 52 / 73

Ackermann function, class ACK, ACK-completeness

Family f0, f1, f2, . . . of functions: f0(n) = n+1 fk+1(n) = fk(fk(. . . fk(n) . . . )) = f (n+1)

k

(n) Ackermann function fA: fA(n) = fn(n). ACK ... class of problems solvable in time fA(g(n)) where g is a primitive recursive function.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 52 / 73

Ackermann function, class ACK, ACK-completeness

Family f0, f1, f2, . . . of functions: f0(n) = n+1 fk+1(n) = fk(fk(. . . fk(n) . . . )) = f (n+1)

k

(n) Ackermann function fA: fA(n) = fn(n). ACK ... class of problems solvable in time fA(g(n)) where g is a primitive recursive function. Ackermann-budget halting problem (AB-HP): Instance: Minsky counter machine M. Question: does M halt from the zero initial configuration within fA(size(M)) steps ?

• Fact. AB-HP is ACK-complete.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 52 / 73

Control state reachability in reset counter machines

Reset counter machines (RCMs). nonnegative counters c1, c2, . . . , cd, control states 1, 2, . . . , r, configuration (ℓ, (n1, n2, . . . , nd)), initial conf. (1, (0, 0, . . . , 0)), (nondeterministic) instructions of the types ℓ

inc(ci)

− → ℓ′ (increment ci), ℓ

dec(ci)

− → ℓ′ (decrement ci, if ci > 0), ℓ

reset(ci)

− → ℓ′ (reset ci, i.e., put ci = 0).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 53 / 73

Control state reachability in reset counter machines

Reset counter machines (RCMs). nonnegative counters c1, c2, . . . , cd, control states 1, 2, . . . , r, configuration (ℓ, (n1, n2, . . . , nd)), initial conf. (1, (0, 0, . . . , 0)), (nondeterministic) instructions of the types ℓ

inc(ci)

− → ℓ′ (increment ci), ℓ

dec(ci)

− → ℓ′ (decrement ci, if ci > 0), ℓ

reset(ci)

− → ℓ′ (reset ci, i.e., put ci = 0). CS-reach problem for RCM: Instance: an RCM M, a control state ℓfin . Question: is (1, (0, 0, . . . , 0)) − →∗ (ℓfin, (. . . )) ?

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 53 / 73

Control state reachability in reset counter machines

Reset counter machines (RCMs). nonnegative counters c1, c2, . . . , cd, control states 1, 2, . . . , r, configuration (ℓ, (n1, n2, . . . , nd)), initial conf. (1, (0, 0, . . . , 0)), (nondeterministic) instructions of the types ℓ

inc(ci)

− → ℓ′ (increment ci), ℓ

dec(ci)

− → ℓ′ (decrement ci, if ci > 0), ℓ

reset(ci)

− → ℓ′ (reset ci, i.e., put ci = 0). CS-reach problem for RCM: Instance: an RCM M, a control state ℓfin . Question: is (1, (0, 0, . . . , 0)) − →∗ (ℓfin, (. . . )) ?

• Fact. CS-reach problem for RCM is ACK-complete.

(See [Schnoebelen, MFCS 2010].)

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 53 / 73

Bisimulation equivalence as a game

Assume LTS L = (S, A, (

a

− →)a∈A). In a position (s, t),

1 Attacker chooses either some s

a

− → s′ or some t

a

− → t′.

2 Defender responses by some t

a

− → t′ or some s

a

− → s′, respectively. The new position is (s′, t′).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 54 / 73

Bisimulation equivalence as a game

Assume LTS L = (S, A, (

a

− →)a∈A). In a position (s, t),

1 Attacker chooses either some s

a

− → s′ or some t

a

− → t′.

2 Defender responses by some t

a

− → t′ or some s

a

− → s′, respectively. The new position is (s′, t′). These rounds are repeated. If a player is stuck, then (s)he loses. An infinite play is a win of Defender. We put s ∼ t (s, t are bisimulation equivalent) if Defender has a winning strategy from position (s, t).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 54 / 73

Bisimulation equivalence as a game

Assume LTS L = (S, A, (

a

− →)a∈A). In a position (s, t),

1 Attacker chooses either some s

a

− → s′ or some t

a

− → t′.

2 Defender responses by some t

a

− → t′ or some s

a

− → s′, respectively. The new position is (s′, t′). These rounds are repeated. If a player is stuck, then (s)he loses. An infinite play is a win of Defender. We put s ∼ t (s, t are bisimulation equivalent) if Defender has a winning strategy from position (s, t).

• Observation. For deterministic LTSs, bisimulation equivalence coincides

with trace equivalence.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 54 / 73

Reduction of CS-reach for RCM to FO-bisimilarity

Given an RCM M, i.e., counters c1, c2, . . . , cd, control states 1, 2, . . . , r, and instructions of the types ℓ

inc(ci)

− → ℓ′ (increment ci), ℓ

dec(ci)

− → ℓ′ (decrement ci, if ci > 0), ℓ

reset(ci)

− → ℓ′ (reset ci, i.e., put ci = 0), and ℓfin, we construct G = (N, A, R) and E0, F0 so that (1, (0, 0, . . . , 0)) − →∗ (ℓfin, (. . . )) iff E0 ∼ F0.

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 55 / 73

CS-reachability as bisimulation game

Example with counters c1, c2; we start with the pair (A1(⊥, ⊥, ⊥, ⊥, ), B1(⊥, ⊥, ⊥, ⊥)). The pair after mimicking (1, (0, 0)) − →∗ (ℓ, (2, 1)) might be

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 56 / 73

Attacker’s win

Attacker wins in (Aℓfin(. . . ), Bℓfin(. . . )) due to the rule Aℓfin(x1, x2, x3, x4)

a

− → . . . (while there is no rule for Bℓfin).

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 57 / 73

Counter increment

For ins = ℓ

inc(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, I(x3), x4), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, I(x3), x4),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 58 / 73

Counter increment

For ins = ℓ

inc(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, I(x3), x4), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, I(x3), x4),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 59 / 73

Counter increment

For ins = ℓ

inc(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, I(x3), x4), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, I(x3), x4),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 60 / 73

Counter reset

For ins = ℓ

reset(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, ⊥, ⊥), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, ⊥, ⊥),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 61 / 73

Counter reset

For ins = ℓ

reset(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, ⊥, ⊥), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, ⊥, ⊥),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 62 / 73

Counter reset

For ins = ℓ

reset(c2)

− → ℓ′ we have rules Aℓ(x1, x2, x3, x4)

ins

− → Aℓ′(x1, x2, ⊥, ⊥), Bℓ(x1, x2, x3, x4)

ins

− → Bℓ′(x1, x2, ⊥, ⊥),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 63 / 73

Counter decrement

For ins = ℓ

dec(c2)

− → ℓ′ we have two phases; the first-phase rules are Aℓ

ins

− → A(ℓ′,2), Aℓ

ins

− → B(ℓ′,2,a), Aℓ

ins

− → B(ℓ′,2,b), Bℓ

ins

− → B(ℓ′,2,a), Bℓ

ins

− → B(ℓ′,2,b),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 64 / 73

Counter decrement

For ins = ℓ

dec(c2)

− → ℓ′ we have two phases; the first-phase rules are Aℓ

ins

− → A(ℓ′,2), Aℓ

ins

− → B(ℓ′,2,a), Aℓ

ins

− → B(ℓ′,2,b), Bℓ

ins

− → B(ℓ′,2,a), Bℓ

ins

− → B(ℓ′,2,b),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 65 / 73

Counter decrement

For ins = ℓ

dec(c2)

− → ℓ′ we have two phases; the first-phase rules are Aℓ

ins

− → A(ℓ′,2), Aℓ

ins

− → B(ℓ′,2,a), Aℓ

ins

− → B(ℓ′,2,b), Bℓ

ins

− → B(ℓ′,2,a), Bℓ

ins

− → B(ℓ′,2,b),

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 66 / 73

Counter decrement (option a)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,a)(x1, x2, x3, x4)

a

− → Bℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,a)(x1, x2, x3, x4)

b

− → x3,

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 67 / 73

Counter decrement (option a)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,a)(x1, x2, x3, x4)

a

− → Bℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,a)(x1, x2, x3, x4)

b

− → x3,

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 68 / 73

Counter decrement (option a)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,a)(x1, x2, x3, x4)

a

− → Bℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,a)(x1, x2, x3, x4)

b

− → x3,

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 69 / 73

Counter decrement (option b)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,b)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,b)(x1, x2, x3, x4)

b

− → x4, I(x1)

c

− → x1

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 70 / 73

Counter decrement (option b)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,b)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,b)(x1, x2, x3, x4)

b

− → x4, I(x1)

c

− → x1

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 71 / 73

Counter decrement (option b)

A(ℓ′,2)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), Aℓ′,2(x1, x2, x3, x4)

b

− → x3, B(ℓ′,2,b)(x1, x2, x3, x4)

a

− → Aℓ′(x1, x2, x3, I(x4)), B(ℓ′,2,b)(x1, x2, x3, x4)

b

− → x4, I(x1)

c

− → x1

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 72 / 73

Concluding remarks

We have shown (Trace) equivalence of deterministic first-order grammars is in TOWER. Bisimulation equivalence of first-order grammars is Ackermann-hard. Questions/problems/related results: more precise complexity bounds ... subcases (simple grammars, one-counter automata, ...) higher orders ... ....

Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 73 / 73