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 M = ( Q , Σ , Γ , δ, q 0 , Z 0 ) finite control unit q 3 ∗ (5 + 7) YES/NO (empty stack acceptance) B stack (LIFO) A B ⊥ Decidability of L ( M 1 ) ? = L ( M 2 ) 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¨ odel 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). Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 4 / 73

Part 1 Equivalence of det-FO-grammars (or of DPDA) is in TOWER. Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 5 / 73

(Det-)labelled transition systems (LTSs); trace equivalence Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 6 / 73

a FO-grammar G = ( N , A , R ) ... rules A ( x 1 , . . . , x m ) − → E Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 11 / 73

(D)pda from a first-order term perspective a Q = { q 1 , q 2 , q 3 } (pushing) rule q 2 A − → q 1 BC configuration q 2 ABA b ε − → q 2 − → q 3 (popping) rule q 2 A q 2 C 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 of size n we have EL ( T , U ) ≤ tower ( g ( n )). tower (0) = 1 tower ( n +1) = 2 tower ( 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 of size n we have EL ( T , U ) ≤ tower ( g ( n )). tower (0) = 1 tower ( n +1) = 2 tower ( 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 ∼ Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 17 / 73

Congruence properties of ∼ k and ∼ Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 18 / 73

Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 19 / 73

“Stair subsequence” of pairs (on balanced witness path) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 27 / 73

Stair subsequence of pairs (written horizontally) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 28 / 73

( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 29 / 73

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

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

Equivalences of pushdown systems are hard Petr Jan car Dept of Computer Science Technical University Ostrava (FEI V SB-TUO), Czech Republic www.cs.vsb.cz/jancar FoSSaCS14, part of ETAPS 2014 Grenoble, 11 Apr 2014 Petr Jan car (TU

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