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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

  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

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

  7. (Det-)labelled transition systems (LTSs); trace equivalence Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 7 / 73

  8. (Det-)labelled transition systems (LTSs); trace equivalence Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 8 / 73

  9. (Det-)labelled transition systems (LTSs); trace equivalence Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 9 / 73

  10. (Det-)labelled transition systems (LTSs); trace equivalence Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 10 / 73

  11. 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

  12. 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 12 / 73

  13. 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 13 / 73

  14. 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 14 / 73

  15. (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

  16. 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

  17. 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

  18. Congruence properties of ∼ k and ∼ Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 17 / 73

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

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

  21. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 20 / 73

  22. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 21 / 73

  23. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 22 / 73

  24. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 23 / 73

  25. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 24 / 73

  26. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 25 / 73

  27. Balancing (the crucial tool for “synchronizing”) Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 26 / 73

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

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

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

  31. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 30 / 73

  32. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 31 / 73

  33. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 32 / 73

  34. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 33 / 73

  35. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 34 / 73

  36. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 35 / 73

  37. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 36 / 73

  38. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 37 / 73

  39. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 38 / 73

  40. ( ℓ, n )-(sub)sequences, with 2 ℓ pairs Petr Janˇ car (TU Ostrava) Equivalences of pushdown systems Grenoble, 11 Apr 2014 39 / 73

  41. 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

  42. 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

  43. 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

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