Some open problems in deciding bisimulation equivalence Petr Janˇ car Dept of Computer Science Technical University Ostrava (FEI Vˇ SB-TUO), Czech Republic www.cs.vsb.cz/jancar Open Problems in Concurrency Theory Bertinoro, Italy, 18 –21 June, 2014 Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 1 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). for (unnormed) BPA in [ ExpTime ... 2-ExpTime ] Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). for (unnormed) BPA in [ ExpTime ... 2-ExpTime ] S´ enizergues (SIAM J.Comput 2005): bisimilarity decidable for (an equivalent of) FO-grammars Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). for (unnormed) BPA in [ ExpTime ... 2-ExpTime ] S´ enizergues (SIAM J.Comput 2005): bisimilarity decidable for (an equivalent of) FO-grammars new proof J. ICALP’14 (arxiv.org/abs/1405.7923) Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). for (unnormed) BPA in [ ExpTime ... 2-ExpTime ] S´ enizergues (SIAM J.Comput 2005): bisimilarity decidable for (an equivalent of) FO-grammars new proof J. ICALP’14 (arxiv.org/abs/1405.7923) Ackermann-hard (J. FoSSaCS’14); TOWER-hard when no ε -transitions (Benedikt, G¨ oller, Kiefer, Murawski at LiCS’13). Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Outline bisimulation equivalence on labelled transition systems (LTSs) here generated by sequential systems (sorry :-) ): context-free grammars (BPA processes) pushdown automata (pushdown processes) first-order grammars, or FO-grammars (also pushdown processes) a line of research started by Baeten, Bergstra, Klop (JACM 1993): bisimilarity decidable for normed BPA the current best time-complexity bound O ( n 4 polylog ( n )) (PhD thesis W. Czerwinski 2012). for (unnormed) BPA in [ ExpTime ... 2-ExpTime ] S´ enizergues (SIAM J.Comput 2005): bisimilarity decidable for (an equivalent of) FO-grammars new proof J. ICALP’14 (arxiv.org/abs/1405.7923) Ackermann-hard (J. FoSSaCS’14); TOWER-hard when no ε -transitions (Benedikt, G¨ oller, Kiefer, Murawski at LiCS’13). branching bisimilarity (van Glabbeek, Weijland, JACM 1996); recent interesting twists by Y. Fu (ICALP’13) and others: BPA, PDA Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 2 / 70
Labelled transition systems; bisimulation equivalence Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 3 / 70
Labelled transition systems; bisimulation equivalence Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 4 / 70
Bisimulation equivalence as a game a Assume LTS L = ( S , A , ( − → ) a ∈A ). In a round starting with a position ( s , t ), → s ′ or some t a a 1 Attacker chooses either some s → t ′ . − − → t ′ or some s a a 2 Defender responses by some t → s ′ , respectively. − − The new position is ( s ′ , t ′ ). The rounds are repeated. If a player is stuck, then (s)he loses. An infinite play is a win of Defender. Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 5 / 70
Bisimulation equivalence as a game a Assume LTS L = ( S , A , ( − → ) a ∈A ). In a round starting with a position ( s , t ), → s ′ or some t a a 1 Attacker chooses either some s → t ′ . − − → t ′ or some s a a 2 Defender responses by some t → s ′ , respectively. − − The new position is ( s ′ , t ′ ). The rounds are repeated. If a player is stuck, then (s)he loses. An infinite play is a win of Defender. We have s ∼ t iff Defender has a winning strategy from position ( s , t ), and s ∼ k t iff Defender can survive k rounds. Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 5 / 70
Bisimulation equivalence as a game a Assume LTS L = ( S , A , ( − → ) a ∈A ). In a round starting with a position ( s , t ), → s ′ or some t a a 1 Attacker chooses either some s → t ′ . − − → t ′ or some s a a 2 Defender responses by some t → s ′ , respectively. − − The new position is ( s ′ , t ′ ). The rounds are repeated. If a player is stuck, then (s)he loses. An infinite play is a win of Defender. We have s ∼ t iff Defender has a winning strategy from position ( s , t ), and s ∼ k t iff Defender can survive k rounds. Observation. For deterministic LTSs, bisimulation equivalence coincides with trace equivalence. Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 5 / 70
Labelled transition systems; bisimulation equivalence Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 6 / 70
Labelled transition systems; bisimulation equivalence Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 7 / 70
Labelled transition systems; bisimulation equivalence Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 8 / 70
a FO-grammar G = ( N , A , R ) ... rules A ( x 1 , . . . , x m ) − → E Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 9 / 70
a FO-grammar G = ( N , A , R ) ... rules A ( x 1 , . . . , x m ) − → E Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 10 / 70
a FO-grammar G = ( N , A , R ) ... rules A ( x 1 , . . . , x m ) − → E Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 11 / 70
a FO-grammar G = ( N , A , R ) ... rules A ( x 1 , . . . , x m ) − → E Petr Janˇ car (TU Ostrava) Deciding bisimulation equivalence Bertinoro, 20 June 2014 12 / 70
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