Some techniques and results in deciding bisimilarity

Some techniques and results in deciding bisimilarity Petr Jančar Dept of Computer Science Technical University Ostrava (FEI VŠB-TU) Czech Republic www.cs.vsb.cz/jancar Talk at the Verification Seminar, ULB, Brussels, 16 December 2005 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 1 / 50

Outline Behavioural equivalences; bisimulation equivalence. Undecidability of equivalences on (labelled place/transition) Petri nets (STACS 1994); imperfect Minsky machine simulation. Undecidability of the reachability set equality. Semilinear witnesses; decidable cases. Bisimilarity on process rewrite systems, and prefix rewrite systems. PSPACE-completeness of bisimilarity on Basic Parallel Processes (LiCS 2003); distance-to-disabling functions (dd-functions). Undecidability of bisimilarity on Type -1 systems (Jančar and Srba, FoSSaCS 2006); Defender s choice technique. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 2 / 50

Vending machines coffee tea V 1 def = 5k.5k.( coffee.collect.v 1 + tea.collect.v 1 ) 5k V 2 def = 5k.5k.coffee.collect.V 2 + 5k.5k.tea.collect.V 2 collect Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 4 / 50

Vending machines - cont. V 1 def = 5k.5k.( coffee.collect.v 1 + tea.collect.v 1 ) V 1 5k 5k coffee tea collect V 2 def = 5k.5k.coffee.collect.V 2 + 5k.5k.tea.collect.V 2 5k 5k 5k 5k collect coffee tea Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 5 / 50

Behavioural equivalences and preorders; simulation Does a system implement another one? Are they equivalent? (system = labelled transition system) Language (trace) equivalences often too coarse coffee coin coin tea coffee coin A binary relation R over STATES is a simulation if whenever (s, t) R then for every action a if s a s a then t t for some t such that (s, t ) R. s is simulated by t if there is a simulation R (s, t). The union of (all) simulations is (the maximal) simulation tea Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 6 / 50

Bisimulation equivalence Milner, Park (1980s) A binary relation R over STATES is a bisimulation if (it is a symmetric simulation, i.e.) whenever (s, t) R then for every action a if s a s a then t t for some t such that (s, t ) R. whenever (s, t) R then for every action a if t a t a then s s for some s such that (s, t ) R. s is bisimilar with t if there is a bisimulation R (s, t). The union of (all) bisimulations is (the maximal) bisimulation. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 7 / 50

Linear Time / Branching Time Spectrum Bisimulation equivalence 2-nested simulation equivalence Ready simulation equivalence Possible-futures equivalence Ready trace equivalence Simulation equivalence Readiness equivalence Failure trace equivalence Failures equivalence Completed trace equivalence Trace equivalence Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 8 / 50

Minsky counter machines A Minsky counter machine C is given by a fixed number of (nonnegative integer) counters c 1, c 2,..., c m a program (in fact, a set of labelled instructions) 1 : com 1 ; 2 : com 2 ;... ; n : com n, where com n is instruction HALT com i (i = 1, 2,..., n 1) are commands of two types: c j := c j + 1; goto k if c j = 0 then goto k 1 else (c j := c j 1; goto k 2 ) Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 10 / 50

Undecidability of behavioural equivalences for Petri nets Fact It is undecidable if a 2-counter machine C halts on the zero input (i.e., when starting with c 1 = c 2 = 0). We show an algorithm A : so that C A N C 1, NC 2 if C halts (on input zero) then the behaviours of N1 C, NC 2 differ drastically (one can perform a trace which the other cannot) if C does not halt then the behaviours of N1 C, NC 2 in a strict sense (the nets are bisimilar) are the same Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 11 / 50

Reduction of Halting Problem to Petri net equivalences c 2 s 1 s 2 s 3 s 4 s 5 s n... s 6 c 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 13 / 50

Reduction of Halting Problem to Petri net equivalences c 2 s 1 s 2 s 3 s 4 s 5 s n... s 6 c 1 + + Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 13 / 50

Reduction of Halting Problem to Petri net equivalences c 2 s 1 s 2 s 3 s 4 s 5 s n... s 6 c 1 + + + + Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 13 / 50

Reduction of Halting Problem to Petri net equivalences c 2 s 1 s 2 s 3 s 4 s 5 s n... s 6 c 1 + + + + p 2 p 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 13 / 50

Reduction - cont... Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 14 / 50

Reduction for the reachability set equality c 2 s 1 s 2 s 3 s 4 s 5 s n... s 6 c 1 + + p 2 p 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 16 / 50

R(N) R(N blue ); when R(N blue ) R(N)? N t SC Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 17 / 50

R(N) R(N blue ); when R(N blue ) R(N)? N t SC COD r 1 AUX r 2 2 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 17 / 50

R(N) R(N blue ); when R(N blue ) R(N)? N t SC t COD r 1 AUX r 2 2 COD for decreasing (green t) Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 17 / 50

Reduction for the reachability set equality cont. If C halts, after m steps, then N blue has a possibility to simulate C correctly, and thus reach a dead marking where SC = m, and COD is in binary 001000010001100100, with length m+1, where 1s correspond to the decreasing (green) transitions Such a marking is unreachable in N ; therefore R(N blue ) R(N). If C does not halt then R(N blue ) R(N) (in fact, R(N blue ) = R(N)). Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 18 / 50

Presburger definable sets = semilinear sets A set L N k is linear if there are: a basis b N k and periods p 1, p 2,..., p n N k so that L = { b + c 1 p 1 + c 2 p 2 + + c n p n c 1, c 2,..., c n N } A set S N k is semilinear iff it is a finite union of linear sets. Ginsburg, Spanier 1966: Presburger-definable subsets of N k are precisely the semilinear sets. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 20 / 50

Reachability set equality revisited For 2 unbounded places, R(N) is (effectively) semilinear; hence R(N 1 ) R(N 2 ) (and R(N 1 ) = R(N 2 )) decidable. For 5 unbounded places, R(N 1 ) = R(N 2 ) (and R(N 1 ) R(N 2 )) undecidable. What about 3 and 4 unbounded places?? Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 21 / 50

Simulation on one-counter nets semilinear p(m) is not simulated by q(n)... red p(m) is simulated by q(n)... black n. 1 0 0 1 2... m?? Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 22 / 50

Process rewrite systems a E 1 1 F1 a E 2 2 F2... E k a k Fk PRS PAD PAN PDA PA PN BPA BPP FS Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 24 / 50

Bisimilarity over context-free processes decidable Context-free grammar (in Greibach NF) finitely many rules of the type A 1 ab 1 B 2... B k Processes BPA (sequential) Processes BPP (parallel) A 1 A 2... A n b a B 1 B 2... B k A 2... A n a a b... A 1 a B 1 B 2... B k Christensen, Hirshfeld, Moller (1993) decidable (nonprimitive recursive upper bound) Srba (2002) PSPACE-hard Jančar (2003) PSPACE-complete Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 25 / 50

DPDA language equivalence, PDA bisimilarity G. Sénizergues (1997) The language equivalence problem for deterministic pushdown automata is decidable C. Stirling (2002) Simplified the proof substantially, and showed that it is primitive recursive They also showed that bisimilarity for nondeterministic pushdown automata decidable. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 26 / 50

Distance-to-disabling functions (dd-functions) dd a : S IN ω dist(s, t) = min { length(w) s w t } dd a (s) = min { dist(s, t) t has no a-successor } Given a tuple F = (d 1,..., d k ), each transition s a t determines a change F(t) F(s) = ( d 1 (t) d 1 (s),..., d k (t) d k (s) ). dd (a,f,δ) (s) = min { dist(s, t) r : if t a r then F(r) F(t) δ }. Observation. s t = d(s) = d(t) for all dd-functions d. The direction = holds for image finite systems. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 28 / 50

Basic Parallel Processes Finitely many rules of the type A 1 a B1 B 2... B k A 1 a B 1 B 2... B k BPP-nets: variables = places Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 29 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 3 x 2 4 x 3 x 4 x 5 x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 3 x 2 1 4 x 3 x 4 x 5 x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 3 x 2 1 4 x 3 x 4 x 5 1 x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 3 x 2 1 4 x 3 2 x 4 x 5 1 x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 6 3 x 2 1 4 x 3 2 x 4 x 5 1 x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

DD-functions on BPP-nets norm Q (M) = min { dist(m, M ) M has no tokens on Q } x 1 6 3 x 2 1 4 x 3 2 x 4 ω x 5 1 x 6 ω norm Q (x 1, x 2, x 3, x 4, x 5, x 6 ) = 6x 1 + x 2 + 2x 3 + ω x 4 + x 5 + ω x 6 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 30 / 50

Bisimilarity on Basic Parallel Processes in PSPACE M 1 M 2 important Q : norm Q (M 1 ) = norm Q (M 2 ) The following puzzling question is thus answered positively: X 2 X 3 = X X 2? Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 31 / 50

Weak bisimilarity on Basic Parallel Processes τ X X 2 τ X X τ τ τ τ b c b c Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 32 / 50

Weak bisimilarity on Basic Parallel Processes τ X 2 X 3 τ X X τ τ τ τ b c b c Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 32 / 50

Bisimilarity on prefix rewrite systems Normed Processes Unnormed Processes Type -2 Σ 1 1 -complete Σ1 1 -complete Type -1b Π 0 1 -complete Σ1 1 -complete Type -1a Π 0 1 -complete Π0 1 -complete Type 0, and decidable decidable Type 1 1 2 EXPTIME-hard EXPTIME-hard Type 2 P 2-EXPTIME P-hard PSPACE-hard Type 3 P-complete P-complete The main new result: Bisimilarity is undecidable on Type -1a systems. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 34 / 50

Pushdown graphs; generated by Type 0 systems px px qa b c a pxa qε qε px b pxa b pxaa b pxaaa b... c c c c q a qa a qaa a qaaa a... a Type 0 system: finite sets of rules w 1 w 2 (the same class of generated graphs) An involved result: Bisimilarity is decidable (Sénizergues (1998,2005), then Stirling) Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 35 / 50

Type -1 systems (or Type -1a systems); rules R a w X XA A b c a XA ε ε X b XA b XAA b XAAA b... c c c c ε a A a AA a AAA Stirling, Sénizergues: is bisimilarity decidable? Sénizergues decidability result for equational graphs of finite out-degree (equivalent to the case R a w with prefix-free R) Maybe in the normed case? (u is normed if each path from u can be prolonged to reach ε) Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 36 / 50 a...

Hierarchy of prefix rewriting Type Form of Rewrite Rules Type -2 a R 1 R 2 Type -1a/-1b R a w / w a R Type 0 w a w Type 1 1 2 px a qw Type 2 X a w Type 3 X a Y, X a ε Type -2 Type -1a Type -1b Type 0 = Type 1 1 2 Type 2 Type 3 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 37 / 50

Bisimilarity is undecidable on Type -1a systems Normed Processes Unnormed Processes Type -2 Σ 1 1 -complete Σ1 1 -complete Type -1b Π 0 1 -complete Σ1 1 -complete Type -1a Π 0 1 -complete Π0 1 -complete Type 0, and decidable decidable Type 1 1 2 EXPTIME-hard EXPTIME-hard Type 2 P 2-EXPTIME P-hard PSPACE-hard Type 3 P-complete P-complete Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 38 / 50

Inf-PCP (a version of Post Correspondence Problem) A PCP-instance : u 1 u 2... u n v 1 v 2... v n u i, v i : nonempty words in an alphabet An infinite initial solution: a sequence i 1, i 2, i 3,... from {1, 2,..., n} such that i 1 =1 and u i1 u i2 u i3 u i4 = v i1 v i2 v i3 v i4 Given a Turing machine M, with instructions (q 0, a) (q 1, b, +1),..., and an input word, say w = aabab, we can construct PCP-instance so that: M does not halt on w there is an infinite initial solution. # q 0 a... a b... #q 0 aabab# bq 1... a b... #q 0 aa b... #q 0 aabab#bq 1 ab... So neg-hp is reducible to inf-pcp; inf-pcp is Π 0 1 -complete. Note: we can even require u i v i Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 39 / 50

Inf-PCP is reducible to bisimilarity on (normed) Type -1a u 1 u 2... u n v 1 v 2... v n u i, v i {A, B} +, u i v i Observation: The following conditions are equivalent u i1 u i2 u i3 = v i1 v i2 v i3 m: u i1 u i2 u im is a prefix of v i1 v i2 v im m: (u i1 u i2 u im ) R is a suffix of (v i1 v i2 v im ) R m: (u im ) R (u im 1 ) R (u i1 ) R is a suffix of (v im ) R (v im 1 ) R (v i1 ) R A game: Defender stepwise generates a sequence... I im... I i3 I i2 I i1 (with i 1 = 1) Attacker has a possibility to stop this process and win whenever (u im ) R (u im 1 ) R (u i1 ) R is not a suffix of (v im ) R (v im 1 ) R (v i1 ) R Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 40 / 50

Implementation of the game; generating rules c X Y X c Y i X c Y i for all i {1, 2,..., n} Y i XI i Y i i X I i for all i {1, 2,..., n} j XI j for all i, j {1, 2,..., n}, i j Y i c XI 1 c X I 1 c YI 1... Y 8 I 1...... Y 8 I 1... Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 41 / 50

Implementation of the game; generating rules c X Y X c Y i X c Y i for all i {1, 2,..., n} Y i XI i Y i i X I i for all i {1, 2,..., n} Y i j XI j for all i, j {1, 2,..., n}, i j c XI 1 c X I 1 c YI 1... Y 8 I 1... 5 8... XI 5 I 1... XI 8 I 1... Y 8 I 1... 5 8 XI 5 I 1... X I 8 I 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 41 / 50

Switch-to-checking rules XI 7 I 15 I 3 I 8 I 1 X I 7 I 15 I 3 I 8 I 1 (u i7 ) R (u i15 ) R (u i3 ) R (u i8 ) R (u i1 ) R (v i7 ) R (v i15 ) R (v i3 ) R (v i8 ) R (v i1 ) R X d C X (I )I i d C w X (I )I i d C w for all i {1, 2,..., n} and all suffices w of vi R Notation: I stands for (I 1 + I 2 + + I n ) ; XI 7 I 15 I 3 I 8 I 1 d CI 7 I 15 I 3 I 8 I 1 X I 7 I 15 I 3 I 8 I 1 d C wi 8 I 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 42 / 50

Checking rules CI 7 I 15 I 3 I 8 I 1 C wi 8 I 1 CA a C C A a C CB b C C B b C C e ε C e ε h(ui CI R ) i C tail(ui R ) C h(vi I R ) i C tail(vi R ) for all i {1, 2,..., n} Notation. h(w) = a when head(w) = A, h(w) = b when head(w) = B Observation. CI 7 I 15 I 3 I 8 I 1 is bisimilar to C wi 8 I 1 iff (u i7 ) R (u i15 ) R (u i3 ) R (u i8 ) R (u i1 ) R = w(v i8 ) R (v i1 ) R Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 43 / 50

Summary of the reduction: inf-pcp bisim-type-1a u 1 u 2... u n v 1 v 2... v n generating rules switch-to-checking rules checking rules There is an infinite initial solution XI 1 is bisimilar with X I 1 (Moreover: XI 1, X I 1 are normed processes.) Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 44 / 50

Π 0 1-completeness of bisim-normed-type-1b The only difference wrt Type -1a: switch-to-checking rules XI 7 I 15 I 3 I 8 I 1 X I 7 I 15 I 3 I 8 I 1 (u i7 ) R (u i15 ) R (u i3 ) R (u i8 ) R (u i1 ) R (v i7 ) R (v i15 ) R (v i3 ) R (v i8 ) R (v i1 ) R X X d C d C(A + B) X d C(A + B) XI 7 I 15 I 3 I 8 I 1 d CwI 7 I 15 I 3 I 8 I 1 X I 7 I 15 I 3 I 8 I 1 d C I 7 I 15 I 3 I 8 I 1 Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 45 / 50

High undecidability; Σ 1 1-completeness; rec-pcp A predicate P(x) (over natural numbers) is in Σ 1 1 iff P(x) MF(M, x) for a set variable M and a first-order arithmetical formula F. A well known Σ 1 1-complete problem: is there an infinite computation of a given nondeterministic Turing machine which visits q 0 infinitely often? A PCP-instance : u 1 u 2... u n v 1 v 2... v n u i, v i : nonempty words in an alphabet Recurrent solution: a sequence i 1, i 2, i 3,... from {1, 2,..., n} such that i 1 =1 and u i1 u i2 u i3 u i4 = v i1 v i2 v i3 v i4 1 appears infinitely often in the sequence i 1, i 2, i 3,... Fact. Problem rec-pcp is Σ 1 1 -complete. Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 46 / 50

Σ 1 1-completeness for (unrestricted) Type -1b X X Y c Y c Y I 1 I X c Y I 1 I Y c X c XI Y c XI X X X d C d C(A + B) X d C(A + B) f Z X f Z CA a C C A a C... ZI i i Z Z I i i Z for all i {1, 2,..., n} Z e ε Z e ε Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 47 / 50

Σ 1 1-completeness for normed Type -2 Like for the previous Type -1b, the only difference is the generating rules: c X Y X c Y I 1 I X c Y I 1 I Y c X YI c XI Y I c XI Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 48 / 50

Bisimilarity on prefix rewrite systems Normed Processes Unnormed Processes Type -2 Σ 1 1 -complete Σ1 1 -complete Type -1b Π 0 1 -complete Σ1 1 -complete Type -1a Π 0 1 -complete Π0 1 -complete Type 0, and decidable decidable Type 1 1 2 EXPTIME-hard EXPTIME-hard Type 2 P 2-EXPTIME P-hard PSPACE-hard Type 3 P-complete P-complete Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 49 / 50

Outline Behavioural equivalences; bisimulation equivalence. Undecidability of equivalences on (labelled place/transition) Petri nets (STACS 1994); imperfect Minsky machine simulation. Undecidability of the reachability set equality. Semilinear witnesses; decidable cases. Bisimilarity on process rewrite systems, and prefix rewrite systems. PSPACE-completeness of bisimilarity on Basic Parallel Processes (LiCS 2003); distance-to-disabling functions (dd-functions). Undecidability of bisimilarity on Type -1 systems (Jančar and Srba, FoSSaCS 2006); Defender s choice technique. THE END Petr Jančar (TU Ostrava) Deciding bisimilarity Brussels, 16.12.2005 50 / 50