Undecidability of domino games and hhp-bisimilarity

History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is that they are preserved under action refinement, an operation important for the top-down design of concurrent systems. It is shown that, in contrast to hp-bisimilarity, checking hhp-bisimilarity for finite labelled asynchronous transition systems is undecidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable. It is also shown that the unlabelled domino bisimilarity problem is undecidable, which implies undecidability of hhp-bisimilarity for unlabelled finite asynchronous systems. Moreover, it is argued that the undecidability of hhp-bisimilarity holds for finite elementary net systems and 1-safe Petri nets.

[1]  Vladimiro Sassone,et al.  Higher dimensional transition systems , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[2]  Søren Christensen,et al.  Bisimulation Equivalence is Decidable for all Context-Free Processes , 2022 .

[3]  Sibylle B. Fröschle,et al.  On Plain and Hereditary History-Preserving Bisimulation , 1999, MFCS.

[4]  Robin Milner,et al.  Algebraic laws for nondeterminism and concurrency , 1985, JACM.

[5]  Albert R. Meyer,et al.  Deciding True Concurrency Equivalences on Safe, Finite Nets , 1996, Theor. Comput. Sci..

[6]  Glynn Winskel,et al.  Bisimulation from Open Maps , 1994, Inf. Comput..

[7]  Mogens Nielsen,et al.  Games and Logics for a Noninterleaving Bisimulation , 1995, Nord. J. Comput..

[8]  David Park,et al.  Concurrency and Automata on Infinite Sequences , 1981, Theoretical Computer Science.

[9]  Faron Moller,et al.  A Polynomial Algorithm for Deciding Bisimilarity of Normed Context-Free Processes , 1994, Theor. Comput. Sci..

[10]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

[11]  Sibylle B. Fröschle,et al.  On Plain and Hereditary History-Preserving Bisimulation , 1999 .

[12]  Mogens Nielsen,et al.  Models for Concurrency , 1992 .

[13]  Wolfgang Thomas,et al.  On the Ehrenfeucht-Fraïssé Game in Theoretical Computer Science , 1993, TAPSOFT.

[14]  David Harel,et al.  On the Solvability of Domino Snake Problems , 1994, Theor. Comput. Sci..

[15]  Erich Grädel,et al.  Domino Games and Complexity , 1990, SIAM J. Comput..

[16]  Rob J. van Glabbeek,et al.  The Linear Time-Branching Time Spectrum (Extended Abstract) , 1990, CONCUR.

[17]  Marvin Minsky,et al.  Computation : finite and infinite machines , 2016 .

[18]  Yuri Gurevich,et al.  The Classical Decision Problem , 1997, Perspectives in Mathematical Logic.

[19]  Glynn Winskel,et al.  Petri Nets and Bisimulation , 1996, Theor. Comput. Sci..

[20]  Robert E. Tarjan,et al.  Three Partition Refinement Algorithms , 1987, SIAM J. Comput..

[21]  Ursula Goltz,et al.  Equivalence Notions for Concurrent Systems and Refinement of Actions (Extended Abstract) , 1989, MFCS.

[22]  P. S. Thiagarajan,et al.  Controllers for Discrete Event Systems via Morphisms , 1998, CONCUR.

[23]  Mogens Nielsen,et al.  Hereditary History Preserving Bisimilarity is Undecidable , 1999 .

[24]  Walter Vogler.  Deciding History Preserving Bisimilarity , 1991, ICALP.

[25]  Scott A. Smolka,et al.  CCS expressions, finite state processes, and three problems of equivalence , 1983, PODC '83.

[26]  Jan Friso Groote,et al.  Undecidable Equivalences for Basic Process Algebra , 1994, Inf. Comput..

[27]  Sibylle B. Fröschle.  Decidability of Plain and Hereditary History-Preserving Bisimilarity for BPP , 1999, EXPRESS.

[28]  Faron Moller,et al.  On the computational complexity of bisimulation , 1995, CSUR.