Cubical Sets are Generalized Transition Systems

We show in this article that \labelled" cubical sets (or Higher-Dimensional Automata) are a natural generalization of transition systems and asynchronous transition systems. This generalizes an older result of 14] which was only holding with precubical sets and subcat-egories of the classical (see 29]) categories of transition systems and asynchronous transition systems. This opens up new promises on the actual use of geometric methods (such as 8]) and on comparisons with other methods for veriication of concurrent programs.

[1]  S. Lane.  Categories for the Working Mathematician , 1971 .

[2]  Lisbeth Fajstrup,et al.  Loops, ditopology and deadlocks , 2000, Mathematical Structures in Computer Science.

[3]  Glynn Winskel,et al.  Bisimulation and open maps , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[4]  Marek Antoni Bednarczyk,et al.  Categories of asynchronous systems , 1987 .

[5]  Martin Raußen,et al.  On the classification of dipaths in geometric models for concurrency , 2000, Mathematical Structures in Computer Science.

[6]  Eric Goubault,et al.  Homology of Higher Dimensional Automata , 1992, CONCUR.

[7]  P. J. Higgins,et al.  On the algebra of cubes , 1981 .

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

[9]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[10]  Lisbeth Fajstrup,et al.  Infinitely Running Concurrent Processes with Loops from a Geometric Viewpoint , 2001, Electron. Notes Theor. Comput. Sci..

[11]  M. W. Shields.  Concurrent Machines , 1985, Comput. J..

[12]  Carl A. Gunter,et al.  Semantic Domains , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[13]  Sjoerd E. Crans,et al.  Pasting Schemes for the Monoidal Biclosed Structure on , 1995 .

[14]  Lisbeth Fajstrup,et al.  Detecting Deadlocks in Concurrent Systems , 1996 .

[15]  Manfred Droste,et al.  Petri nets and automata with concurrency relations—an adjunction , 1993 .

[16]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[17]  Peter Gabriel,et al.  Calculus of Fractions and Homotopy Theory , 1967 .

[18]  Antti Valmari,et al.  A stubborn attack on state explosion , 1990, Formal Methods Syst. Des..

[19]  Pierre Wolper,et al.  Using partial orders for the efficient verification of deadlock freedom and safety properties , 1991, Formal Methods Syst. Des..

[20]  Glynn Winskel,et al.  Models for Concurrency , 1994 .

[21]  Vaughan R. Pratt,et al.  Modeling concurrency with geometry , 1991, POPL '91.

[22]  Philippe Gaucher,et al.  Homotopy invariants of higher dimensional categories and concurrency in computer science , 1999, Mathematical Structures in Computer Science.