Symbolic Algorithms for Infinite-State Games

A procedure for the analysis of state spaces is called symbolic if it manipulates not individual states, but sets of states that are represented by constraints. Such a procedure can be used for the analysis of infinite state spaces, provided termination is guaranteed. We present symbolic procedures, and corresponding termination criteria, for the solution of infinite-state games, which occur in the control and modular verification of infinite-state systems. To characterize the termination of symbolic procedures for solving infinite-state games, we classify these game structures into four increasingly restrictive categories: 1. Class 1 consists of infinite-state structures for which all safety and reachability games can be solved. 2. Class 2 consists of infinite-state structures for which all ω-regular games can be solved. 3. Class 3 consists of infinite-state structures for which all nested positive boolean combinations of ω-regular games can be solved. 4. Class 4 consists of infinite-state structures for which all nested boolean combinations of ω-regular games can be solved. We give a structural characterization for each class, using equivalence relations on the state spaces of games which range from game versions of trace equivalence to a game version of bisimilarity. We provide infinite-state examples for all four classes of games from control problems for hybrid systems. We conclude by presenting symbolic algorithms for the synthesis of winning strategies ("controller synthesis") for infinite-state games with arbitrary u-regular objectives, and prove termination over all class-2 structures. This settles, in particular, the symbolic controller synthesis problem for rectangular hybrid systems.

[1]  Robert McNaughton,et al.  Infinite Games Played on Finite Graphs , 1993, Ann. Pure Appl. Logic.

[2]  Thomas A. Henzinger,et al.  Alternating-time temporal logic , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[3]  Nicolas Halbwachs,et al.  Minimal Model Generation , 1990, CAV.

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

[5]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[6]  Joseph Sifakis,et al.  On the Synthesis of Discrete Controllers for Timed Systems (An Extended Abstract) , 1995, STACS.

[7]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[8]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[9]  P. Ramadge,et al.  Supervisory control of a class of discrete event processes , 1987 .

[10]  Thomas A. Henzinger,et al.  Alternating Refinement Relations , 1998, CONCUR.

[11]  Dexter Kozen,et al.  Results on the Propositional µ-Calculus , 1982, ICALP.

[12]  A. Prasad Sistla,et al.  On Model-Checking for Fragments of µ-Calculus , 1993, CAV.

[13]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[14]  Parosh Aziz Abdulla,et al.  Verifying Networks of Timed Processes (Extended Abstract) , 1998, TACAS.

[15]  Andrzej Wlodzimierz Mostowski,et al.  Regular expressions for infinite trees and a standard form of automata , 1984, Symposium on Computation Theory.

[16]  Jim Alves-Foss,et al.  Higher Order Logic Theorem Proving and its Applications 8th International Workshop, Aspen Grove, Ut, Usa, September 11-14, 1995 : Proceedings , 1995 .

[17]  Thomas A. Henzinger,et al.  HYTECH: a model checker for hybrid systems , 1997, International Journal on Software Tools for Technology Transfer.

[18]  Satoshi Yamane,et al.  The symbolic model-checking for real-time systems , 1996, Proceedings of the Eighth Euromicro Workshop on Real-Time Systems.

[19]  J. R. Büchi,et al.  Solving sequential conditions by finite-state strategies , 1969 .

[20]  Wolfgang Thomas,et al.  On the Synthesis of Strategies in Infinite Games , 1995, STACS.

[21]  Thomas A. Henzinger,et al.  Rectangular Hybrid Games , 1999, CONCUR.

[22]  Dexter Kozen,et al.  RESULTS ON THE PROPOSITIONAL’p-CALCULUS , 2001 .