On Reachability and Safety in Infinite-State Systems

We introduce some new models of infinite-state transition systems. The basic model, called a (reversal-bounded) counter machine (CM), is a nondeterministic finite automaton augmented with finitely many reversal-bounded counters (i.e. each counter can be incremented or decremented by 1 and tested for zero, but the number of times it can change mode from nondecreasing to nonincreasing and vice-versa is bounded by a constant, independent of the computation). We extend a CM by augmenting it with some familiar data structures: (i) A pushdown counter machine (PCM) is a CM augmented with an unrestricted pushdown stack. (ii) A tape counter machine (TCM) is a CM augmented with a two-way read/write worktape that is restricted in that the number of times the head crosses the boundary between any two adjacent cells of the worktape is bounded by a constant, independent of the computation (thus, the worktape is finite-crossing). There is no bound on how long the head can remain on a cell. (iii) A queue counter machine (QCM) is a CM augmented with a queue that is restricted in that the number of alternations between non-deletion phase and non-insertion phase on the queue is bounded by a constant. A non-deletion (non-insertion) phase is a period consisting of insertions (deletions) and no-changes, i.e., the queue is idle. We show that emptiness, (binary, forward, and backward) reachability, nonsafety, and invariance for these machines are solvable. We also look at extensions of the models that allow the use of linear-relation tests among the counters and parameterized constants as "primitive" predicates. We investigate the conditions under which these problems are still solvable.

[1]  Javier Esparza,et al.  Decidability of model checking for infinite-state concurrent systems , 1997, Acta Informatica.

[2]  Thomas A. Henzinger,et al.  Symbolic Model Checking for Real-Time Systems , 1994, Inf. Comput..

[3]  Pierre Wolper,et al.  Verifying Systems with Infinite but Regular State Spaces , 1998, CAV.

[4]  Hubert Comon-Lundh,et al.  Multiple Counters Automata, Safety Analysis and Presburger Arithmetic , 1998, CAV.

[5]  Oscar H. Ibarra,et al.  Binary Reachability Analysis of Discrete Pushdown Timed Automata , 2000, CAV.

[6]  Oscar H. Ibarra,et al.  A Technique for Proving Decidability of Containment and Equivalence of Linear Constraint Queries , 1999, J. Comput. Syst. Sci..

[7]  Sheila A. Greibach,et al.  Checking Automata and One-Way Stack Languages , 1969, J. Comput. Syst. Sci..

[8]  Igor Walukiewicz,et al.  Pushdown Processes: Games and Model-Checking , 1996, Inf. Comput..

[9]  Richard Gerber,et al.  Model-checking concurrent systems with unbounded integer variables: symbolic representations, approximations, and experimental results , 1999, TOPL.

[10]  Nicolas Halbwachs,et al.  Verification of Linear Hybrid Systems by Means of Convex Approximations , 1994, SAS.

[11]  Yassine Lakhnech,et al.  Computing Abstractions of Infinite State Systems Compositionally and Automatically , 1998, CAV.

[12]  Thomas A. Henzinger,et al.  A really temporal logic , 1994, JACM.

[13]  Hubert Comon-Lundh,et al.  Timed Automata and the Theory of Real Numbers , 1999, CONCUR.

[14]  Edmund M. Clarke,et al.  Symbolic Model Checking: 10^20 States and Beyond , 1990, Inf. Comput..

[15]  Patrice Godefroid,et al.  Symbolic Verification of Communication Protocols with Infinite State Spaces Using QDDs (Extended Abstract) , 1996, CAV.

[16]  Oscar H. Ibarra,et al.  Counter machines and the safety and disjointness problems for database queries with linear constraints , 2001, Where Mathematics, Computer Science, Linguistics and Biology Meet.

[17]  Pierre Wolper,et al.  Symbolic Verification with Periodic Sets , 1994, CAV.

[18]  M. Minsky.  Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines , 1961 .

[19]  Oscar H. Ibarra,et al.  Conter Machines: Decidable Properties and Applications to Verification Problems , 2000, MFCS.

[20]  Alain Finkel,et al.  Decidability of Reachability Problems for Classes of Two Counters Automata , 2000, STACS.

[21]  Jürgen Dingel,et al.  Model Checking for Infinite State Systems Using Data Abstraction, Assumption-Commitment Style reasoning and Theorem Proving , 1995, CAV.

[22]  Emil L. Post.  A variant of a recursively unsolvable problem , 1946 .

[23]  Rajeev Alur,et al.  Model-Checking in Dense Real-time , 1993, Inf. Comput..

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

[25]  Oscar H. Ibarra,et al.  Reversal-Bounded Multicounter Machines and Their Decision Problems , 1978, JACM.

[26]  Javier Esparza,et al.  Reachability Analysis of Pushdown Automata: Application to Model-Checking , 1997, CONCUR.

[27]  Ju. V. Matijasevic,et al.  ENUMERABLE SETS ARE DIOPHANTINE , 2003 .

[28]  Pierre Wolper,et al.  A direct symbolic approach to model checking pushdown systems , 1997, INFINITY.

[29]  Orna Grumberg,et al.  Abstract interpretation of reactive systems , 1997, TOPL.

[30]  Eitan M. Gurari,et al.  The Complexity of Decision Problems for Finite-Turn Multicounter Machines , 1981, J. Comput. Syst. Sci..

[31]  Ahmed Bouajjani,et al.  On the Automatic Verification of Systems with Continuous Variables and Unbounded Discrete Data Structures , 1994, Hybrid Systems.