Abstract
In concurrency theory, there are several examples where the interleaved model of concurrency can distinguish between execution sequences which are not significantly different. One such example is sequences that differ from each other by stuttering, i.e., the number of times a state can adjacently repeat. Another example is executions that differ only by the ordering of independently executed events. Considering these sequences as different is semantically rather meaningless. Nevertheless, specification languages that are based on interleaving semantics, such as linear temporal logic (LTL), can distinguish between them. This situation has led to several attempts to define languages that cannot distinguish between such equivalent sequences. In this paper, we take a different approach to this problem: we develop algorithms for deciding if a property cannot distinguish between equivalent sequences, i. e., is closed under the equivalence relation. We focus on properties represented by regular languages, Ω-regular languages, or propositional LTL formulae and show that for such properties there is a wide class of equivalence relations for which determining closure is decidable, in fact in PSPACE. Hence, checking the closure of a specification is no more difficult than checking satisfiability of a temporal formula. Among the closure properties we are able to handle, one finds trace closedness, stutter closedness and projective closedness, for all of which we are also able to prove a PSPACE lower bound. Being able to check that a property is closed under an equivalence relation has an immediate application in state-space exploration based verification. Indeed, the knowledge that the specification does not distinguish between equivalent execution sequences allows constructing a reduced state space where it is sufficient that at least one sequence per equivalence class is represented.
This author is supported by a DIMACS postdoctoral fellowship funded by NSF award 91-19999 and the New Jersey Commission on Science and Technology.
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References
Alur, R., Peled, D., Penczek, W.: Model-checking of causality properties. In Proc. 10th IEEE Symposium on Logic in Computer Science, San Diego, California (1995) 90–100.
Arnold, A.: A syntactic congruence for rational Ω-languages. Theoretical Computer Science 39 (1985) 333–335.
Diekert, V., Gastin, P., Petit, A.: Rational and recognizable trace languages. Information and Computation 116 (1995) 134–153.
Godefroid, P.: Using partial orders to improve automatic verification methods. In Proc. 2nd Workshop on Computer Aided Verification, New Brunswick, NJ. Lect. Notes in Comput. Sci., vol. 531, Springer (1990) 176–185.
Katz, S., Peled, D.: Verification of distributed programs using representative interleaving sequences. Distributed Computing 6 (1992) 107–120.
Kwiatkowska, M. Z.: Event fairness and non-interleaving concurrency. Formal Aspects of Computing 1 (1989) 213–228.
Kozen, D.: Lower bounds for natural proof systems. 18th IEEE Symposium on Foundations of Computer Science, Providence, Rhode Island (1977) 254–266.
Lamport, L.: How to make a multiprocessor computer that correctly executes multiprocess programs. IEEE Transactions on Computers 28 (1979) 690–691.
Lamport, L.: What good is temporal logic? In Proc. IFIP Congr. on Information Processing, Elsevier (1983) 657–668.
Mazurkiewicz, A.: Trace theory. In Proc. Advances in Petri Nets 1986, Bad Honnef, Germany. Lect. Notes in Comput. Sci., vol. 255, Springer (1987) 279–324.
Pécuchet, J.-P.: Etude Syntaxique des parties reconnaissable de mots infinis. In Automata, Languages, and Programming: 13th Intern. Coll. Rennes, France. Lect. Notes in Comput. Sci., vol.226, Springer (1986) 294–303.
Peled, D.: On projective and separable properties. In Proc. Colloquium on Trees in Algebra and Programming, Edinburgh, Scotland. Lect. Notes in Comput. Sci., vol. 787, Springer (1994) 291–307.
Peled, D.: All from One, One from All: on Model Checking using representatives, In Proc. 5th International Conference on Computer Aided Verification, Elounda, Greece, Lect. Notes in Comput. Sci., vol.697, Springer (1993) 409–423.
Peled, D.: Combining partial-order reductions with on-the-fly model-checking. Formal Methods in System Design 8 (1996) 39–64.
Peled, D., Pnueli, A.: Proving partial-order properties. Theoretical Computer Science 126 (1994) 143–182.
Pnueli, A.: The temporal logic of programs. In Proc. 18th IEEE Symposium on Foundation of Computer Science, Providence, Rhode Island (1977) 46–57.
Sistla, A. P., Clarke, E. M.: The complexity of propositional linear temporal logics. Journal of the ACM 32 (1985) 733–749.
Sistla, A. P., Vardi, M. Y., Wolper, P.: The complementation problem for Büchi automata with applications to temporal logic. Theoretical Computer Science 49 (1987) 217–237.
Thiagarajan, P. S.: A trace based extension of linear time temporal logic. In Proc. 10th IEEE Symposium on Logic in Computer Science, Paris, France (1994) 438–447.
Thomas, W.: Automata and quantifier hierarchies: formal properties of finite automata and applications. In Proc. of LITP Spring School on Theoretical Computer Science, J. E. Pin, ed. Lect. Notes in Comput. Sci., vol. 386, Springer (1989) 104–119.
Thomas, W.: Automata on infinite objects. In Handbook of Theoretical Computer Science, vol. B, J. van Leeuwen, ed., Elsevier, Amsterdam (1990) 133–191.
Valmari, A.: A stubborn attack on state explosion. Formal Methods in System Design 1 (1992) 297–322.
Vardi, M. Y., Wolper, P.: Automata-theoretic techniques for modal logics of programs. J. Comput. System Sci. 32 (1986) 182–221.
Vardi, M. Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115 (1994) 1–37.
Wolper, P.: Temporal logic can be more expressive. Information and Control 56 (1983) 72–99.
Wolper, P., Godefroid, P.: Partial-order methods for temporal verification. In Proc. CONCUR, 4th Conference on Concurrency Theory, Hildesheim, Germany. Lect. Notes in Comput. Sci., vol.715, Springer (1993) 233–246.
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Peled, D., Wilke, T., Wolper, P. (1996). An algorithmic approach for checking closure properties of Ω-regular languages. In: Montanari, U., Sassone, V. (eds) CONCUR '96: Concurrency Theory. CONCUR 1996. Lecture Notes in Computer Science, vol 1119. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61604-7_78
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