Global and local invariants in transition systems
Given a transition system and a cover P of the set of its states, a set of local invariants with respect to P is defined as a set of predicates in bijection with the blocks of P and in such a way that a local invariant be true every time the system is in a state belonging to the corresponding block of the cover.
This definition is proved to be sufficiently general in the sense that any proof made by using global invariants can be also made by using sets of local invariants. The same result is proved for two more restrictive definitions of the notion of local invariant by using well-known properties of connections between lattices.
Finally, it is shown how the theory of connections can provide a general frame for tackling the problem of decomposing a global assertion into a logically equivalent set of local assertions.
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- [ASH 75]E.A. ASHCROFT “Proving assertions about parallel programs” Journal of Comp. and System Sciences 10, (1975), pp. 110–135.Google Scholar
- [BLI 73]A. BLIKLE “An algebraic approach to mathematical theory of programs” Polish Academy of Sciences, Report nℴ 119, Warszawa, 1973.Google Scholar
- [COU 78]P. COUSOT “Méthodes itératives de construction et d'approximation de points fixes d'opérateurs monotones sur un treillis, analyse sémantique des programmes” Thèse d'Etat, Grenoble, March 1978.Google Scholar
- [COU 80]P. COUSOT and R. COUSOT “Constructing program invariance proof methods” International Workshop on Program Construction, Château de Bonas, INRIA (Ed.) Sept. 1980, pp. 13–21.Google Scholar
- [FLO 67]R.W. FLOYD “Assigning meaning to programs” Proc. Symp. on Applied Mathematics, Vol. 19, I.T. Schwartz (Ed.), A.M.S., 1967, pp. 19–32.Google Scholar
- [HOA 78]C.A.R. HOARE “Some properties of predicate transformers” J.A.C.M., Vol. 25, nℴ 3, July 1978, pp. 461–480.Google Scholar
- [KEL 72]R.M. KELLER “Vector replacement systems: a formalism for modeling asynchronous systems” Princeton University technical report nℴ 117, December 1972.Google Scholar
- [KEL 76]R.M. KELLER “Formal verification of parallel programs” Comm. ACM, Vol. 19, nℴ 7, July 1976, pp. 371–384.Google Scholar
- [LAM 80]L. LAMPORT “Sometime” is sometimes “not never”-On the temporal logic of programs” Proc. of the 7th Annual ACM Symp. on Principles of Programming Languages Las Vegas, Jan. 1980, pp. 174–185.Google Scholar
- [MAN 78]Z. MANNA and R. WALDINGER “Is “sometime” sometimes better than “always” ?” Comm. ACM, Vol. 21, nℴ 2, Feb. 1978, pp. 159–172.Google Scholar
- [MAN 81]Z. MANNA and A. PNUELLI “Verification of concurrent programs: the temporal framework” Int. Summer School Theoretical Foundations of Programming Methodology, July 28 to August 9, 1981, Munich.Google Scholar
- [MAZ 74]A. MAZURKIEWICZ “Proving properties of processes” Algorytmy, XI, nℴ 19, 1974, pp. 5–22.Google Scholar
- [ORE 44]O. ORE “Galois connexions”, Trans. A.M.S., 55 (1944), pp. 493–513.Google Scholar
- [PNU 79]A. PNUELLI “The temporal semantics of concurrent programs” LNCS Vol. 70, Springer Verlag, July 1979, pp. 1–20.Google Scholar
- [QUE 81]J.P. QUEILLE and J. SIFAKIS “Specification and verification of concurrent systems in CESAR: An example” Research report RR 254. IMAG, Grenoble, June 1981.Google Scholar
- [SAN 77]L.E. SANCHIS “Data types as lattices: retractions, closures and projectiores” RAIRO Informatique Théorique, Vol. 11, nℴ 4, 1977, pp. 329–344.Google Scholar
- [SIF 79]J. SIFAKIS “A unified approach for studying the properties of transition systems” Research report RR 179, IMAG, Grenoble, December 1979 (Revised December 1980); to appear in Theoretical Computer Science 1982.Google Scholar
- [SIF 81]J. SIFAKIS “Global and local invariants in transition systems” Research report RR nℴ 274, IMAG, Grenoble, November 1981.Google Scholar