ABSTRACT
Dynamic logic is a generalization of first order logic in which quantifiers of the form “for all χ...” are replaced by phrases of the form “after executing program α...”. This logic subsumes most existing first-order logics of programs that manipulate their environment, including Floyd's and Hoare's logics of partial correctness and Manna and Waldinger's logic of total correctness, yet is more closely related to classical first-order logic than any other proposed logic of programs. We consider two issues: how hard is the validity problem for the formulae of dynamic logic, and how might one axiomatize dynamic logic? We give bounds on the validity problem for some special cases, including a Π02-completeness result for the partial correctness theories of uninterpreted flowchart programs. We also demonstrate the completeness of an axiomatization of dynamic logic relative to arithmetic.
- 1.de Bakker, J.W., and W.P. de Roever. A calculus for recursive program schemes, in Automata, Languages and Programming (ed. Nivat), 167-196. North Holland, 1972.Google Scholar
- 2.Basu, S. K. and R. T. Yeh. Strong Verification of Programs. IEEE Trans. Software Engineering, SE-1, 3, 339-345.Sept. 75. Google ScholarDigital Library
- 3.Burstall, R.M. Program Proving as Hand Simulation with a Little Induction. IFIP 1974, Stockholm.Google Scholar
- 4.Cook, S.A. Axiomatic and Interpretive Semantics for an Algol Fragment. TR-79, Toronto, Feb. 1975.Google Scholar
- 5.Dijkstra, E. A Discipline of Programming. Prentice-Hall, Englewood Cliffs, N.J. 1976. Google ScholarDigital Library
- 6.Harel, D., A. Pnueli and J. Stavi. A complete axiomatic system for proving deductions about recursive programs. Proc. Ninth Ann. ACM Symp. on Theory of Computing, Boulder, Col., May 1977. Google ScholarDigital Library
- 7.Hitchcock, P. and D. Park. Induction Rules and Termination Proofs. In Automata, Languages and Programming (ed. Nivat, M.), IRIA. North-Holland, 1973.Google Scholar
- 8.Hoare, C.A.R. An Axiomatic Basis for Computer Programming. CACM 12, 576-580, 1969. Google ScholarDigital Library
- 9.Hughes, C.E. and M.J. Cresswell. An Introduction to Modal Logic. London: Methuen and Co Ltd. 1972.Google Scholar
- 10.Kripke, S. Semantical considerations on Modal Logic. Acta Philosophica Fennica, 83-94, 1963.Google Scholar
- 11.Luckham, D., D. Park and M. Paterson. On Formalized Computer Programs. J.CSS 3, 2, 119-127. May 1970.Google Scholar
- 12.Manna, Z. and R. Waldinger. Is "sometime" sometimes better than "always"? Intermittent assertions in proving program correctness. Proc. 2nd Int.Conf. on Software Engineering, Oct. 1976. Google ScholarDigital Library
- 13.Pratt, V.R. Semantical Considerations on Floyd-Hoare Logic. 17th IEEE Symposium on Foundations of Computer Science, Oct. 1976.Google ScholarDigital Library
- 14.Rogers, H. Theory of Recursive Functions and Effective Computability. McCraw-Hill, 1967. Google ScholarDigital Library
- 15.Tarski, A. The semantic conception of truth and the foundations of semantics. Philos. and Phenom. Res, 4, 341-376, 1944.Google ScholarCross Ref
Index Terms
- Computability and completeness in logics of programs (Preliminary Report)
Recommendations
Nondeterminism in logics of programs
POPL '78: Proceedings of the 5th ACM SIGACT-SIGPLAN symposium on Principles of programming languagesWe investigate the principles underlying reasoning about nondeterministic programs, and present a logic to support this kind of reasoning. Our logic, an extension of dynamic logic ([22] and [12]), subsumes most existing first-order logics of ...
Inductive Completeness of Logics of Programs
We propose a new approach to delineating logics of programs, based directly on inductive definition of program semantics. The ingredients are elementary and well-known, but their fusion yields a simple yet powerful approach, surprisingly overlooked for ...
Structural Completeness and Superintuitionistic Inquisitive Logics
Logic, Language, Information, and ComputationAbstractIn this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics involving also systems that are not closed under uniform substitution. We just require that each logic must be ...
Comments