An algebraic approach to temporal logic
The sequential calculus is an algebraic calculus, intended for reasoning about phenomena with a duration and their sequencing. It can be specialized to various domains used for reasoning about programs and systems, including Tarski's calculus of binary relations, Kleene's regular expressions, Hoare's CSP and Dijkstra's regularity calculus.
In this paper we use the sequential calculus as a tool for algebraizing temporal logics. We show that temporal operators are definable in terms of sequencing and we show how a specific logic may be selected by introducing additional axioms. All axioms of the complete proof system for discrete linear temporal logic (given in ) are obtained as theorems of sequential algebra.
Our work embeds temporal logic into an algebra naturally equipped with sequencing constructs, and in which recursion is definable. This could be a first step towards a design calculus for transforming temporal specifications by stepwise refinement into executable programs.
This paper is an extended abstract of a technical report  containing full proofs (available by ftp). Most proofs have been omitted and simplifying assumptions were made to make the presentation easier.
- 1.R. Berghammer, P. Kempf, G. Schmidt, and T. Ströhlein. Relation algebra and logic of programs. In Algebraic Logic, volume 54 of Colloquia Mathematica Societatis János Bolyai. Budapest University, 1988.Google Scholar
- 2.R. Berghammer and B. von Karger. Formal derivation of CSP programs from formal specifications. submitted to MPC 95, 1995.Google Scholar
- 3.S. M. Brien. A time-interval calculus. In R. Bird, C. Morgan, and J. Woodcock, editors, Mathematics of Program Construction, LNCS 669. Springer-Verlag, 1992.Google Scholar
- 4.E. W. Dijkstra. The unification of three calculi. In M. Broy, editor, Program Design Calculi, pages 197–231. Springer Verlag, 1993.Google Scholar
- 5.B. v. Karger. Temporal logic as a sequential calculus. Procos technical report [kiel bvk 17], Christian-Albrechts-Univ., Inst. f. Inf. und Prakt. Math., Kiel, 1994. available via WWW, http://WWW.informatik.uni-kiel.de/~procos/kiel.html.Google Scholar
- 6.B. v. Karger and C.A.R. Hoare. Sequential calculus. To appear in Information Processing Letters, 1995.Google Scholar
- 7.S. Kleene. Representation of events in nerve nets and finite automata. In Shannon and McCarthy, editors, Automata Studies, pages 3–42. Princeton University Press, 1956.Google Scholar
- 8.R. Maddux. A working relational model: The derivation of the Dijkstra-Scholten predicate transformer semantics from Tarski's axioms of the Peirce-Schröder calculus of relations. Manuscript, 1992.Google Scholar
- 9.Z. Manna and A. Pnueli. The Temporal logic of Reactive and Concurrent Systems-Specification. Springer-Verlag, 1991.Google Scholar
- 10.B. Moszkowski. Some very compositional temporal properties. Technical Report TR-466, University of Newcastle, 1993. Accepted for Procomet 1994, San Miniato.Google Scholar
- 11.E. Olderog and C. Hoare. Specification oriented semantics for communicating processes. Acta Inf., 23:9–66, 1986.Google Scholar
- 12.G. Schmidt and T. Ströhlein. Relations and Graphs. EATCS Monographs on Theoretical Computer Science. Springer, 1991.Google Scholar
- 13.A. Tarski. On the calculus of relations. Journal of Symbolic Logic, 6(3):73–89, 1941.Google Scholar
- 14.C. Zhou, C.A.R. Hoare, and A. P. Ravn. A calculus of durations. Information Processing Letters, 40:269–276, 1992.Google Scholar