Temporal logic (vol. 1): mathematical foundations and computational aspects
Temporal logic (vol. 1): mathematical foundations and computational aspects
On the temporal analysis of fairness
POPL '80 Proceedings of the 7th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
Future temporal logic needs infinitely many modalities
Information and Computation
The temporal logic of programs
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
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Kamp's theorem established the expressive completeness of the temporal modalities Until and Since for the First-Order Monadic Logic of Order (FOMLO) over Real and Natural time flows. Over Natural time, a single future modality (Until) is sufficient to express all future FOMLO formulas. These are formulas whose truth value at any moment is determined by what happens from that moment on. Yet this fails to extend to Real time domains: Here no finite basis of future modalities can express all future FOMLO formulas. In this paper we show that finiteness can be recovered if we slightly soften the requirement that future formulas must be totally past-independent: We allow formulas to depend just on the very very near-past, and maintain the requirement that they be independent of the rest - actually - of most of the past. We call them ‘almost future' formulas, and show that there is a finite basis of almost future modalities which is expressively complete over the Reals for the almost future fragment of FOMLO.