An Axiomatization of Lamport's Temporal Logic of Actions

This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonproot educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of the Systems Research Center of Digital Equipment Corporation in Palo Alto, California; an acknowledgment o f t h e authors and individual contributors to the work; and all applicable portions of the copyright notice. Copying, reproducing, or republishing for any other purpose shall require a license with payment of fee to the Systems Research Center. All rights reserved. iii Author's Abstract Lamport recently invented a temporal logic of actions suitable for expressing concurrent programs and for reasoning about their computations. In this logic, actions have s y n tactic representations, which can be combined and analyzed. The basic construct for relating actions and computations is ; a computation satisses the formula A if either the computation has halted or the rst action in the computation is an A action. In addition, the language includes the temporal operators 2 always" and 3 eventually", and thus it is easy to write both safety and liveness formulas. However, the temporal logic of actions is not very expressive i n s o m e r e-spects just expressive enough. One cannot deene the next" and the un-til" operators of many previous temporal logics. This is actually a feature, in that formulas with until" are too often incomprehensible, and next" violates the important principle of invariance under stuttering. A proof system for the logic of actions might be obtained by translating into previous, richer formalisms. In this translation we forfeit the logic and its advantages. A new suit of rules for temporal reasoning with actions is therefore wanted. A complete axiomatization can provide some guidance in choosing and understanding the rules used in practice, and in particular the laws for reasoning about programs. In this paper, we study a proof system for a propositional logic, ptla. After an informal introduction, we deene the syntax and semantics of ptla precisely, and then present our proof system and prove its completeness.