"Sometime" is sometimes "not never": on the temporal logic of programs

• Authors: Leslie Lamport
• Venue: POPL '80 Proceedings of the 7th ACM SIGPLAN-SIGACT symposium on Principles of programming languages
• Year: 1980

Pnueli [15] has recently introduced the idea of using temporal logic [18] as the logical basis for proving correctness properties of concurrent programs. This has permitted an elegant unifying formulation of previous proof methods. In this paper, we attempt to clarify the logical foundations of the application of temporal logic to concurrent programs. In doing so, we will also clarify the relation between concurrency and nondeterminism, and identify some problems for further research.In this paper, we consider logics containing the temporal operators "henceforth" (or "always") and "eventually" (or "sometime"). We define the semantics of such a temporal logic in terms of an underlying model that abstracts the fundamental concepts common to almost all the models of computation which have been used. We are concerned mainly with the semantics of temporal logic, and will not discuss in any detail the actual rules for deducing theorems.We will describe two different temporal logics for reasoning about a computational model. The same formulas appear in both logics, but they are interpreted differently. The two interpretations correspond to two different ways of viewing time: as a continually branching set of possibilities, or as a single linear sequence of actual events. The temporal concepts of "sometime" and "not never" ("not always not") are equivalent in the theory of linear time, but not in the theory of branching time -- hence, our title. We will argue that the logic of linear time is better for reasoning about concurrent programs, and the logic of branching time is better for reasoning about nondeterministic programs.The logic of linear time was used by Pnueli in [15], while the logic of branching time seems to be the one used by most computer scientists for reasoning about temporal concepts. We have found this to cause some confusion among our colleagues, so one of our goals has been to clarify the formal foundations of Pnueli's work.The following section gives an intuitive discussion of temporal logic, and Section 3 formally defines the semantics of the two temporal logics. In Section 4, we prove that the two temporal logics are not equivalent, and discuss their differences. Section 5 discusses the problems of validity and completeness for the temporal logics. In Section 6, we show that there are some important properties of the computational model that cannot be expressed with the temporal operators "henceforth" and "eventually", and define more general operators.