Linear temporal logic

Abstract: Publisher Summary This chapter discusses temporal and modal logic. The chapter describes a multiaxis classification of systems of temporal logic. The chapter describes the framework of linear temporal logic. In both its propositional and first-order forms, linear temporal logic has been widely employed in the specification and verification of programs. The chapter describes the competing framework of branching temporal logic, which has seen wide use. It also explains how temporal logic structures can be used to model concurrent programs using non-determinism and fairness. The chapter also discusses other modal and temporal logics in computer science. The chapter describes the formal syntax and semantics of Propositional Linear Temporal Logic (PLTL). The chapter also describes the formal syntax and semantics for two representative systems of propositional branching-time temporal logics.

Abstract: Temporal logic comes in two varieties: linear-time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system; branching-time temporal logic allows explicit existential and universal quantification over all paths. We introduce a third, more general variety of temporal logic: alternating-time temporal logic offers selective quantification over those paths that are possible outcomes of games, such as the game in which the system and the environment alternate moves. While linear-time and branching-time logics are natural specification languages for closed systems, alternating-time logics are natural specification languages for open systems. For example, by preceding the temporal operator "eventually" with a selective path quantifier, we can specify that in the game between the system and the environment, the system has a strategy to reach a certain state. The problems of receptiveness, realizability, and controllability can be formulated as model-checking problems for alternating-time formulas. Depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators, we obtain the two alternating-time temporal logics ATL and ATLa.ATL and ATLa are interpreted over concurrent game structures. Every state transition of a concurrent game structure results from a choice of moves, one for each player. The players represent individual components and the environment of an open system. Concurrent game structures can capture various forms of synchronous composition for open systems, and if augmented with fairness constraints, also asynchronous composition. Over structures without fairness constraints, the model-checking complexity of ATL is linear in the size of the game structure and length of the formula, and the symbolic model-checking algorithm for CTL extends with few modifications to ATL. Over structures with weak-fairness constraints, ATL model checking requires the solution of 1-pair Rabin games, and can be done in polynomial time. Over structures with strong-fairness constraints, ATL model checking requires the solution of games with Boolean combinations of Buchi conditions, and can be done in PSPACE. In the case of ATLa, the model-checking problem is closely related to the synthesis problem for linear-time formulas, and requires doubly exponential time.

Abstract: This paper is motivated by the need for a formal specification method for real-time systems. In these systemsquantitative temporal properties play a dominant role. We first characterize real-time systems by giving a classification of such quantitative temporal properties. Next, we extend the usual models for temporal logic by including a distance function to measure time and analyze what restrictions should be imposed on such a function. Then we introduce appropriate temporal operators to reason about such models by turning qualitative temporal operators into (quantitative) metric temporal operators and show how the usual quantitative temporal properties of real-time systems can be expressed in this metric temporal logic. After we illustrate the application of metric temporal logic to real-time systems by several examples, we end this paper with some conclusions.

Abstract: The differences between and appropriateness of branching versus linear time temporal logic for reasoning about concurrent programs are studied. These issues have been previously considered by Lamport. To facilitate a careful examination of these issues, a language, CTL*, in which a universal or existential path quantifier can prefix an arbitrary linear time assertion, is defined. The expressive power of a number of sublanguages is then compared. CTL* is also related to the logics MPL of Abrahamson and PL of Harel, Kozen, and Parikh. The paper concludes with a comparison of the utility of branching and linear time temporal logics.