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It is shown that the decision problem for the temporal logic with until and since connectives over real-numbers time is PSPACE-complete.

Abstract. We give a sound and complete axiomatization for the full computation tree logic, CTL*, of R-generable models. The language of CTL*, which is a propositional temporal language, is built recursively from the atomic propositions using the next X and until U operators of PLTL, and the existential path switching operator E of CTL as well as classical connectives. This language is appropriate for describing any situation with paths as countable sequences of states, the propositions having a truth evaluation at states and the possibility of at least some states lying on more than one path. We will be interested in the logic obtained by restricting attention to Kripke structures with states, a total accessibility relation between them and the set of all paths which arise by moving from state to state along the accessibility relation (which is usually called R). This standard semantics for CTL* is thus called the semantics over R-generable models.The main uses of CTL* in computer science are for developing and checking the correctness of complex reactive systems. See [Emerson, 1990] for a survey. CTL* is also used widely as a framework for comparing other languages more appropriate for specific reasoning tasks of this type. See the description in . These include the purely linear and purely branching sub-languages as well as languages such as [Bernholtz and Grumberg, 1994] which allow a limited amount of interplay between these two aspects.Validity of formulas of CTL* is known to be decidable. This was proved in [Emerson and Sistla, 1984] using an automata-theoretic approach. Essentially one

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

Propositional linear time temporal logic (LTL) is the standard temporal logic for computing applications and many reasoning techniques and tools have been developed for it. Tableaux for deciding satisfiability have existed since the 1980s. However, the tableaux for this logic do not look like traditional tree-shaped tableau systems and their processing is often quite complicated.In this paper, we introduce a novel style of tableau rule which supports a new simple traditionalstyle tree-shaped tableau for LTL. We prove that it is sound and complete. As well as being simple to understand, to introduce to students and to use, it is also simple to implement and is competitive against state of the art systems. It is particularly suitable for parallel implementations.

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