The aim of this paper is to summarize and analyze some results obtained in 2000-2001 about decidable and undecidable fragments of various first-order temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community' to a number of interesting open problems.

Until recently, First-Order Temporal Logic (FOTL) has been only partially understood. While it is well known that the full logic has no finite axiomatisation, a more detailed analysis of fragments of the logic was not previously available. However, a breakthrough by Hodkinson et al., identifying a finitely axiomatisable fragment, termed the monodic fragment, has led to improved understanding of FOTL. Yet, in order to utilise these theoretical advances, it is important to have appropriate proof techniques for this monodic fragment. In this paper, we modify and extend the clausal temporal resolution technique, originally developed for propositional temporal logics, to enable its use in such monodic fragments. We develop a specific normal form for monodic formulae in FOTL, and provide a complete resolution calculus for formulae in this form. Not only is this clausal resolution technique useful as a practical proof technique for certain monodic classes, but the use of this approach provides us with increased understanding of the monodic fragment. In particular, we here show how several features of monodic FOTL can be established as corollaries of the completeness result for the clausal temporal resolution method. These include definitions of new decidable monodic classes, simplification of existing monodic classes by reductions, and completeness of clausal temporal resolution in the case of

Recently, a hierarchy of spatio-temporal languages based on the propositional temporal logic PTL and the spatial languages RCC-8, BRCC-8 and S4u has been introduced. Although a number of results on their computational properties were obtained, the most important questions were left open.

We present TeMP---the first experimental system for testing validity of monodic temporal logic formulae. The prover implements fine-grained temporal resolution. The core operations required by the procedure are performed by an efficient resolution-based prover for classical first-order logic.

As a remedy for the bad computational behaviour of first-order temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining so-called monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of first-order logic like the two-variable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for first-order fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL.

The clausal resolution method for propositional linear-time temporal logics is well known and provides the basis for a number of temporal provers. The method is based on an intuitive clausal form, called SNF, comprising three main clause types and a small number of resolution rules. In this paper, we show how the normal form can be radically simplified and, consequently, how a simplified clausal resolution method can be defined for this important variety of logic.

First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of first-order temporal logic over expanding domains. We first define a normal form for monodic formulae and show how arbitrary monodic formulae can be translated into the normal form, while preserving satisfiability. We then introduce novel resolution calculi that can be applied to formulae in this normal form and state correctness and completeness results for the method. We illustrate the method on a comprehensive example. The method is based on classical first-order resolution and can, thus, be efficiently implemented.

this paper. 2 Basics of PLTL Let P be a set of propositional variables. The set of formulae of propositional linear time logic PLTL (over P) is inductively defined as follows: (i) ? is a formula of PLTL, (ii) every propositional variable of P is a formula of PLTL, (iii) if ' and / are formulae of PLTL, then :' and (' /) are formulae of PLTL, and (iv) if ' and / are formulae of PLTL, then #' (in the next moment of time ' is true), 3' (sometimes in the future ' is true), 2' (always in the future ' is true), (' U /) (' is true until / is true), and (' W /) (' is true unless / is true) are formulae of PLTL. Other Boolean connectives including ?, , !, and $ are defined using ?, :, and

complexity; clausal temporal resolution. In this paper, we consider a tractable sub-class of propositional linear time temporal logic, and provide a complete clausal resolution calculus for it. The fragment is important as it captures simple Büchi automata. We also show that, just as the emptiness check for a Büchi automaton is tractable, the complexity of deciding unsatisfiability, via resolution, of our logic is polynomial (rather than exponential). Consequently, a Büchi automaton can be represented within our logic, and its emptiness can be tractably decided via deductive methods. This may have a significant impact upon approaches to verification, since techniques such as model checking inherently depend on the ability to check emptiness of an appropriate Büchi automaton. Thus, we also discuss how such a logic might form the basis for practical deductive temporal verification. 1

Abstract. First-order temporal logic is a concise and powerful notation, withmany potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order tem-poral logics has identified important enumerable and even decidable fragments including the guarded fragment with equality. In this paper, we specialise themonodic resolution method to the guarded monodic fragment with equality and first-order temporal logic over expanding domains. We introduce novel resolu-tion calculi that can be applied to formulae in the normal form associated with the clausal resolution method, and state correctness and completeness results. 1 Introduction First-order temporal logic (FOTL) is a powerful notation with many applications informal methods. Unfortunately, this power leads to high complexity, most notably the lack of recursive axiomatisations for general FOTL. Recently, significant work has beencarried out in defining monodic