In this paper a normal form, called Separated Normal Form (SNF), for temporal logic formulae is described. A simple propositional temporal logic, based on a discrete linear model structure, is introduced and a procedure for transforming an arbitrary formula of this logic into SNF is described. It is shown that the transformation process preserves satisfiability and ensures that any model of the transformed formula is a model of the original one. This normal form not only provides a simple and concise representation for temporal formulae, but is also used as the basis for both a resolution proof method and an execution mechanism for this type of temporal logic. In addition to outlining these applications, we show how the normal form can be extended to deal with first-order temporal logic. 1

A resolution based proof system for a temporal logic of knowledge is presented and shown to be correct. Such logics are useful for proving properties of distributed and multi-agent systems. Examples are given to illustrate the proof system. An extension of the basic system to the multimodal case is given and illustrated using the `muddy children problem'. 1 Introduction Temporal logics have been shown to have many applications in computer science and artificial intelligence. For example, they are used in the specification and verification of reactive systems [28], in temporal query languages [8], executable logics [18] and for reasoning about action [36]. For some applications, however, logics containing connectives that operate over just the one modal dimension of time do not provide sufficient expressive power. For such applications, it is necessary to provide connectives that allow us to represent the properties of different modal dimensions in the same logic. In this paper, we co...

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

this paper we present a breadth-rst search style algorithm which enables practical implementation of the resolution method for temporal logics developed by Fisher [15]. Fisher's method has been shown correct [36], deals with the full range of past and futuretime temporal operators and has only one temporal resolution rule making it suitable for mechanisation. The resolution procedure is characterised by translation to a normal form, the application of a classical style resolution rule to derive contradictions that occur at the same point in time (termed step resolution), together with a new resolution rule, which derives contradictions over temporal sequences (termed temporal resolution). As it is the latter that is the most expensive part of the algorithm, involving search through graphs, as well as the most novel, it is on the application of the temporal resolution rule that we concentrate. We suggest a breadth-rst search approach to the application of the temporal resolution rule and through analysis of its operation and output, explain why it is an improvement on search mechanisms suggested previously [12]

Temporal logics are extensions of classical logic with operators that deal with time. They have been used in a wide variety of areas within Computer Science and Artificial Intelligence, for example robotics [14], databases [15], hardware verification [8] and agent-based systems [12].

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

In this paper we propose a hypothesis-driven design of the empirical analysis of different decision procedures which we refer to as scientific benchmarking. The approach is to start by choosing the benchmark problems for which, on the basis of analytical considerations, we expect a particular decision procedure to exhibit a behaviour different from another decision procedure. Then empirical tests are performed in order to verify the particular hypothesis concerning the decision procedures under consideration. As a case study, we apply this methodology to compare different decision procedures for propositional temporal logic. We define two classes of randomly generated temporal logic formulae which we use to investigate the behaviour of two tableaux-based temporal logic approaches using the Logics Workbench, a third tableaux-based approach using the STeP system, and temporal resolution using a new prover called TRP. 1

this paper we will consider the combination of modal (including temporal) logics, identifying leading edge research that we, and others, have carried out. Such combined systems have a wide variety of applications that we will describe, but also have significant problems, often concerning interactions that occur between the separate modal dimensions. However, we begin by reviewing why we might want to use modal logics at all.

In this paper we extend our clausal resolution method for linear temporal logics to a branching-time framework. The branching-time temporal logics considered are Computation Tree Logic (CTL), often regarded as the simplest useful logic of this class, and Extended CTL (ECTL), which is CTL extended with fairness operators. The key elements of the resolution method, namely the normal form, the concept of step resolution and a novel temporal resolution rule, are introduced and justified with respect to both these logics. A completeness argument is provided, together with an example of the use of the temporal resolution method. Finally, we consider future work, in particular extension of the method yet further, to CTL , and implementation of the approach by utilising techniques developed for linear-time temporal resolution.

Both proof, via clausal resolution, and execution, via the imperative future approach, depend on the use of a normal form for temporal formulae. While the systems developed have centred around the use of an unrestricted normal form, we here consider a Horn clause-like version of the normal form and its effect on both execution and resolution. This refined normal form is as expressive as the original, and represents a natural way to describe systems, yet allows both execution and resolution to be implemented more efficiently in practice. 1