Verifying that a temporal logic specification satisfies a temporal property requires some form of theorem proving. However, although proof procedures exist for such logics, many are either unsuitable for automatic implementation or only deal with small fragments of the logic. In this thesis the algorithms for, and strategies to guide, a fully automated temporal resolution theorem prover are given, proved correct and evaluated. An approach to applying resolution, a proof method for classical logics suited to mechanisation, to temporal logics has been developed by Fisher. The method involves translation to a normal form, classical style resolution within states and temporal resolution over states. It has only one temporal resolution rule and is therefore particularly suitable as the basis of an automated temporal resolution theorem prover. As the application of the temporal resolution rule is the most costly part of the method, involving search amongst graphs, different algorithms on w...

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...

. In this paper we give and evaluate the algorithms for a fully automated temporal resolution theorem prover. An approach to applying resolution, a proof method for classical logics suited to mechanisation, to temporal logics has been developed by Fisher. As the application of the temporal resolution rule is the most costly part of the method, involving search amongst graphs, we propose different algorithms on which to base an implementation. The paper concludes with a comparison of their performance. 1 Introduction Temporal logics have been used extensively for the specification and verification of properties of concurrent systems, see for example [Hai82, Lam83, MP92]. However proof procedures for such logics have tended to be tableau [Wol85, Gou84] or automata [VW86] based rather than based on resolution [Rob65]. In many cases the resolution based decision procedures that have been developed for temporal logics [CdC84, Ven86, AM90] are unsuitable for implementation because they only...

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]

The generation of too much information prohibits efficient resolution proof search in classical logics. Hence subsumption is used to discard redundant information and strategies have been developed to guide the proof search avoiding irrelevant information. The extension of the resolution method to temporal logics, for example that by Fisher for propositional linear-time temporal logics, further magnifies this problem. Here we provide an algorithm to efficiently remove irrelevant information prior to the application of Fisher's temporal resolution rule, show it retains the completeness of the temporal resolution system and demonstrate its efficiency on a set of examples. 1 Introduction Resolution [20] is a decision procedure for classical propositional and first-order logic claimed to be machine oriented. However, the proof of theorems may be slowed or even not finish due to the generation of too much information [28]. This information could either be irrelevant, i.e. when a clause i...

. In this paper, we present algorithms developed in order to implement a clausal resolution method for discrete, linear temporal logics, presented in [Fis91]. As part of this method, temporal formulae are rewritten into a normal form and both `non-temporal' and `temporal' inference rules are applied. Through the use of a graph-based representation for the normal form, "efficient" search algorithms can be applied to detect sets of formulae for which temporal resolution is applicable. Further, rather than constructing the full graph structure, our algorithms only explore and construct as little of the graph as possible. These algorithms have been implemented and have been combined with sub-programs performing translation to normal form and non-temporal resolution to produce an integrated resolution based temporal theorem-prover. 1 Introduction Although resolution has been widely used as a decision procedure in classical logics, decision procedures in temporal logic have usually been tab...

A variety of proof methods have been developed to support the effective mechanisation of temporal logic. While clausal temporal resolution has been successfully employed for a range of problems, a number of improvements are still required. In particular, as there is no consistent control strategy underlying the method, a large amount of irrelevant information may sometimes be generated. Following on from classical resolution, where the Set of Support strategy has been used very successfully, we here introduce, justify and apply a temporal version of this strategy, thus allowing the supporting set to be carried over between the different phases of the resolution method. This not only restricts the production of irrelevant information but, under certain conditions, retains the completeness of the refutation process. 1

An approach to applying clausal resolution, a proof method for classical logics suited to mechanisation, to temporal logics has been developed by Fisher. The method involves translation to a normal form, classical style resolution within states and temporal resolution between states. The method consists of only one temporal resolution rule and is therefore particularly suitable as the basis of an automated temporal resolution theorem prover. As the application of this temporal resolution rule is the most costly part of the method, involving search amongst graphs, it is on this area we focus. A breadth-first search approach to the application of this rule is presented and shown to be correct. Analysis of its operation is carried out and test results for its comparison to a previously developed depth-first style algorithm given. 1 Introduction Temporal logics have been used extensively for the specification and verification of properties of concurrent systems, see for example [18, 19, ...

This chapter considers theorem proving for discrete temporal logics. We are interested in deciding or at least enumerating the formulas of the logic which are valid, that is, are true in all circumstances. Most of the techniques for temporal theorem-proving have been extensions for methods developed for classical logics but completely novel techniques have also been developed. Initially we concentrate on discrete linear-time temporal logics, describing axiomatic, tableau, automata and resolution based approaches. The application of these approaches to other temporal logics is discussed. 1 Introduction Readers of this handbook will be aware of the wide variety of useful tasks which require reasoning about time. There are many applications of temporal reasoning tasks to problems of knowledge changing, to planning, to processing natural language, to managing the interchange of information, and to developing complex systems. There are a wide variety of temporal logics available in which s...