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...

Abstract. In this paper we overview one specific approach to the formal development of multi-agent systems. This approach is based on the use of temporal logics to represent both the behaviour of individual agents, and the macro-level behaviour of multi-agent systems. We describe how formal specification, verification and refinement can all be developed using this temporal basis, and how implementation can be achieved by directly executing these formal representations. We also show how the basic framework can be extended in various ways to handle the representation and implementation of agents capable of more complex deliberation and reasoning.

Temporal logics of knowledge are useful for reasoning about situations where the knowledge of an agent or component is important, and where change may occur in this knowledge over time. Here we use temporal logics of knowledge to reason about security protocols. We show how to specify the Needham-Schroeder protocol using temporal logics of knowledge and prove various properties using a resolution calculus for this logic. 1

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.

Temporal logics of knowledge are useful in order to specify complex systems in which agents are both dynamic and have information about their surroundings. We present a resolution method for propositional temporal logic combined with multi-modal S5 and illustrate its use on examples. This paper corrects a previous proposal for resolution in multi-modal temporal logics of knowledge. Keywords: temporal and modal logics, non-classical resolution, theorem-proving 1 Introduction Combinations of logics have been useful for specifying and reasoning about complex situations, for example multi-agent systems [21, 24], accident analysis [15], and security protocols [18]. For example, logics to formalise multi-agent systems often incorporate a dynamic component representing change of over time; an informational component to capture the agent's knowledge or beliefs; and a motivational component for notions such as goals, wishes, desires or intentions. Often temporal or dynamic logic is used for...

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

There is increasing use of combinations of modal logics in both foundational and applied research areas. This article provides an introduction to both the principles of such combinations and to the variety of techniques that have been developed for them. In addition, the article outlines many key research problems yet to be tackled within this callenging area of work.

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. Although a complete and correct resolution-style calculus has already been suggested for this specific fragment, this calculus involves constructions too complex to be of a practical value. In this paper, we develop a machineoriented clausal resolution method which features radically simplified proof search. We first define a normal form for monodic formulae and then introduce a novel resolution calculus that can be applied to formulae in this normal form. The calculus is based on classical first-order resolution and can, thus, be efficiently implemented. We prove correctness and completeness results for the calculus and illustrate it on a comprehensive example. An implementation of the method is briefly discussed.

In this paper we propose algorithms to implement a branching time temporal resolution theorem prover. The branching time temporal logic considered is Computation Tree Logic (CTL), often regarded as the simplest useful logic of this class. Unlike the majority of the research into temporal logic, we adopt a resolution-based approach. The method applies step and temporal resolution rules to the set of formulae in a normal form. Whilst step resolution is similar to the classical resolution rule, the temporal resolution rule resolves a formula, ', that must eventually occur with a set of formulae that together imply that ' can never occur. Thus the method is dependent on the efficient detection of such sets of formulae. We present algorithms to search for these sets of formulae, give a correctness argument, and examples of their operation.

Temporal logics of knowledge are useful for reasoning about situations where the knowledge of an agent or component is important, and where change in this knowledge may occur over time. Here we use temporal logics of knowledge to reason about the game Cluedo. We show how to specify Cluedo using temporal logics of knowledge and prove statements about the knowledge of the players using a clausal resolution calculus for this logic.