Practical Proof Methods for Combined Modal and Temporal Logics

For a number of years, temporal and modal logics have been applied outside pure logic in areas such as formal methods, theoretical computer science and artiicial intelligence. In our research we are particularly interested in the use of modal logics in the characterisation of complex components within software systems as intelligent or rational agents. This approach allows the system designer to analyse applications at a much higher level of abstraction. In order to reason about such agents, a number of theories of rational agency have been developed , for example the BDI (Rao and Georgee 1991) and KARO (van Linder, van der Hoek, and Meyer 1996) frameworks. The leading agent theories and formal methods in this area all share similar logical properties, more precisely, they all exhibit (i) an informational component, being able to represent an agent's beliefs (by the modal logic KD45) or knowledge (by the modal logic S5), (ii) a dynamic component, allowing the representation of dynamic activity (by temporal or dynamic logic), and, (iii) a motivational component, often representing the agents desires, intentions or goals (by the modal logic KD). While many of the basic properties of such combinations of modal and temporal or dynamic logics are well understood very little work has been carried out on practical proof methods for such logics. Our aim in recent work has been to develop proof methods that are general enough to capture a wide range of combinations of temporal and modal logics, but still provide viable means for eeective theorem proving. Currently, we are investigating an approach with the following properties: { The approach covers the combination of discrete, linear, temporal logic with extensions of multi-modal K m by any combination of the axiom schemata 4, 5, B, D, and T. { Instead of combining two calculi operating according to the same underlying principles, like for example two tableaux-based calculi, we combine two different approaches to theorem-proving in modal and temporal logics, namely the translation approach for modal logics (using rst-order resolution) and the SNF approach for temporal logics (using modal resolution). { The particular translation we use has only recently been proposed by de Niv-elle (1999) and can be seen as a special case of the T-encoding introduced by Ohlbach (1998). It allows for conceptually simple decision procedures for