Until recently, First-Order Temporal Logic (FOTL) has been little 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 the 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 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 are 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 monodic logics with expanding domains, a case with much significance in both theory and practice. ⋆ On leave from Steklov Institute of Mathematics at St.Petersburg A (variable) assignment a over D is a function from the set of individual variables to D. For every moment of time, n, there is a corresponding first-order structure M n = D, I n , where I n = I(n). Intuitively, FOTL formulae are interpreted in sequences of worlds, M 0 , M 1 , . . . with truth values in different worlds being connected by means of temporal operators.The truth relation M n |= a φ in a structure M, for an assignment a, is defined inductively in the usual way under the following understanding of temporal operators:M is a model for a formula φ (or φ is true in M) if there exists an assignment a such that M 0 |= a φ. A formula is satisfiable if it has a model. A formula is valid if it is true in any temporal structure under any assignment. This logic is complex. It is known that even "small" fragments of FOTL, such as the two-variable monadic fragment (all predicates are unary), are not recursively enumerable [25,18]. However, the set of valid monodic formulae is known to be finitely axiomatisable [32].Definition 1 (Monodic Formula). An FOTL-formula φ is called monodic if any subformulae of the form T ψ, where T is one of ❣ , , ♦, contains at most one free variable.The addition of either equality or function symbols to the monodic fragment leads to the loss of recursive enumerability [32]. Moreover, it was proved in [7] that the two variable monadic monodic fragment with equality is not recursively enumerable. However, in [17] it was shown that the guarded monodic fragment with...

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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 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. By careful encoding, parts of the calculus can be implemented using 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.