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32
Monodic temporal resolution
 ACM Transactions on Computational Logic
, 2003
"... Until recently, FirstOrder 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 f ..."
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Cited by 27 (15 self)
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Until recently, FirstOrder 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
TeMP: A Temporal Monodic Prover
 In Proc. IJCAR04, LNAI
, 2004
"... We present TeMPthe first experimental system for testing validity of monodic temporal logic formulae. The prover implements finegrained temporal resolution. The core operations required by the procedure are performed by an efficient resolutionbased prover for classical firstorder logic. ..."
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Cited by 20 (11 self)
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We present TeMPthe first experimental system for testing validity of monodic temporal logic formulae. The prover implements finegrained temporal resolution. The core operations required by the procedure are performed by an efficient resolutionbased prover for classical firstorder logic.
An AutomataTheoretic Approach to Constraint LTL
, 2003
"... We consider an extension of lineartime temporal logic (LTL) with constraints interpreted over a concrete domain. We use a new automatatheoretic technique to show pspace decidability of the logic for the constraint systems (Z, <, =) and (N, <, =). Along the way, we give an automatatheoretic ..."
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Cited by 20 (7 self)
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We consider an extension of lineartime temporal logic (LTL) with constraints interpreted over a concrete domain. We use a new automatatheoretic technique to show pspace decidability of the logic for the constraint systems (Z, <, =) and (N, <, =). Along the way, we give an automatatheoretic proof of a result of [BC02] when the constraint system D satisfies the completion property. Our decision procedures extend easily to handle extensions of the logic with past operators and constants, as well as an extension of the temporal language itself to monadic second order logic. Finally, we show that the logic...
Temporalising Tableaux
 STUDIA LOGICA
, 2004
"... As a remedy for the bad computational behaviour of firstorder temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining socalled monodic fragments of FOTL. In this paper, we are concerned with ..."
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Cited by 17 (5 self)
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As a remedy for the bad computational behaviour of firstorder temporal logic (FOTL), it has recently been proposed to restrict the application of temporal operators to formulas with at most one free variable thereby obtaining socalled monodic fragments of FOTL. In this paper, we are concerned with constructing tableau algorithms for monodic fragments based on decidable fragments of firstorder logic like the twovariable fragment or the guarded fragment. We present a general framework that shows how existing decision procedures for firstorder fragments can be used for constructing a tableau algorithm for the corresponding monodic fragment of FOTL.
Towards the Implementation of FirstOrder Temporal Resolution: the Expanding Domain Case
"... Firstorder 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 firstorder temporal logics has identified important enumerable and even decidable frag ..."
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Cited by 11 (7 self)
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Firstorder 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 firstorder temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of firstorder 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 firstorder resolution and can, thus, be efficiently implemented.
A Note on Relativised Products of Modal Logics
 Advances in Modal Logic
, 2003
"... this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cu ..."
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Cited by 10 (6 self)
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this paper. each frame of the class.) For example, K is the logic of all nary product frames. It is not hard to see that S5 is the logic of all nary products of universal frames having the same worlds, that is, frames hU; R i i with R i = U U . We refer to product frames of this kind as cubic universal product S5 frames. Note that the `ireduct' F U 1 U n ; R i of F 1 F n is a union of n disjoint copies of F i . Thus, F and F i validate the same formulas, and so L n L 1 L n : There is a strong interaction between the modal operators of product logics. Every nary product frame satis es the following two properties, for each pair i 6= j, i; j = 1; : : : ; n: Commutativity : 8x8y8z xR i y ^ yR j z ! 9u (xR j u ^ uR i z) ^ xR j y ^ yR i z ! 9u (xR i u ^ uR j z) Church{Rosser property : 8x8y8z xR i y ^ xR j z ! 9u (yR j u ^ zR i u) This means that the corresponding modal interaction formulas 2 i 2 j p $ 2 j 2 i p and 3 i 2 j p ! 2 j 3 i p belong to every ndimensional product logic. The geometrically intuitive manydimensional structure of product frames makes them a perfect tool for constructing formalisms suitable for, say, spatiotemporal representation and reasoning (see e.g. [33, 34]) or reasoning about the behaviour of multiagent systems (see e.g. [4]). However, the price we have to pay for the use of products is an extremely high computational complexityeven the product of two NPcomplete logics can be nonrecursively enumerable (see e.g. [29, 27]). In higher dimensions practically all products of `standard' modal logics are undecidable and non nitely axiomatisable [16]
Spatial Logics with Connectedness Predicates
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2010
"... We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of thes ..."
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Cited by 7 (2 self)
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We consider quantifierfree spatial logics, designed for qualitative spatial representation and reasoning in AI, and extend them with the means to represent topological connectedness of regions and restrict the number of their connected components. We investigate the computational complexity of these logics and show that the connectedness constraints can increase complexity from NP to PSpace, ExpTime and, if component counting is allowed, to NExpTime.
Mechanising FirstOrder Temporal Resolution
, 2003
"... Firstorder 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 firstorder temporal logics has identified important enumerable and even decidable frag ..."
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Cited by 7 (5 self)
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Firstorder 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 firstorder temporal logics has identified important enumerable and even decidable fragments. Although a complete and correct resolutionstyle 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 firstorder 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.