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On the Decision Problem for TwoVariable FirstOrder Logic
, 1997
"... We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity ..."
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Cited by 48 (1 self)
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We identify the computational complexity of the satisfiability problem for FO², the fragment of firstorder logic consisting of all relational firstorder sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finitemodel property, which means that if an FO²sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIMEcomplete.
BDDBased Decision Procedures for the Modal Logic K
 Journal of Applied Nonclassical Logics
, 2005
"... We describe BDDbased decision procedures for the modal logic K. Our approach is inspired by the automatatheoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of typeswhich can be viewed as an onthefly emptiness of the automaton. We use ..."
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Cited by 17 (1 self)
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We describe BDDbased decision procedures for the modal logic K. Our approach is inspired by the automatatheoretic approach, but we avoid explicit automata construction. Instead, we compute certain fixpoints of a set of typeswhich can be viewed as an onthefly emptiness of the automaton. We use BDDs to represent and manipulate such type sets, and investigate different kinds of representations as well as a "levelbased" representation scheme. The latter turns out to speed up construction and reduce memory consumption considerably. We also study the effect of formula simplification on our decision procedures. To proof the viability of our approach, we compare our approach with a representative selection of other approaches, including a translation of to QBF. Our results indicate that the BDDbased approach dominates for modally heavy formulae, while searchbased approaches dominate for propositionally heavy formulae.
Circumscription and Implicit Definability
 Journal of Automated Reasoning
, 1985
"... : We explore some connections between the technique of circumscription in artificial intelligence and the notion of implicit definition in mathematical logic. Implicit definition can be taken as the informal intent, but not necessarily the formal result, of circumscription. This raises some question ..."
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Cited by 12 (2 self)
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: We explore some connections between the technique of circumscription in artificial intelligence and the notion of implicit definition in mathematical logic. Implicit definition can be taken as the informal intent, but not necessarily the formal result, of circumscription. This raises some questions for logical theory and suggests some implications for artificial intelligence practice. The principal implication is that when circumscription "works" its conclusions can be explicitly described. Keywords: Circumscription, implicit definition, Beth's theorem, assumptions, artificial intelligence. Acknowledgements: This paper would not exist if not for the initial observation, patient explanations, proofs and algorithms of Richard Statman. I am grateful to him for this help, and to Nils Nilsson and Joseph Schatz for comments and suggestions. I, of course, am responsible for any errors in the following. This research was supported by the Defense Advanced Research Projects Agency (DOD), ARPA ...