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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.
The TwoVariable Guarded Fragment with Transitive Relations
 In Proc. LICS'99
, 1999
"... We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occu ..."
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Cited by 34 (1 self)
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We consider the restriction of the guarded fragment to the twovariable case where, in addition, binary relations may be specified as transitive. We show that (i) this very restricted form of the guarded fragment without equality is undecidable and that (ii) when allowing nonunary relations to occur only in guards, the logic becomes decidable. The latter subclass of the guarded fragment is the one that occurs naturally when translating multimodal logics of the type K4, S4 or S5 into rstorder logic. We also show that the loosely guarded fragment without equality and with a single transitive relation is undecidable.
Complexity Results for FirstOrder TwoVariable Logic with Counting
, 2000
"... Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete ..."
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Cited by 28 (1 self)
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Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIMEcomplete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order twovariable logic L 2 is NEXPTIMEcomplete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in nondeterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of twovariables logic L 2 by a week access to car...
Maslov's Class K Revisited
 In Proc. CADE16
, 1999
"... . This paper gives a new treatment of Maslov's class K in the framework of resolution. More specifically, we show that K and the class DK consisting of disjunction of formulae in K can be decided by a resolution refinement based on liftable orderings. We also discuss relationships to other solvable ..."
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Cited by 15 (11 self)
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. This paper gives a new treatment of Maslov's class K in the framework of resolution. More specifically, we show that K and the class DK consisting of disjunction of formulae in K can be decided by a resolution refinement based on liftable orderings. We also discuss relationships to other solvable and unsolvable classes. 1 Introduction Maslov's class K [13] is one of the most important solvable fragments of firstorder logic. It contains a variety of classical solvable fragments including the Monadic class, the initially extended Skolem class, the Godel class, and the twovariable fragment of firstorder logic FO 2 [4]. It also encompasses a range of nonclassical logics, like a number of extended modal logics, many description logics used in the field of knowledge representation [11, 4, chap. 7], and some reducts of representable relational algebras. For this reason practical decision procedures for the class K are of general interest. According to Maslov [13] the inverse method pro...
Decidable Fragments of ManySorted Logic
 LPAR 2007
, 2007
"... We investigate the possibility of developing a decidable logic which allows expressing a large variety of real world specifications. The idea is to define a decidable subset of manysorted (typed) first order logic. The motivation is that types simplify the complexity of mixed quantifiers when th ..."
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Cited by 6 (0 self)
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We investigate the possibility of developing a decidable logic which allows expressing a large variety of real world specifications. The idea is to define a decidable subset of manysorted (typed) first order logic. The motivation is that types simplify the complexity of mixed quantifiers when they quantify over different types. We noticed that many real world verification problems can be formalized by quantifying over different types in such a way that the relations between types remain simple. Our main result is a decidable fragment of manysorted firstorder logic that captures many real world specifications.
Decidability of Cylindric Set Algebras of Dimension Two and FirstOrder Logic With Two Variables
, 1997
"... The aim of this paper is to give a new proof for the decidability and finite model property of firstorder logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse ..."
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The aim of this paper is to give a new proof for the decidability and finite model property of firstorder logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse 2 ). The new proof also shows the known results that the universal theory of Pse 2 is decidable and that every finite Pse 2 can be represented on a finite base. Since the class Cs 2 of cylindric set algebras of dimension 2 forms a reduct of Pse 2 , these results extend to Cs 2 as well. We hasten to remark that the results proved here are not new, and indeed there are several rather different proofs available (references below). We felt justified publishing this new proof, since we believe it is simpler than the proofs known, and accessible to both algebraists and logicians. The proof uses only very elementary ideas from universal algebra and model theory and one heavy combinatorial theor...
Model Building in the CrossRoads of Consequence and NonConsequence Relations
"... this paper we overcome this limitation by using Ramc to guide (by generating consequences or nonconsequences, by detecting countermodels, simplifying clauses, . . . ) the enumeration of the set of eqinterpretations. We prove that by combining Ramc with a newrule (mbsplitting) any eqinterpretat ..."
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Cited by 1 (0 self)
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this paper we overcome this limitation by using Ramc to guide (by generating consequences or nonconsequences, by detecting countermodels, simplifying clauses, . . . ) the enumeration of the set of eqinterpretations. We prove that by combining Ramc with a newrule (mbsplitting) any eqinterpretation can eventually be reached.
Using Resolution as Decision Procedure
, 2005
"... Logic made a major step in 1879, when Gottlob Frege published his ’Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens’, see [26]. In this paper, which marks the start of modern logic, Frege introduced a formal language, (which he called Begriffsschrift), in which ..."
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Logic made a major step in 1879, when Gottlob Frege published his ’Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens’, see [26]. In this paper, which marks the start of modern logic, Frege introduced a formal language, (which he called Begriffsschrift), in which mathematical statements
A DECIDABLE SUBCLASS OF THE MINIMAL GODEL CLASS WITH IDENTITY
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