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15
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 78 (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 39 (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 NEXPTIM ..."
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Cited by 32 (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 solv ..."
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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 8 (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.
All proper normal extensions of S5square have the polynomial size model property
 Studia Logica
, 2000
"... It is shown that all proper normal extensions of the bimodal system S5 have the polysize model property. In fact, every normal proper extension L of S5 complete with respect to a class of finite frames FL . To each such class corresponds a natural number b(L) { the bound of L. For every L, there e ..."
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Cited by 4 (1 self)
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It is shown that all proper normal extensions of the bimodal system S5 have the polysize model property. In fact, every normal proper extension L of S5 complete with respect to a class of finite frames FL . To each such class corresponds a natural number b(L) { the bound of L. For every L, there exists a polynomial P ( ) of degree b(L)+1 such that every Lsatisfiable formula ' is satisfiable on an Lframe whose universe is bounded by P (j'j), for j'j the number of subformulas of '. It is shown that this bound is optimal.
Random models and the godel case of the decision problem
 Journal of Symbolic Logic
, 1983
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at. ..."
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at.
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 ..."
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Cited by 2 (0 self)
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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.
Succinct definitions in the first order theory of graphs. Accepted for publication
 in Annals of Pure and Applied Logic. (arxiv.org/abs/math.LO/0401307) Oleg Pikhurko, Joel Spencer and Oleg Verbitsky
"... We say that a first order sentence A defines a graph G if A is true on G but false on any graph nonisomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (r ..."
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We say that a first order sentence A defines a graph G if A is true on G but false on any graph nonisomorphic to G. Let L(G) (resp. D(G)) denote the minimum length (resp. quantifier rank) of a such sentence. We define the succinctness function s(n) (resp. its variant q(n)) to be the minimum L(G) (resp. D(G)) over all graphs on n vertices. We prove that s(n) and q(n) may be so small that for no general recursive function f we can have f(s(n)) ≥ n for all n. However, for the function q ∗ (n) = maxi≤n q(i), which is the least monotone nondecreasing function bounding q(n) from above, we have q ∗ (n) = (1 + o(1))log ∗ n, where log ∗ n equals the minimum number of iterations of the binary logarithm sufficient to lower n below 1. We show an upper bound q(n) < log ∗ n + 5 even under the restriction of the class of graphs to trees. Under this restriction, for q(n) we also have a matching lower bound.