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18
On the Restraining Power of Guards
 Journal of Symbolic Logic
, 1998
"... Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional moda ..."
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Cited by 119 (2 self)
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Guarded fragments of firstorder logic were recently introduced by Andr'eka, van Benthem and N'emeti; they consist of relational firstorder formulae whose quantifiers are appropriately relativized by atoms. These fragments are interesting because they extend in a natural way many propositional modal logics, because they have useful modeltheoretic properties and especially because they are decidable classes that avoid the usual syntactic restrictions (on the arity of relation symbols, the quantifier pattern or the number of variables) of almost all other known decidable fragments of firstorder logic. Here, we investigate the computational complexity of these fragments. We prove that the satisfiability problems for the guarded fragment (GF) and the loosely guarded fragment (LGF) of firstorder logic are complete for deterministic double exponential time. For the subfragments that have only a bounded number of variables or only relation symbols of bounded arity, satisfiability is EXPTI...
Integrity Constraints for XML
, 1999
"... this paper, we extend XML DTDs with several classes of integrity constraints and investigate the complexity of reasoning about these constraints. The constraints range over keys, foreign keys, inverse constraints as well as ID constraints for capturing the semantics of object identities. They imp ..."
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Cited by 83 (12 self)
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this paper, we extend XML DTDs with several classes of integrity constraints and investigate the complexity of reasoning about these constraints. The constraints range over keys, foreign keys, inverse constraints as well as ID constraints for capturing the semantics of object identities. They improve semantic specifications and provide a better reference mechanism for native XML applications. They are also useful in information exchange and data integration for preserving the semantics of data originating in relational and objectoriented databases. We establish complexity and axiomatization results for the (finite) implication problems associated with these constraints. In addition, we study implication of more general constraints, such as functional, inclusion and inverse constraints defined in terms of navigation paths
TwoVariable Logic with Counting is Decidable
, 1996
"... We prove that the satisfiability problem for C² is decidable. C² is firstorder logic with only two variables in the presence of arbitrary counting quantifiers 9 ?m , m ? 1. It considerably extends L², plain firstorder with only two variables, which is known to be decidable by a result of Mort ..."
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Cited by 59 (3 self)
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We prove that the satisfiability problem for C² is decidable. C² is firstorder logic with only two variables in the presence of arbitrary counting quantifiers 9 ?m , m ? 1. It considerably extends L², plain firstorder with only two variables, which is known to be decidable by a result of Mortimer. Unlike L², C² does not have the finite model property. As C² extends L² by expressive means for counting, significant applications arise from the fact that C² embeds corresponding counting extensions of modal logics.
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.
On Logics with Two Variables
 Theoretical Computer Science
, 1999
"... This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable ..."
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Cited by 41 (8 self)
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This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not firstorder...
Undecidability Results on TwoVariable Logics
 IN PROCEEDINGS OF 14TH SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE STACS`97, LECTURE NOTES IN COMPUTER SCIENCE NR. 1200
, 1998
"... It is a classical result of Mortimer that L², firstorder logic with two variables, is decidable for satisfiability. We show that going beyond L² by adding any one of the following leads to an undecidable logic: ffl very weak forms of recursion, viz. (i) transitive closure operations (ii) (restrict ..."
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Cited by 27 (4 self)
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It is a classical result of Mortimer that L², firstorder logic with two variables, is decidable for satisfiability. We show that going beyond L² by adding any one of the following leads to an undecidable logic: ffl very weak forms of recursion, viz. (i) transitive closure operations (ii) (restricted) monadic fixedpoint operations ffl weak access to cardinalities, through the Hartig (or equicardinality) quantifier ffl a choice construct known as Hilbert's "operator. In fact all these extensions of L² prove to be undecidable both for satisfiability, and for satisfiability in finite models. Moreover most of them are hard for \Sigma 1 1 , the first level of the analytical hierachy, and thus have a much higher degree of undecidability than firstorder logic.
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
Why Are Modal Logics So Robustly Decidable?
"... Modal logics are widely used in a number of areas in computer science, in particular ..."
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Cited by 20 (1 self)
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Modal logics are widely used in a number of areas in computer science, in particular
Two Variable FirstOrder Logic over Ordered Domains
 Journal of Symbolic Logic
, 1998
"... The satisfiability problem for the twovariable fragment of firstorder logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wel ..."
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Cited by 14 (0 self)
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The satisfiability problem for the twovariable fragment of firstorder logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations. It is shown that FO 2 over ordered, respectively wellordered, domains or in the presence of one wellfounded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO 2 , namely nondeterministic exponential time. In contrast FO 2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings, wellorderings, or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO 2 an...