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On the Relative Expressiveness of Description Logics and Predicate Logics
 ARTIFICIAL INTELLIGENCE JOURNAL
, 1996
"... It is natural to view concept and role definitions in Description Logics as expressing monadic and dyadic predicates in Predicate Calculus. We show that the descriptions built using the constructors usually considered in the DL literature are characterized exactly as the predicates definable by form ..."
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Cited by 145 (3 self)
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It is natural to view concept and role definitions in Description Logics as expressing monadic and dyadic predicates in Predicate Calculus. We show that the descriptions built using the constructors usually considered in the DL literature are characterized exactly as the predicates definable by formulas in ¨L³, the subset of First Order Predicate Calculus with monadic and dyadic predicates which allows only three variable symbols. In order to handle “number bounds”, we allow numeric quantifiers, and for transitive closure of roles we use infinitary disjunction. Using previous results in the literature concerning languages with limited numbers of variables, we get as corollaries the existence of formulae of FOPC which cannot be expressed as descriptions. We also show that by omitting role composition, descriptions express exactly the formulae in ¨L², which is known to be decidable.
A Descriptive Approach to LanguageTheoretic Complexity
, 1996
"... Contents 1 Language Complexity in Generative Grammar 3 Part I The Descriptive Complexity of Strongly ContextFree Languages 11 2 Introduction to Part I 13 3 Trees as Elementary Structures 15 4 L 2 K;P and SnS 25 5 Definability and NonDefinability in L 2 K;P 35 6 Conclusion of Part I 57 DRAFT ..."
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Cited by 52 (3 self)
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Contents 1 Language Complexity in Generative Grammar 3 Part I The Descriptive Complexity of Strongly ContextFree Languages 11 2 Introduction to Part I 13 3 Trees as Elementary Structures 15 4 L 2 K;P and SnS 25 5 Definability and NonDefinability in L 2 K;P 35 6 Conclusion of Part I 57 DRAFT 2 / Contents Part II The Generative Capacity of GB Theories 59 7 Introduction to Part II 61 8 The Fundamental Structures of GB Theories 69 9 GB and Nondefinability in L 2 K;P 79 10 Formalizing XBar Theory 93 11 The Lexicon, Subcategorization, Thetatheory, and Case Theory 111 12 Binding and Control 119 13 Chains 131 14 Reconstruction 157 15 Limitations of the Interpretation 173 16 Conclusion of Part II 179 A Index of Definitions 183 Bibliography DRAFT 1<
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.
Decidability and Expressiveness for FirstOrder Logics of Probability
 Information and Computation
, 1989
"... We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show t ..."
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Cited by 40 (6 self)
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We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is \Pi 2 1 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, \Pi 1 1 hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is \Pi 2 1 complete with as little as one unary predicate in the language, even without equality. With equalit...
On the Classical Decision Problem
 Perspectives in Mathematical Logic
, 1993
"... this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References ..."
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Cited by 36 (0 self)
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this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References
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 29 (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...
FiniteModel Theory  A Personal Perspective
 Theoretical Computer Science
, 1993
"... Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph ..."
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Cited by 20 (0 self)
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Finitemodel theory is a study of the logical properties of finite mathematical structures. This paper is a very personalized view of finitemodel theory, where the author focuses on his own personal history, and results and problems of interest to him, especially those springing from work in his Ph.D. thesis. Among the topics discussed are:
Common Knowledge and Update in Finite Environments
 Information and Computation
, 1997
"... Logics of knowledge have been shown to provide a useful approach to the high level specification and analysis of distributed systems. It has been proposed that such systems can be developed using knowledgebased protocols, in which agents' actions have preconditions that test their state of knowledg ..."
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Cited by 20 (5 self)
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Logics of knowledge have been shown to provide a useful approach to the high level specification and analysis of distributed systems. It has been proposed that such systems can be developed using knowledgebased protocols, in which agents' actions have preconditions that test their state of knowledge. Both computerassisted analysis of the knowledge properties of systems and automated compilation of knowledgebased protocols require the development of algorithms for the computation of states of knowledge. This paper studies one of the computational problems of interest, the model checking problem for knowledge formulae in the S5 n Kripke structures generated by finite state environments in which states determine an observation for each agent. Agents are assumed to have perfect recall, and may operate synchronously or asynchronously. It is shown that, in this setting, model checking of common knowledge formulae is intractable, but efficient incremental algorithms are developed for formu...
Decision Procedures using Model Building techniques
 In Computer Science Logic (9th Int. Workshop CSL'95
, 1996
"... . Few year ago we have developed an Automated Deduction approach to model building. The method, called RAMC 1 looks simultaneously for inconsistencies and models for a given formula. The capabilities of RAMC have been extended both for model building and for unsatisfiability detection by including ..."
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Cited by 18 (4 self)
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. Few year ago we have developed an Automated Deduction approach to model building. The method, called RAMC 1 looks simultaneously for inconsistencies and models for a given formula. The capabilities of RAMC have been extended both for model building and for unsatisfiability detection by including in it the use of semantic strategies. In the present work we go further in this direction and define more general and powerful semantic rules. These rules are an extension of Slagle 's semantic resolution. The robustness of our approach is evidenced by proving that the method is also a decision procedure for a wide range of classes decidable by semantic resolution and in particular by hyperresolution. Moreover, the method builds models for satisfiable formulae in these classes, in particular, for satisfiable formulae that do not have any finite model. 1 Introduction Model building and model checking are extremely important topics in Logic and Computer Science. Few years ago we have develop...
Symbolic Model Checking for EventDriven RealTime Systems
 In IEEE RealTime Systems Symposium
, 1993
"... In this article, we consider symbolic model checking for eventdriven realtime systems. We first propose a Synchronous RealTime Event Logic (SREL) for capturing the formal semantics of synchronous, eventdriven realtime systems. The concrete syntax of these systems is given in terms of a graphica ..."
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Cited by 15 (0 self)
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In this article, we consider symbolic model checking for eventdriven realtime systems. We first propose a Synchronous RealTime Event Logic (SREL) for capturing the formal semantics of synchronous, eventdriven realtime systems. The concrete syntax of these systems is given in terms of a graphical programming language called Modechart, by Jahanian and Mok, which can be translated into SREL structures. We then present a symbolic modelchecking algorithm for SREL. In particular, we give an efficient algorithm for constructing OBDDs (Ordered Binary Decision Diagrams) for linear constraints among integer variables. This is very important in a BDDbased symbolic model checker for realtime systems, since timing and event occurrence constraints are used very often in the specification of these systems. We have incorporated our construction algorithm into the SMV v2.3 from CarnegieMellon University and have been able to achieve one to two orders of magnitude in speedup and space saving when compared to the implementation of timing and eventcounting functions by integer arithmetics provided by SMV. Categories and Subject Descriptors: F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs—mechanical verification; specification techniques;