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32
Computing With FirstOrder Logic
, 1995
"... We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtaine ..."
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Cited by 52 (13 self)
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We study two important extensions of firstorder logic (FO) with iteration, the fixpoint and while queries. The main result of the paper concerns the open problem of the relationship between fixpoint and while: they are the same iff ptime = pspace. These and other expressibility results are obtained using a powerful normal form for while which shows that each while computation over an unordered domain can be reduced to a while computation over an ordered domain via a fixpoint query. The fixpoint query computes an equivalence relation on tuples which is a congruence with respect to the rest of the computation. The same technique is used to show that equivalence of tuples and structures with respect to FO formulas with bounded number of variables is definable in fixpoint. Generalizing fixpoint and while, we consider more powerful languages which model arbitrary computation interacting with a database using a finite set of FO queries. Such computation is modeled by a relational machine...
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.
Infinitary Logics and 01 Laws
 Information and Computation
, 1992
"... We investigate the in nitary logic L 1! , in which sentences may have arbitrary disjunctions and conjunctions, but they involve only a nite number of distinct variables. We show that various xpoint logics can be viewed as fragments of L 1! , and we describe a gametheoretic characterizat ..."
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Cited by 42 (4 self)
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We investigate the in nitary logic L 1! , in which sentences may have arbitrary disjunctions and conjunctions, but they involve only a nite number of distinct variables. We show that various xpoint logics can be viewed as fragments of L 1! , and we describe a gametheoretic characterization of the expressive power of the logic. Finally, we study asymptotic probabilities of properties 1! on nite structures. We show that the 01 law holds for L 1! , i.e., the asymptotic probability of every sentence in this logic exists and is equal to either 0 or 1. This result subsumes earlier work on asymptotic probabilities for various xpoint logics and reveals the boundary of 01 laws for in nitary logics.
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
Relational Queries over Interpreted Structures
 Journal of the ACM
"... We rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries. We identify parts of the classical theory that go through `as before' when interpreted structure is present, parts that go through only for cl ..."
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Cited by 22 (12 self)
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We rework parts of the classical relational theory when the underlying domain is a structure with some interpreted operations that can be used in queries. We identify parts of the classical theory that go through `as before' when interpreted structure is present, parts that go through only for classes of nicelybehaved structures, and parts that only arise in the interpreted case. The first category includes a number of results on language equivalence and expressive power characterizations for the activedomain semantics for a variety of logics. Under this semantics, quantifiers range over elements of a relational database. The main kind of results we prove here are generic collapse results: for generic queries, adding operations beyond order, does not give us extra power. The second category includes results on the natural semantics, under which quantifiers range over the entire interpreted structure. We prove, for a variety of structures, naturalactive collapse results, s...
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:
On sufficient conditions for unsatisfiability of random formulas
 JOURNAL OF THE ACM
, 2004
"... A descriptive complexity approach to random 3SAT is initiated. We show that unsatisfiability of any significant fraction of random 3CNF formulas cannot be certified by any property that is expressible in Datalog. Combined with the known relationship between the complexity of constraint satisfactio ..."
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Cited by 10 (2 self)
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A descriptive complexity approach to random 3SAT is initiated. We show that unsatisfiability of any significant fraction of random 3CNF formulas cannot be certified by any property that is expressible in Datalog. Combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in Datalog, our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always. Our result is a consequence of designing a winning strategy for one of the players in the existential pebble game. The winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3CNF formula. The second contribution of our work is the connection between finite model theory and propositional proof complexity. To make this connection explicit, we establish a tight relationship between the number of pebbles needed to win the game and the width of the Resolution refutations. As a consequence to our result and the known sizewidth relationship in Resolution, we obtain new proofs of the exponential lower bounds for Resolution refutations of random 3CNF formulas and the Pigeonhole Principle.
Asymptotic Conditional Probabilities: The Nonunary Case
 J. SYMBOLIC LOGIC
, 1993
"... Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction ..."
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Cited by 9 (2 self)
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Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for firstorder sentences. Given firstorder sentences ' and `, we consider the structures with domain f1; : : : ; Ng that satisfy `, and compute the fraction of them in which ' is true. We then consider what happens to this fraction as N gets large. This extends the work on 01 laws that considers the limiting probability of firstorder sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii [Lio69], if there is a nonunary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kii also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is welldefined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.
Normal Forms for SecondOrder Logic over Finite Structures, and Classification of NP Optimization Problems
 Annals of Pure and Applied Logic
, 1996
"... We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination ..."
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Cited by 9 (5 self)
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We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination of existential formulas, and (ii) over nite successor structures, the KolaitisThakur hierarchy of minimization problems collapses completely and the KolaitisThakur hierarchy of maximization problems collapses partially. The normal form theorem for 1 2 fails if 1 2 is replaced with 1 1 or if innite structures are allowed. 1 Introduction We consider secondorder logic with equality (unless otherwise stated explicitly) and without function symbols of positive arity. Predicates are denoted by capitals and individual variables by lower case letters; a bold face version of a letter denotes a tuple of corresponding symbols. For brevity, we say that a formula reduces t...
Generalized Quantifiers and 01 Laws
 PROC. 10TH IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE
, 1995
"... We study 01 laws for extensions of firstorder logic by Lindstrom quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q]  the extension of firstorder logic by means of the quantifier Q  to have a 01 law. We use these conditions to show, ..."
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Cited by 9 (1 self)
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We study 01 laws for extensions of firstorder logic by Lindstrom quantifiers. We state sufficient conditions on a quantifier Q expressing a graph property, for the logic FO[Q]  the extension of firstorder logic by means of the quantifier Q  to have a 01 law. We use these conditions to show, in particular, that FO[Rig], where Rig is the quantifier expressing rigidity, has a 01 law. We also show that FO[Ham], where Ham is the quantifier expressing Hamiltonicity, does not have a 01 law. Blass and Harary pose the question whether there is a logic which is powerful enough to express Hamiltonicity or rigidity and which has a 01 law. It is a consequence of our results that there is no such regular logic (in the sense of abstract model theory) in the case of Hamiltonicity, but there is one in the case of rigidity. We also consider sequences of vectorized quantifiers, and show that the extensions of firstorder logic obtained by adding such sequences generated by quantifiers that are...