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27
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.
A large deviation result on the number of small subgraphs of a random graph
 Combinatorics, Probability and Computing
, 2001
"... �������� � Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions • For what λ does YH have subGaussian behaviour, namely P r(YH − �(YH)  ≥ (λvar(YH)) 1/2) ..."
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Cited by 24 (1 self)
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�������� � Fix a small graph H and let YH denote the number of copies of H in the random graph G(n, p). We investigate the degree of concentration of YH around its mean, motivated by the following questions • For what λ does YH have subGaussian behaviour, namely P r(YH − �(YH)  ≥ (λvar(YH)) 1/2) ≤ e −cλ. where c is a positive constant. • What is P r(YH ≥ (1 + ɛ)�(YH)), for a constant ɛ, i.e, when the upper tail is of order Ω(�(YH))? • Fixing λ = ω(1) in advance, find a reasonably small tail T = T (λ) such that P r(YH − �(YH)  ≥ T) ≤ e −λ. We prove a general concentration result which contains a partial answer to each of these questions. Given a graph H with k vertices {v1,..., vk} and m edges. Consider a bigger graph G and a subgraph H ′ of G. The density of H is the ratio m/k; H is balanced
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:
Almost Everywhere Equivalence Of Logics In Finite Model Theory
 BULLETIN OF SYMBOLIC LOGIC
, 1996
"... We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More pr ..."
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Cited by 16 (1 self)
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We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L 0 are two logics and # is an asymptotic measure on finite structures, then L j a.e. L 0 (#) means that there is a class C of finite structures with #(C ) = 1 and such that L and L 0 define the same queries on C. We carry out a systematic investigation of j a.e. with respect to the uniform measure and analyze the j a.e. equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework.
Randomness and semigenericity
 Transactions of the American Mathematical Society
, 1997
"... Abstract. Let L contain only the equality symbol and let L + be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L + structures with ‘edge probability ’ n−α.ByTα, the almost sure theory of random L +structures we mean the collection of L + ..."
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Cited by 13 (6 self)
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Abstract. Let L contain only the equality symbol and let L + be an arbitrary finite symmetric relational language containing L. Suppose probabilities are defined on finite L + structures with ‘edge probability ’ n−α.ByTα, the almost sure theory of random L +structures we mean the collection of L +sentences which have limit probability 1. Tα denotes the theory of the generic structures for Kα (the collection of finite graphs G with δα(G) =G−α·  edges of G  hereditarily nonnegative). 0.1. Theorem. T α, the almost sure theory of random L +structures, is the same as the theory Tα of the Kαgeneric model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable. This paper unites two apparently disparate lines of research. In [8], Shelah and Spencer proved a 01law for first order sentences about random graphs with edge probability n −α where α is an irrational number between 0 and 1. Answering a question raised by Lynch [5], we extend this result from graphs to hypergraphs
How complex are random graphs in first order logic
"... It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in a such formula can serve as a measure for the “first order complexity ” of G. Here, this parameter is studied for random g ..."
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Cited by 12 (9 self)
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It is not hard to write a first order formula which is true for a given graph G but is false for any graph not isomorphic to G. The smallest number D(G) of nested quantifiers in a such formula can serve as a measure for the “first order complexity ” of G. Here, this parameter is studied for random graphs. We determine it asymptotically when the edge probability p is constant; in fact, D(G) is of order log n then. For very sparse graphs its magnitude is Θ(n). On the other hand, for certain (carefully chosen) values of p the parameter D(G) can drop down to the very slow growing function log ∗ n, the inverse of the TOWERfunction. The general picture, however, is still a mystery. 1
Queries Are Easier Than You Thought (probably)
, 1992
"... The optimization of a large class of queries is explored, using a powerful normal form recently proven. The queries include the fixpoint and while queries, and an extension of while with arithmetic. The optimization method is evaluated using a probabilistic analysis. In particular, the average compl ..."
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Cited by 11 (5 self)
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The optimization of a large class of queries is explored, using a powerful normal form recently proven. The queries include the fixpoint and while queries, and an extension of while with arithmetic. The optimization method is evaluated using a probabilistic analysis. In particular, the average complexity of fixpoint and while is considered and some surprising results are obtained. They suggest that the worstcase complexity is sometimes overly pessimistic for such queries, whose average complexity is often much more reasonable than the provably rare worst case. Some computational properties of queries are also investigated. A probabilistic notion of boundedness is defined, and it is shown that all programs in the class considered are bounded almost everywhere. An effective way of using this fact is provided. 1 Introduction The complexity of query languages has traditionally been investigated using worstcase bounds. We argue that this approach provides an overly pessimistic picture o...
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.