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15
Infinitary Logic and Inductive Definability over Finite Structures
 Information and Computation
, 1995
"... The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abi ..."
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Cited by 56 (6 self)
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The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...
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:
Asymptotic Conditional Probabilities: The Unary Case
, 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 of ..."
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Cited by 11 (3 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[31] and Grove, Halpern, and Koller [22], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that ` can use a nonunary predicate symbol. Liogon'kii [31] shows that if we condition on formulas ` involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be effectively computed. This is the case even if we place no corresponding restrictions on '. We extend this result here to the case where ` involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quantifier nesting, or whether the vocabulary is finite or infinite. We completely characterize the complexity of the problem in the different cases, and show related results for the associated approximation problem.
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...
Finite Variable Logics
, 1993
"... In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this conne ..."
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Cited by 7 (0 self)
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In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for L ! 1! . Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout. 1 Some extensions of firstorder logic Quisani: Hello. Who are you? I am Yuri's imaginary student, and I usually talk to him at this time. Author: I'm afraid he may be a bit late. I am a computer scientist from London, England. I have some imaginary students myself, so maybe I can help. I was reading your earlier conversation on 01 laws [Gu3]. Quisani: I remember it. We examined the...
More about Recursive Structures: Descriptive Complexity and ZeroOne Laws
 in Proc. 11th IEEE Symp. on Logic in Computer Science
, 1996
"... : This paper continues our work on infinite, recursive structures. We investigate the descriptive complexity of several logics over recursive structures, including firstorder, secondorder, and fixpoint logic, exhibiting connections between expressibility of a property and its computational complex ..."
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Cited by 4 (0 self)
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: This paper continues our work on infinite, recursive structures. We investigate the descriptive complexity of several logics over recursive structures, including firstorder, secondorder, and fixpoint logic, exhibiting connections between expressibility of a property and its computational complexity. We then address 01 laws, proposing a version that applies to recursive structures, and using it to prove several nonexpressibility results. 0 Introduction Infinite recursive structures, with recursive graphs as a special case, have been studied quite extensively in the past. Most interesting properties of recursive graphs have been shown to be undecidable, and many are actually outside the arithmetic hierarchy; see, e.g., [AMS, Be1, Be2, BG, H, HH1]. In [HH2] we considered recursive structures to be generalizations of finite relational data bases, and investigated the class of computable queries over them, the motivation being borrowed from [CH1]. A computable query is a (partial) r...
Random Graphs and the Parity Quantifier
, 2009
"... The classical zeroone law for firstorder logic on random graphs says that for every firstorder property ϕ in the theory of graphs and every p ∈ (0, 1), the probability that the random graph G(n, p) satisfies ϕ approaches either 0 or 1 as n approaches infinity. It is well known that this law fails ..."
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Cited by 2 (0 self)
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The classical zeroone law for firstorder logic on random graphs says that for every firstorder property ϕ in the theory of graphs and every p ∈ (0, 1), the probability that the random graph G(n, p) satisfies ϕ approaches either 0 or 1 as n approaches infinity. It is well known that this law fails to hold for any formalism that can express the parity quantifier: for certain properties, the probability that G(n, p) satisfies the property need not converge, and for others the limit may be strictly between 0 and 1. In this work, we capture the limiting behavior of properties definable in first order logic augmented with the parity quantifier, FO[⊕], over G(n, p), thus eluding the above hurdles. Specifically, we establish the following “modular convergence law”: For every FO[⊕] sentence ϕ, there are two explicitly computable rational numbers a0, a1, such that for i ∈ {0, 1}, as n approaches infinity, the probability that the random graph G(2n + i, p) satisfies ϕ approaches ai. Our results also extend appropriately to FO equipped with Modq quantifiers for prime q.
On the Expressive Power of Logics on Finite Models
, 2003
"... Structures" [Mos74], where they are called inductive relations. It should also be pointed out that in Immerman's book on "Descriptive Complexity" LFP is denoted by FO(LFP) (the closure of FO under least fixedpoints) and LFP 1 is denoted by LFP(FO) (least fixedpoints of firstorder formulas). ..."
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Structures" [Mos74], where they are called inductive relations. It should also be pointed out that in Immerman's book on "Descriptive Complexity" LFP is denoted by FO(LFP) (the closure of FO under least fixedpoints) and LFP 1 is denoted by LFP(FO) (least fixedpoints of firstorder formulas).
Computational Model Theory: An Overview
 LOGIC JOURNAL OF THE IGPL
, 1998
"... The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developmen ..."
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The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. It is this connection between complexity theory and finitemodel theory that we term computational model theory. In this overview paper we o#er one perspective on computational model theory. Two important observationsunderly our perspective: (1) while computationaldevices work on encodingsof problems, logic is applied directly to the underlying mathematical structures, and this "mismatch" complicates the relationship between logic and complexity significantly, and (2) firstorder logic has severely limited expressive power on finite structures, and one way to increase the...
Finite Models and Finitely Many Variables
 Banach Center Publications
, 1999
"... We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relation ..."
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Cited by 1 (0 self)
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We consider L  first order logic restricted to k variables, and interpreted in finite structures. The study of classes of finite structures axiomatisable with finitely many variables has assumed importance through connections with computational complexity. In particular, we investigate the relationship between the size of a finite structure and the number of distinct types it realizes, with respect to L . Some open questions, formulated as finitary LöwenheimSkolem properties, are presented regarding this relationship. This is also investigated through finitary versions of an EhrenfeuchtMostowski property.