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17
Logic and the Challenge of Computer Science
, 1988
"... Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objec ..."
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Cited by 155 (16 self)
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Nowadays computer science is surpassing mathematics as the primary field of logic applications, but logic is not tuned properly to the new role. In particular, classical logic is preoccupied mostly with infinite static structures whereas many objects of interest in computer science are dynamic objects with bounded resources. This chapter consists of two independent parts. The first part is devoted to finite model theory; it is mostly a survey of logics tailored for computational complexity. The second part is devoted to dynamic structures with bounded resources. In particular, we use dynamic structures with bounded resources to model Pascal.
Toward Logic Tailored for Computational Complexity
 COMPUTATION AND PROOF THEORY
, 1984
"... Whereas firstorder logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the firstorder theory of finite structures and alternatives to firstorder logic, especially polynomial time logic. ..."
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Cited by 76 (6 self)
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Whereas firstorder logic was developed to confront the infinite it is often used in computer science in such a way that infinite models are meaningless. We discuss the firstorder theory of finite structures and alternatives to firstorder logic, especially polynomial time logic.
Definability by constantdepth polynomialsize circuits
 Information and Control
, 1986
"... A function of boolean arguments is symmetric if its value depends solely on the number of l's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constantdepth polynomialsize sequences of boolean circuits, and discus ..."
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Cited by 15 (0 self)
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A function of boolean arguments is symmetric if its value depends solely on the number of l's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constantdepth polynomialsize sequences of boolean circuits, and discuss the complete characterization. (We treat both uniform and nonuniform sequences of circuits.) Our results imply that these circuits can compute functions that are not definable in firstorder logic. In the second part of the paper we generalize from circuits computing symmetric functions to circuits recognizing firstorder structures. By imposing fairly natural restrictions we develop a circuit model with precisely the power of firstorder logic: a class of structures is firstorder definable if and only if it can be recognized by a constantdepth polynomialtime sequence of such circuits. © 1986 Academic Press, Inc.
Tailoring Recursion for Complexity
 J. SYMBOLIC LOGIC
, 1995
"... We design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analogue of firstorder logic and describe algebras of the function ..."
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Cited by 11 (1 self)
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We design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analogue of firstorder logic and describe algebras of the functions computable in nondeterministic logarithmic space, deterministic and nondeterministic polynomial time, and for the functions computable by AC¹circuits.
Logic meets algebra: the case of regular languages
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2007
"... The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view ..."
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Cited by 8 (1 self)
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The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and blockproducts of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.
Averagecase complexity of detecting cliques
, 2010
"... The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain mod ..."
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Cited by 4 (0 self)
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The computational problem of testing whether a graph contains a complete subgraph of size k is among the most fundamental problems studied in theoretical computer science. This thesis is concerned with proving lower bounds for kClique, as this problem is known. Our results show that, in certain models of computation, solving kClique in the average case requires Ω(n k/4) resources (moreover, k/4 is tight). Here the models of computation are boundeddepth Boolean circuits and unboundeddepth monotone circuits, the complexity measure is the number of gates, and the input distributions are random graphs with an appropriate density of edges. Such random graphs (the wellstudied ErdősRényi graphs) are widely believed to be a source of computationally hard instances for clique problems, a hypothesis first articulated by Karp in 1976. This thesis gives the first unconditional lower bounds supporting this hypothesis. Significantly, our result for boundeddepth Boolean circuits breaks out of the traditional
Y = 2x Vs. Y = 3x
 In Proc. 8th IEEE Symp. on Logic in Computer Science
, 1994
"... We show that no formula of first order logic using linear ordering and the logical relation y = 2x can define the property that the size of a finite model is divisible by 3. This answers a longstanding question which may be of relevance to certain open problems in circuit complexity. Introduction ..."
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Cited by 3 (0 self)
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We show that no formula of first order logic using linear ordering and the logical relation y = 2x can define the property that the size of a finite model is divisible by 3. This answers a longstanding question which may be of relevance to certain open problems in circuit complexity. Introduction Descriptive complexity theory originated with a fundamental result of Fagin [8] which characterized queries computable in nondeterministic polynomial time as classes of models of existential sentences of second order logic. Subsequently, the basic complexity classes L, NL, P, and PSPACE, were also tied to logical languages, particularly with certain extensions of first order logic by various kinds of inductive definitions (see Immerman [13] for a survey). Pure first order logic per se appeared more recently in the context of low level parallel complexity classes in the paper by Gurevich and Lewis [11]. Immerman [12] characterized the nonuniform complexity class AC 0 by the first order logi...
Invariant Definability and P/poly
, 1999
"... . We look at various uniform and nonuniform complexity classes within P=poly and its variations L=poly, NL=poly, NP=poly and PSpace=poly, and look for analogues of the AjtaiImmerman theorem which characterizes AC0 as the nonuniformly First Order Definable classes of finite structures. We have pr ..."
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. We look at various uniform and nonuniform complexity classes within P=poly and its variations L=poly, NL=poly, NP=poly and PSpace=poly, and look for analogues of the AjtaiImmerman theorem which characterizes AC0 as the nonuniformly First Order Definable classes of finite structures. We have previously observed that the AjtaiImmerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC0 . Invariant definability is a notion closely related to but different from implicit definability and \Deltadefinability. Its exact relationship to these other notions of definability has been determined in [Mak97]. Our first results are a slight generalization of similar results due to Molzan and can be stated as follows: let C be one of L; NL;P, NP, PSpace and L be a logic which captures C on ordered structures. Then the nonuniform Linvariantly definable classes of (not necessarily ordered) finite structures are...
Extensions of an Idea of McNaughton
 Math. Systems Theory
, 1993
"... Two important measures of the computational complexity of a regular language are the type of finite automaton needed to recognize it and the type of logical expression needed to describe it. Important connections between these measures were studied by Buchi and McNaughton as early as 1960. In this s ..."
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Two important measures of the computational complexity of a regular language are the type of finite automaton needed to recognize it and the type of logical expression needed to describe it. Important connections between these measures were studied by Buchi and McNaughton as early as 1960. In this survey we describe the logical formalism used, outline these early results, and describe modern extensions of this work. In particular, we show how the formalism is extended by the use of new quantifiers and atomic predicates to express many of the fundamental classes of boolean circuit complexity. 2. Introduction A formal language may be thought of as either a subset of the set of all strings or as a property which some strings have and some do not. A natural question about the latter notion is how easy or hard it is to express a given property in the language of logic. By placing a metric of some sort on this expressibility, we create a complexity theory, which we can then compare to the ...
Languages Defined With Modular Counting Quantifiers
 Information and Computation
, 2001
"... . We prove that a regular language defined by a boolean combination of generalized \Sigma 1sentences built using modular counting quantifiers can be defined by a boolean combination of \Sigma 1sentences in which only regular numerical predicates appear. The same statement, with "\Sigma 1 &quo ..."
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. We prove that a regular language defined by a boolean combination of generalized \Sigma 1sentences built using modular counting quantifiers can be defined by a boolean combination of \Sigma 1sentences in which only regular numerical predicates appear. The same statement, with "\Sigma 1 " replaced by "firstorder" is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 : The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case. A preliminary version of this paper appeared in the Proceedings of the 1998 STACS conference. 1 Background 1.1 Lower bounds questions for smalldepth circuit families This paper was motivated by some open problems about the computational power of families of boolean circuits. As it turns out, we will not mention circuits at all after this introductory section. Nonetheless, our main result represents a pos...