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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 153 (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.
Fixpoint Logics, Relational Machines, and Computational Complexity
 In Structure and Complexity
, 1993
"... We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have t ..."
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Cited by 37 (5 self)
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We establish a general connection between fixpoint logic and complexity. On one side, we have fixpoint logic, parameterized by the choices of 1storder operators (inflationary or noninflationary) and iteration constructs (deterministic, nondeterministic, or alternating). On the other side, we have the complexity classes between P and EXPTIME. Our parameterized fixpoint logics capture the complexity classes P, NP, PSPACE, and EXPTIME, but equality is achieved only over ordered structures. There is, however, an inherent mismatch between complexity and logic  while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures. To overcome this mismatch, we develop a theory of relational complexity, which bridges tha gap between standard complexity and fixpoint logic. On one hand, we show that questions about containments among standard complexity classes can be translated to questions about containments among relational complex...
Fixedpoint logics on planar graphs
, 1998
"... We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2 ..."
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Cited by 35 (12 self)
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We study the expressive power of inflationary fixedpoint logic IFP and inflationary fixedpoint logic with counting IFP+C on planar graphs. We prove the following results:(1) IFP captures polynomial time on 3connected planar graphs, and IFP+C captures polynomial time on arbitrary planar graphs. (2) Planar graphs can be characterized up to isomorphism in a logic with finitely many variables and counting. This answers a question of Immerman [7].(3) The class of planar graphs is definable in IFP. This answers a question of Dawar and Grädel [16].
A logic of nonmonotone inductive definitions
 ACM transactions on computational logic
, 2007
"... Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated i ..."
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Cited by 29 (16 self)
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Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive Definitions (IDlogic). The semantics of the logic is strongly influenced by the wellfounded semantics of logic programming. This paper discusses the formalisation of different forms of (non)monotone induction by the wellfounded semantics and illustrates the use of the logic for formalizing mathematical and commonsense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the wellfounded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 18 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
A zeroone law for logic with a fixedpoint operator
 Inform. and Control
"... The logic obtained by adding the leastfixedpoint operator to firstorder logic was proposed as a query language by Aho and Ullman (in "Proc. 6th ACM Sympos. on Principles of Programming Languages, " 1979, pp. 110120) and has been studied, particularly in connection with finite models, b ..."
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Cited by 14 (6 self)
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The logic obtained by adding the leastfixedpoint operator to firstorder logic was proposed as a query language by Aho and Ullman (in "Proc. 6th ACM Sympos. on Principles of Programming Languages, " 1979, pp. 110120) and has been studied, particularly in connection with finite models, by numerous authors. We extend to this logic, and to the logic containing the more powerful iterativefixedpoint operator, the zeroone law proved for firstorder logic in (Glebskii, Kogan, Liogonki, and Talanov (1969), Kibernetika 2, 3142; Fagin (1976), J. Symbolic Logic 41, 5058). For any sentence q ~ of the extended logic, the proportion of models of q ~ among all structures with universe {1, 2,..., n} approaches 0 or 1 as n tends to infinity. We also show that the problem of deciding, for any cp, whether this proportion approaches 1 is complete for exponential time, if we consider only q)'s with a fixed finite vocabulary (or vocabularies of bounded arity) and complete for doubleexponential time if ~0 is unrestricted. In addition, we establish some related results. © 1985 Academic Press, Inc.
Model Expansion as a Framework for Modelling and Solving Search Problems
"... We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite mod ..."
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Cited by 3 (3 self)
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We propose a framework for modelling and solving search problems using logic, and describe a project whose goal is to produce practically effective, general purpose tools for representing and solving search problems based on this framework. The mathematical foundation lies in the areas of finite model theory and descriptive complexity, which provide us with many classical results, as well as powerful techniques, not available to many other approaches with similar goals. We describe the mathematical foundations; explain an extension to classical logic with inductive definitions that we consider central; give a summary of complexity and expressiveness properties; describe an approach to implementing solvers based on grounding; present grounding algorithms based on an extension of the relational algebra; describe an implementation of our framework which includes use of inductive definitions, sorts and order; and give experimental results comparing the performance of our implementation with ASP solvers and another solver based on the same framework. 1.
Search Problems as Model Finding
"... ABSTRACT. Arguments for logicbased knowledge representation often emphasize the primacy of entailment in reasoning, and traditional logicbased formulations of AI tasks were frequently in terms of entailment. More recently, practical progress in satisfiabilitybased methods has encouraged formulati ..."
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ABSTRACT. Arguments for logicbased knowledge representation often emphasize the primacy of entailment in reasoning, and traditional logicbased formulations of AI tasks were frequently in terms of entailment. More recently, practical progress in satisfiabilitybased methods has encouraged formulation of problems as model finding. Here, we argue for the formalization of search problems, which abound in AI as well as other areas, as a particular form of model finding called model expansion. An important conceptual part of this proposal is the formalization of the problem instance as a structure, rather than as a formula. Adopting this view leads naturally to taking descriptive complexity theory as the starting point for developing a theory of languages for representing search problems. We explain the formalization of search as model expansion, and the reasons we consider it an appropriate basis for such a theory. We emphasize the role of model expansion for first order logic, with extensions, in specifying NP search problems, and describe the formalization of arithmetic in this context. This paper is dedicated to Hector J. Levesque.