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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 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.
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
"... A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We ..."
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Cited by 88 (2 self)
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A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuitsize lower bounds ...
A Descriptive Approach to LanguageTheoretic Complexity
, 1996
"... Contents 1 Language Complexity in Generative Grammar 3 Part I The Descriptive Complexity of Strongly ContextFree Languages 11 2 Introduction to Part I 13 3 Trees as Elementary Structures 15 4 L 2 K;P and SnS 25 5 Definability and NonDefinability in L 2 K;P 35 6 Conclusion of Part I 57 DRAFT ..."
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Cited by 55 (3 self)
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Contents 1 Language Complexity in Generative Grammar 3 Part I The Descriptive Complexity of Strongly ContextFree Languages 11 2 Introduction to Part I 13 3 Trees as Elementary Structures 15 4 L 2 K;P and SnS 25 5 Definability and NonDefinability in L 2 K;P 35 6 Conclusion of Part I 57 DRAFT 2 / Contents Part II The Generative Capacity of GB Theories 59 7 Introduction to Part II 61 8 The Fundamental Structures of GB Theories 69 9 GB and Nondefinability in L 2 K;P 79 10 Formalizing XBar Theory 93 11 The Lexicon, Subcategorization, Thetatheory, and Case Theory 111 12 Binding and Control 119 13 Chains 131 14 Reconstruction 157 15 Limitations of the Interpretation 173 16 Conclusion of Part II 179 A Index of Definitions 183 Bibliography DRAFT 1<
Descriptive and Computational Complexity
 COMPUTATIONAL COMPLEXITY THEORY, PROC. SYMP. APPLIED MATH
, 1989
"... Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computatio ..."
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Cited by 47 (0 self)
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Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions  that of checking and that of expressing  are related. However it is startling how closely tied they are when the second question refers to expressing the property in firstorder logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in firstorder logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in secondorder existential logic
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 43 (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.
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 38 (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...
Feasible Computation through Model Theory
, 1993
"... The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as ..."
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Cited by 35 (7 self)
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The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of firstorder logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...
Monotone versus Positive
, 1987
"... In connection with the least fixed point operator the following question was raised: Suppose that a firstorder formula ‘P(P) is (semantically) monotone in a predicate symbol P on finite structures. Is (P(P) necessarily equivalent on finite structures to a firstorder formula with only positive occ ..."
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Cited by 35 (3 self)
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In connection with the least fixed point operator the following question was raised: Suppose that a firstorder formula ‘P(P) is (semantically) monotone in a predicate symbol P on finite structures. Is (P(P) necessarily equivalent on finite structures to a firstorder formula with only positive occurrences of P? In this paper, this question is answered negatively. Moreover, the counterexample naturally gives a uniform sequence of constantdepth, polynomialsize, monotone Boolean circuits that is not equivalent to any (however nonuniform) sequence of constantdepth, polynomialsize, positive Boolean circuits.
Linear Time Computable Problems and FirstOrder Descriptions
, 1996
"... this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories o ..."
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Cited by 35 (2 self)
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this article is a proof that each FO problem can be solved in linear time if only relational structures of bounded degree are considered. The basic idea of the proof is a localization technique based on a method that was originally developed by Hanf (Hanf 1965) to show that the elementary theories of two structures are equal under certain conditions, i.e., that two structures agree on all firstorder sentences. Fagin, Stockmeyer and Vardi (Fagin et al. 1993) developed a variant of this technique, which is applicable in descriptive complexity theory to classes of finite relational structures of uniformly bounded degree. Variants of this result can also be found in Gaifman (1982) (see also Thomas (1991)). The essential content of this result, which is also called the HanfSphere Lemma, is that two relational structures of bounded degree satisfy the same firstorder sentences of a certain quantifierrank if both contain, up to a certain number m, the same number of isomorphism types of substructures of a bounded radius r. In addition, a technique of model interpretability from Rabin (1965) (see also Arnborg et al. (1991), Seese (1992), Compton and Henson (1987) and Baudisch et al. (1982)) is adapted to descriptive complexity classes, and proved to be useful for reducing the case of an arbitrary class of relational structures to a class of structures consisting only of the domain and one binary irreflexive and symmetric relation, i.e., the class of simple graphs. It is shown that the class of simple graphs is lintimeuniversal with respect to firstorder logic, which shows that many problems on descriptive complexity classes, described in languages extending firstorder logic for arbitrary structures, can be reduced to problems on simple graphs. This paper is organized as f...