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TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
The complexity of propositional proofs
 Bulletin of Symbolic Logic
"... Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorit ..."
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Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents
First Order Logic, Fixed Point Logic and Linear Order
, 1995
"... The Ordered conjecture of Kolaitis and Vardi asks whether fixedpoint logic differs from firstorder logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures o ..."
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The Ordered conjecture of Kolaitis and Vardi asks whether fixedpoint logic differs from firstorder logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of finite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a modeltheoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexitytheoretic implications of this line of research.
Rudimentary Reductions Revisited
 Information Processing Letters 40
, 1991
"... We show that logbounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtimeuniform AC . ..."
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We show that logbounded rudimentary reductions #de#ned and studied by Jones in 1975# characterize Dlogtimeuniform AC .
Complexity Doctrines
, 1995
"... vii Introduction ix 1 Tensor and Linear Time 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Almost Equational Specification : : : : : : : : : : : : : : : : : 3 1.1.1 Sketches : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.1.2 Orthogonality : : : : : : : : : : ..."
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vii Introduction ix 1 Tensor and Linear Time 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 Almost Equational Specification : : : : : : : : : : : : : : : : : 3 1.1.1 Sketches : : : : : : : : : : : : : : : : : : : : : : : : : : 3 1.1.2 Orthogonality : : : : : : : : : : : : : : : : : : : : : : : 5 1.1.3 Essentially Algebraic Specification : : : : : : : : : : : : 8 1.2 Tensor and System T : : : : : : : : : : : : : : : : : : : : : : : 10 1.2.1 Serial Composition : : : : : : : : : : : : : : : : : : : : 10 1.2.2 Parallel Composition : : : : : : : : : : : : : : : : : : : 11 1.2.3 Unary Numbers : : : : : : : : : : : : : : : : : : : : : : 13 1.2.4 System T : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.3 Comprehensions and Tiers : : : : : : : : : : : : : : : : : : : : 15 1.3.1 Comprehensions : : : : : : : : : : : : : : : : : : : : : : 15 1.3.2 Extents : : : : : : : : : : : : : : : : : : : : : : : : : : 18 1.3.3 Dyadic Numbers : : : : : : : : : : : : : : : ...
On Quasilinear Time Complexity Theory
, 1994
"... This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy. ..."
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This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomialtime hierarchy carry over to the quasilineartime hierarchy.
ON THE CORRESPONDENCE BETWEEN ARITHMETIC THEORIES AND PROPOSITIONAL PROOF SYSTEMS
"... Bounded arithmetic is closely related to propositional proof systems, and this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and Pudlák ..."
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Bounded arithmetic is closely related to propositional proof systems, and this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and Pudlák [42]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties.
An exploration of the partial respects in which an axiom system recognizing solely addition as a total function can verify its own consistency
 Journal of Symbolic Logic
"... Abstract. This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: 1) α treats multiplication as ..."
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Abstract. This article will study a class of deduction systems that allow for a limited use of the modus ponens method of deduction. We will show that it is possible to devise axiom systems α that can recognize their consistency under a deduction system D provided that: 1) α treats multiplication as a 3way relation (rather than as a total function), and that 2)D does not allow for the use of a modus ponens methodology above essentially the levels of Π1 and Σ1 formulae. Part of what will make this boundarycase exception to the Second Incompleteness Theorem interesting is that we will also characterize generalizations of the Second Incompleteness Theorem that take force when we only slightly weaken the assumptions of our boundarycase exceptions in any of several further directions.