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116
Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations
, 1997
"... We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, i ..."
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Cited by 139 (5 self)
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We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.
An Exponential Lower Bound to the Size of Bounded Depth Frege . . .
, 1994
"... We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for whic ..."
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Cited by 66 (9 self)
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We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.
Lower bounds on Hilbert's Nullstellensatz and propositional proofs
 PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
, 1996
"... The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polyno ..."
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Cited by 65 (18 self)
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The socalled weak form of Hilbert's Nullstellensatz says that a system of algebraic equations over a field, Qj(x) = 0, does not have a solution in the algebraic closure if and only if 1 is in the ideal generated by the polynomials (?,(*) • We shall prove a lower bound on the degrees of polynomials P,(x) such that £, P,(x)Qt(x) = 1. This result has the following application. The modular counting principle states that no finite set whose cardinality is not divisible by q can be partitioned into ^element classes. For each fixed cardinality N, this principle can be expressed as a propositional formula Count^fo,...) with underlying variables xe, where e ranges over <7element subsets of N. Ajtai [4] proved recently that, whenever p,q are two different primes, the propositional formulas Count $ n+I do not have polynomial size, constantdepth Frege proofs from substitution instances of Count/?, where m^O (modp). We give a new proof of this theorem based on the lower bound for Hilbert's Nullstellensatz. Furthermore our technique enables us to extend the independence results for counting principles to composite numbers p and q. This improved lower bound together with new upper bounds yield an exact characterization of when Count, can be proved efficiently from Countp, for all values of p and q.
A tutorial on Stålmarck's proof procedure for propositional logic
 Formal Methods in System Design
, 1998
"... We explain Stalmarck's proof procedure for classical propositional logic. The method is implemented in a commercial tool that has been used successfully in real industrial verification projects. Here, we present the proof system underlying the method, and motivate the various design decisio ..."
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Cited by 65 (1 self)
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We explain Stalmarck's proof procedure for classical propositional logic. The method is implemented in a commercial tool that has been used successfully in real industrial verification projects. Here, we present the proof system underlying the method, and motivate the various design decisions that have resulted in a system that copes well with the large formulas encountered in industrialscale verification. 1
Notes on Polynomially Bounded Arithmetic
"... We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The p ..."
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Cited by 60 (1 self)
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We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general modeltheoretical investigations on fragments of bounded arithmetic. Contents 0 Introduction and motivation. 1 1 Preliminaries. 3 1.1 The polynomially bounded hierarchy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2 The axioms of secondorder bounded arithmetic. : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.3 Rudimentary functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 1.4 Other fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.5 Polynomial time computable functions. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.6 Relations among fragments. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.7 Relations with Buss' bounded arithmetic. : : : :...
The Taming of the Cut. Classical Refutations with Analytic Cut
 JOURNAL OF LOGIC AND COMPUTATION
, 1994
"... The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of vi ..."
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Cited by 54 (1 self)
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The method of analytic tableaux is a direct descendant of Gentzen's cutfree sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cutfree systems are anomalous from the prooftheoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the "computational scandal" that such systems cannot polynomially simulate the truthtables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being "close" to Smullyan's tableau method, is not cutfree but includes a class...
Lower Bounds For The Polynomial Calculus
, 1998
"... We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumpt ..."
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Cited by 51 (5 self)
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We show that polynomial calculus proofs (sometimes also called Groebner proofs) of the pigeonhole principle PHP n must have degree at least (n=2)+1 over any field. This is the first nontrivial lower bound on the degree of polynomial calculus proofs obtained without using unproved complexity assumptions. We also show that for some modifications of PHP n , expressible by polynomials of at most logarithmic degree, our bound can be improved to linear in the number of variables. Finally, we show that for any Boolean function f n in n variables, every polynomial calculus proof of the statement "f n cannot be computed by any circuit of size t," must have degree t=n). Loosely speaking, this means that low degree polynomial calculus proofs do not prove NP 6 P=poly.
A New Proof of the Weak Pigeonhole Principle
, 2000
"... The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further re ..."
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Cited by 38 (3 self)
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The exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, Paris, Wilkie and Woods [16] showed how to prove the weak pigeonhole principle with boundeddepth, quasipolynomialsize proofs. Their argument was further refined by Kraj'icek [9]. In this paper, we present a new proof: we show that the the weak pigeonhole principle has quasipolynomialsize LK proofs where every formula consists of a single AND/OR of polylog fanin. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth. 1 Introduction The pigeonhole principle is a fundamental axiom of mathematics, stating that there is no onetoone mapping from m pigeons to n holes when m ? n. It expresses Department of Mathematics and Computer Science, Clarkson University, Potsdam, NY 136995815, U.S.A. alexis@clarkson.edu. Research supported by NSF grant CCR9877150. y Department of Computer Science, University o...
Resolution and the weak pigeonhole principle
 IN CSL
, 1997
"... We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for treelike resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule. ..."
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Cited by 38 (3 self)
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We give new upper bounds for resolution proofs of the weak pigeonhole principle. We also give lower bounds for treelike resolution proofs. We present a normal form for resolution proofs of pigeonhole principles based on a new monotone resolution rule.
Two party immediate response disputes: properties and efficiency
 Artificial Intelligence
, 2001
"... Abstract. Two Party Immediate Response Disputes (TPIdisputes) are one class of dialogue or argument game in which the protagonists take turns producing counter arguments to the ‘most recent ’ argument advanced by their opponent. Argument games have been found useful as a means of modelling dialecti ..."
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Cited by 38 (17 self)
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Abstract. Two Party Immediate Response Disputes (TPIdisputes) are one class of dialogue or argument game in which the protagonists take turns producing counter arguments to the ‘most recent ’ argument advanced by their opponent. Argument games have been found useful as a means of modelling dialectical discourse and in providing semantic bases for proof theoretic aspects of reasoning. In this article we consider a formalisation of TPIdisputes in the context of finite Argument Systems. Our principal concern may, informally, be phrased as follows: given a specific argument system, À and argument, x within À, what can be stated concerning the number of rounds a dispute might take for one of its protagonists to accept that x has some defence respectively cannot be defended?