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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 35 (2 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...
OPEN QUESTIONS IN REVERSE MATHEMATICS
, 2010
"... The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discu ..."
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Cited by 8 (0 self)
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The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discuss them in the context of related work. The list is definitely not comprehensive, and my
Characterising Definable Search Problems in Bounded Arithmetic via Proof Notations
, 2009
"... The complexity class of Π p kPolynomial Local Search (PLS) problems with Π p ℓgoal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by tran ..."
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The complexity class of Π p kPolynomial Local Search (PLS) problems with Π p ℓgoal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the ParisWilkietranslation. For ℓ ≤ k, the Σb ℓ+1definable search problems of T k+1 2 are exactly characterised by Π p kPLS problems with Πp ℓgoals. These Π p kPLS problems can be defined in a weak base theory such as S1 2, and proved to be total in T k+1 2. Furthermore, the Π p kPLS definitions can be Skolemised with simple polynomial time functions. The Skolemised Π p kPLS definitions give rise to a new ∀Σb1(α) principle conjectured to separate Tk 2(α) from T k+1 2 (α). 1
Parallel computable higher type functionals (Extended Abstract)
 In Proceedings of IEEE 34th Annual Symposium on Foundations of Computer Science, Nov 35
, 1993
"... ) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different way ..."
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Cited by 4 (4 self)
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) Peter Clote A. Ignjatovic y B. Kapron z 1 Introduction to higher type functionals The primary aim of this paper is to introduce higher type analogues of some familiar parallel complexity classes, and to show that these higher type classes can be characterized in significantly different ways. Recursiontheoretic, prooftheoretic and machinetheoretic characterizations are given for various classes, providing evidence of their naturalness. In this section, we motivate the approach of our work. In proof theory, primitive recursive functionals of higher type were introduced in Godel's Dialectica [13] paper, where they were used to "witness" the truth of arithmetic formulas. For instance, a function f witnesses the formula 8x9y\Phi(x; y), where \Phi is quantifierfree, provided that 8x\Phi(x; f(x)); while a type 2 functional F witnesses the formula 8x9y8u9v\Phi(x; y; u; v), provided that 8x8u\Phi(x; f(x); u; F (x; f(x); u)): Godel's formal system T is a variant of the finit...
Cutting plane and Frege proofs
 Information and Computation
, 1995
"... The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the nonexistence of solutions for families of integer linear inequalities. We define the system CP + , a modification of the cutting plane system, and show that CP + can polynomi ..."
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The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the nonexistence of solutions for families of integer linear inequalities. We define the system CP + , a modification of the cutting plane system, and show that CP + can polynomially simulate Frege systems F . In [8], it is shown that F polynomially simulates CP + , so both systems are polynomially equivalent. To establish this result, propositional formulas are represented in a natural manner, and an effective version of cut elimination is proved for the system CP + . Additionally, an alternative proof is given which directly translates F proofs into CP + . Thus, within a polynomial factor, one can simulate classical propositional logic proofs using the cut rule by refutationstyle proofs involving linear inequalities with the ceiling operation. Since there are polynomial size cutting plane CP proofs for many elementary combinatorial principles (pigeonhole p...
On the computational complexity of cutreduction
, 2009
"... Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theorie ..."
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Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way. 1 Introduction and Related Work Since Gentzen’s invention of the “Logik Kalkül ” LK and the proof of his “Hauptsatz ” [Gen35a, Gen35b], cutelimination has been studied in many papers on proof theory. Mints ’ invention of continuous normalisation [Min78, KMS75] isolates operational aspects of normalisation, that is, the manipulations on (in
A note on universal measures for weak implicit computational complexity
 Proc. of the 9th International Conference, LPAR 2002 (Tbilisi), LNCS
, 2002
"... Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate “universal measures ” by “dynamic ordinals”, and “weak implicit computational complexity ” by “bounded arithmetic”. Concretely, we will describe the connection between ..."
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Abstract. This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate “universal measures ” by “dynamic ordinals”, and “weak implicit computational complexity ” by “bounded arithmetic”. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories.
Resolution refutations and propositional proofs with heightrestriction
 Proc. of the 16th International Workshop, CSL 2002 (Edinburgh), LNCS
, 2002
"... Abstract. Height restricted resolution (proofs or refutations) is a natural restriction of resolution where the height of the corresponding proof tree is bounded. Height restricted resolution does not distinguish between tree and sequencelike proofs. We show that polylogarithmicheight resolutio ..."
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Abstract. Height restricted resolution (proofs or refutations) is a natural restriction of resolution where the height of the corresponding proof tree is bounded. Height restricted resolution does not distinguish between tree and sequencelike proofs. We show that polylogarithmicheight resolution is strongly connected to the bounded arithmetic theory S12(α). We separate polylogarithmicheight resolution from quasipolynomial size treelike resolution. Inspired by this we will study infinitely many sublinearheight restrictions given by functions n 7 → 2i (log(i+1) n)O(1) for i ≥ 0. We show that the resulting resolution systems are connected to certain bounded arithmetic theories, and that they form a strict hierarchy of resolution proof systems. To this end we will develop some proof theory for height restricted proofs.
Dynamic ordinals – universal measures for implicit computational complexity
, 2002
"... We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordin ..."
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We extend the definition of dynamic ordinals to generalised dynamic ordinals. We compute generalised dynamic ordinals of all fragments of relativised bounded arithmetic by utilising methods from Boolean complexity theory, similar to Krajíček in [14]. We indicate the role of generalised dynamic ordinals as universal measures for implicit computational complexity. I.e., we describe the connections between generalised dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories. In particular, through the determination of generalised dynamic ordinals we reobtain wellknown independence results for relativised bounded arithmetic theories.