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 bounded-depth, quasipolynomial-size 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 quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. 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 one-to-one 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 CCR-9877150. y Department of Computer Science, University o...

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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. Recursion-theoretic, proof-theoretic and machine-theoretic 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 quantifier-free, 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...

The cutting plane refutation system CP for propositional logic is an extension of resolution and is based on showing the non-existence 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 refutation-style proofs involving linear inequalities with the ceiling operation. Since there are polynomial size cutting plane CP proofs for many elementary combinatorial principles (pigeonhole p...

Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations, and 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

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 re-obtain well-known independence results for relativised bounded arithmetic theories.

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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