Originating from work in operations research the cutting plane refutation system CP is an extension of resolution, where unsatisfiable propositional logic formulas in conjunctive normal form are recognized by showing the non-existence of boolean solutions to associated families of linear inequalities. Polynomial size CP proofs are given for the undirected s-t connectivity principle. The subsystems CPq of CP , for q 2, are shown to be polynomially equivalent to CP , thus answering problem 19 from the list of open problems of [8]. We present a normal form theorem for CP2-proofs and thereby for arbitrary CP -proofs. As a corollary, we show that the coefficients and constant terms in arbitrary cutting plane proofs may be exponentially bounded by the number of steps in the proof, at the cost of an at most polynomial increase in the number of steps in the proof. The extension CPLE + , introduced in [9] and there shown to p- simulate Frege systems, is proved to be polynomially equivalen...

It is shown that any sequence #n of tautologies which expresses the validity of a fixed combinatorial principle either is "easy" i.e. has polynomial size tree-resolution proofs or is "di#cult" i.e requires exponential size tree-resolution proofs. It is shown that the class of tautologies which are hard (for tree-resolution) is identical to the class of tautologies which are based on combinatorial principles which are violated for infinite sets. Actually it is

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This paper is based on my lecture [26]. It examines the problem of proving non-trivial lower bounds for the length of proofs in propositional logic from the perspective of methods available rather than surveying known partial results (i.e., lower bounds for weaker proof systems). We discuss neither motivations for proving lower bounds for propositional logic nor relations to other problems in logic or complexity theory. The reader is referred to [20] for the background information (as well as for all details missing in this paper). The paper is aimed at curious non-specialists. The style of our exposition is accordingly informal at places and we do not burden the text (especially in the introduction) with exhausting references not directly related to our main objective. The reader starving for details can find them, together with all original references, in [20] (see also expository articles [25, 32]). Introduction

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...

Formal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs. © 2001 Academic Press Key Words: formal proofs; logical flow graphs; cut elimination; bridge groups; Baumslag–Solitar groups; Gersten groups.

We define the notion of a combinatorics of a first order structure, and a relation of covering between first order structures and propositional proof systems. Namely, a first order structure M combinatorially satisfies an L-sentence \Phi iff \Phi holds in all L-structures definable in M. The combinatorics Comb(M) of M is the set of all sentences combinatorially satisfied in M. Structure M covers a propositional proof system P iff M combinatorially satisfies all \Phi for which the associated sequence of propositional formulas h\Phi i n, encoding that \Phi holds in Lstructures of size n, have polynomial size P-proofs. That is, Comb(M) contains all \Phi feasibly verifiable in P. Finding M that covers P but does not combinatorially satisfy \Phi thus gives a super polynomial lower bound for the size of P-proofs of h\Phi in. We show that any proof system admits a class of structures covering it; these structures are expansions of models of bounded arithmetic. We also give, using structures covering proof systems R \Lambda

A classical quantified modal logic is used to define a "feasible" arithmetic A 1 2 whose provable functions are exactly the polynomial-time computable functions. Informally, one understands # as "# is feasibly demonstrable". A 1 2 di#ers from a system A2 that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e. -free) formulas. Thus, A 1 2 is defined without any reference to bounding terms, and admitting induction over formulas having arbitrarily many alternations of unbounded quantifiers. The system also uses only a very small set of initial functions. To obtain the characterization, one extends the Curry-Howard isomorphism to include modal operations. This leads to a realizability translation based on recent results in higher-type ramified recursion. The fact that induction formulas are not restricted in their logical complexity, allows one to use the Friedman A translation directly. The development also leads us to propose a new Frege ru...

Many times in mathematics there is a natural dichotomy between describing some object from the inside and from the outside. Imagine algebraic varieties for instance; they can be described from the outside as solution sets of polynomial equations, but one can also try to understand how it is for actual points to move around inside them, perhaps to parameterize them in some way. The concept of formal proofs has the interesting feature that it provides opportunities for both perspectives. The inner perspective has been largely overlooked, but in fact lengths of proofs lead to new ways to measure the information content of mathematical objects. The disparity between minimal lengths of proofs with and without "lemmas" provides an indication of internal symmetry of mathematical objects and their descriptions.

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