Results 1  10
of
19
Cutelimination and redundancyelimination by resolution
 Journal of Symbolic Computation
, 2000
"... A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
A new cutelimination method for Gentzen’s LK is defined. First cutelimination is generalized to the problem of redundancyelimination. Then the elimination of redundancy in LKproofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LKproof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LKproof ψ ′ with atomic cuts; ψ ′ can be constructed from the resolution proof and ψ by a projection method. In the last step the atomic cuts are eliminated and a cutfree proof is obtained. The complexity of the method is analyzed and it is shown that a nonelementary speedup over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudocuts) can be transformed into formal ones by the method of redundancyelimination; moreover, the method can even be used to transform incorrect proofs into correct ones. 1.
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Kreisel's `Unwinding Program
 In Odifreddi [53
, 1996
"... Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
Abstract

Cited by 11 (10 self)
 Add to MetaCart
This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
Proof Transformation by CERES
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM) 2006, VOLUME 4108 OF LECTURE NOTES IN ARTIFICIAL INTELLIGENCE
, 2006
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set o ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an LKproof with only atomic cuts. In this paper we present an extension of CERES to a calculus LKDe which is stronger than the Gentzen calculus LK (it contains rules for introduction of definitions and equality rules). This extension makes it much easier to formalize mathematical proofs and increases the performance of the cutelimination method. The system CERES already proved efficient in handling very large proofs.
CERES: An Analysis of Fürstenberg’s Proof of the Infinity of Primes
, 2008
"... The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzen’s famous cutelimination theorem stating that s ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
The distinction between analytic and synthetic proofs is a very old and important one: An analytic proof uses only notions occurring in the proved statement while a synthetic proof uses additional ones. This distinction has been made precise by Gentzen’s famous cutelimination theorem stating that synthetic proofs can be transformed into analytic ones. CERES (cutelimination by resolution) is a cutelimination method that has the advantage of considering the original proof in its full generality which allows the extraction of different analytic arguments from it. In this paper we will use an implementation of CERES to analyze Fürstenberg’s topological proof of the infinity of primes. We will show that Euclid’s original proof can be obtained as one of the analytic arguments from Fürstenberg’s proof. This constitutes a proofofconcept example for a semiautomated analysis of realistic mathematical proofs providing new information about them.
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
CutElimination: Experiments with CERES
, 2005
"... Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of ..."
Abstract

Cited by 9 (8 self)
 Add to MetaCart
Cutelimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cutelimination method CERES (cutelimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skeleton of a proof with only atomic cuts. In this paper we present a systematic experiment with the implementation of CERES on a proof of reasonable size and complexity. It turns out that the proof with cuts can be transformed into two mathematically different proofs of the theorem. In particular, the application of positive and negative hyperresolution yield different mathematical arguments. As an unexpected sideeffect the derived clauses of the resolution refutation proved particularly interesting as they can be considered as meaningful universal lemmas. Though the proof under investigation is intuitively simple, the experiment demonstrates that new (and relevant) mathematical information on proofs can be obtained by computational methods. It can be considered as a first step in the development of an experimental culture of computeraided proof analysis in mathematics.