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Methods of CutElimination
 PROJECTION, LECTURE
"... This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers ad ..."
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This short report presents the main topics of methods of cutelimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cutelimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.
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 ..."
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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 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 ..."
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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.
Sufficient conditions for cut elimination with complexity analysis
 Annals of Pure and Applied Logic
, 2007
"... Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The con ..."
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Cited by 9 (4 self)
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Sufficient conditions for first order based sequent calculi to admit cut elimination by a SchütteTait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related with the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.
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 ..."
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Cited by 9 (8 self)
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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.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
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Cited by 7 (5 self)
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Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Herbrand sequent extraction
 IN INTELLIGENT COMPUTER MATHEMATICS
, 2008
"... Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extr ..."
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Cited by 5 (3 self)
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Computer generated proofs of interesting mathematical theorems are usually too large and full of trivial structural information, and hence hard to understand for humans. Techniques to extract specific essential information from these proofs are needed. In this paper we describe an algorithm to extract Herbrand sequents from proofs written in Gentzen’s sequent calculus LK for classical firstorder logic. The extracted Herbrand sequent summarizes the creative information of the formal proof, which lies in the instantiations chosen for the quantifiers, and can be used to facilitate its analysis by humans. Furthermore, we also demonstrate the usage of the algorithm in the analysis of a proof of the equivalence of two different definitions for the mathematical concept of lattice, obtained with the proof transformation system CERES.
On the complexity of proof deskolemization
 J. Symbolic Logic
"... Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of for ..."
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Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of formulas is a standard technique in logic. It consists of replacing existential quantifiers by new function symbols whose arguments reflect the dependencies of the quantifier. The Skolemization of a formula is satisfiabilityequivalent to the original formula. This transformation has a number of applications, it is for example crucial for automated theorem
Towards an Algorithmic Construction of CutElimination Procedures
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2009
"... We investigate cutelimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cutelimination. We show that, for commutative singleconclusion sequent calculi containing generalized knotted structural rules and arbitrary logical rules, the ..."
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We investigate cutelimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cutelimination. We show that, for commutative singleconclusion sequent calculi containing generalized knotted structural rules and arbitrary logical rules, the problem can be decided by resolutionbased methods. A general cutelimination proof for these calculi is also provided.
On the form of witness terms
 ARCH. MATHEMATICAL LOGIC
, 2010
"... We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree ..."
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Cited by 4 (3 self)
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We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree grammar computing witness terms is given and a class of proofs is shown to have only elementary cutelimination.