Sufficient conditions for first order based sequent calculi to admit cut elimination by a Schütte-Tait 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.

Cut-elimination 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 cut-elimination method CERES (cut-elimination 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 LK-proof 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 cut-elimination method. The system CERES already proved efficient in handling very large proofs.

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 cut-elimination theorem stating that synthetic proofs can be transformed into analytic ones. CERES (cut-elimination by resolution) is a cut-elimination 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 proof-of-concept example for a semi-automated analysis of realistic mathematical proofs providing new information about them.

Cut-elimination 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 cut-elimination method CERES (cut-elimination 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 side-effect 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 computer-aided proof analysis in mathematics.

This short report presents the main topics of methods of cut-elimination which will be presented in the course at the ESSLLI'99. It gives a short introduction addressing the problem of cut-elimination in general. Furthermore we give a brief description of several methods and refer to other papers added to the course material.

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 two-sorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result

Abstract. In this paper we define the concept of a profile, which is a characteristic clause set, corresponding to an LK-proof in first-order logic, which is invariant under rule permutations. It is shown (via cutelimination) that the profile is even invariant under a large class of proof transformations (called “simple transformations”), which includes transformations to negation normal form. As proofs having the same profile show the same behavior w.r.t. cut-elimination (which can be formally defined via the method CERES), proofs obtained by simple transformations can be considered as equal in this sense. A comparison with related results based on proof nets is given: in particular it is shown that proofs having the same profile define a larger equivalence class than those having the same proof net. 1

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 first-order 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.

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 cut-free 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 satisfiability-equivalent to the original formula. This transformation has a number of applications, it is for example crucial for automated theorem

We investigate cut-elimination in propositional substructural logics. The problem is to decide whether a given calculus admits (reductive) cut-elimination. We show that, for commutative single-conclusion sequent calculi containing generalized knotted structural rules and arbitrary logical rules, the problem can be decided by resolution-based methods. A general cut-elimination proof for these calculi is also provided.