A new cut-elimination method for Gentzen’s LK is defined. First cut-elimination is generalized to the problem of redundancy-elimination. Then the elimination of redundancy in LK-proofs is performed by a resolution method in the following way: A set of clauses C is assigned to an LK-proof ψ and it is shown that C is always unsatisfiable. A resolution refutation of C then serves as a skeleton of an LK-proof 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 cut-free proof is obtained. The complexity of the method is analyzed and it is shown that a nonelementary speed-up over Gentzen’s method can be achieved. Finally an application to automated deduction is presented: it is demonstrated how informal proofs (containing pseudo-cuts) can be transformed into formal ones by the method of redundancy-elimination; moreover, the method can even be used to transform incorrect proofs into correct ones. 1.

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.

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.

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.

Sufficient conditions for propositional based logics to enjoy cut elimination are established. These conditions are satisfied by a wide class of logics encompassing among others classical and intuitionistic logic, modal logic S4, and classical and intuitionistic linear logic and some of their fragments. The class of logics is characterized by the type of rules and provisos used in their sequent calculi. The conditions can be checked in finite time and define relations between the rules and the provisos so that the calculus can enjoy cut elimination. A general proof of cut elimination is presented for any calculus satisfying those conditions.

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 introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications. 1

In this thesis we study a formal first order system T (tind) in the standard language L, of Gentzen’s LK, see [Tak87]. T (tind) extends LK by the following valid first-order inference rule (A is quantifier-free). Γ, A(a), Λ → ∆, A(s(a)), Θ Γ, A(0), Λ → ∆, A(s n (0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T (tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T (tind). Let ∃¯xF (¯x) be a Σ1 formula; provable by Π. Then there exists a disjunction � N i1 · · · � N il M1(s i1 (0),..., s il(0)) ∨ · · · ∨ Mm(s i1 (0),..., s il(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T (tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the end-formula as usual. The main result is that we can bound the length of the {s, 0}-matrix of the above disjunctions in this way.

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. Cut-elimination can be applied to mine real mathematical proofs, i.e. for extracting explicit and algorithmic information. The system CERES (cut-elimination by resolution) is based on automated deduction and was successfully applied to the analysis of nontrivial mathematical proofs. In this paper we focus on the input-output environment of CERES, and show how users can interact with the system and extract new mathematical knowledge. 1

We present methods for removing top-level cuts from a sequent calculus or Tait-style proof without significantly increasing the space used for storing the proof. For propositional logic, this requires converting a proof from tree-like to dag-like form, but it most doubles the number of lines in the proof. For first-order logic, the proof size can grow exponentially, but the proof has a succinct description and is polynomial-timeuniform. We usedirect, globalconstructionsthat give polynomial time methods for removing all top-level cuts from proofs. Byexploitingprenexrepresentations,this extendsto removingall cuts, with final proof size bounded superexponentially in the alternation of quantifiers in cut formulas. 1