Cut-elimination Result Research Articles

PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS

AbstractWe present novel proof systems for various FDE-based modal logics. Among the systems considered are a number of Belnapian modal logics introduced in Odintsov & Wansing (2010) and Odintsov & Wansing (2017), as well as the modal logic KN4 with strong implication introduced in Goble (2006). In particular, we provide a Hilbert-style axiom system for the logic $BK^{\square - } $ and characterize the logic BK as an axiomatic extension of the system $BK^{FS} $. For KN4 we provide both an FDE-style axiom system and a decidable sequent calculus for which a contraction elimination and a cut elimination result are shown.

Cut elimination for entailment relations

Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature.

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

Term Sequent Logic

We consider a term sequent logic for the lambda-calculus. Term sequents are a judgement form similar to the logical judgement form of entailment between sentences, but denoting equality or reducibility between terms. Using term sequents, it is possible to treat lambda-terms almost like logical sentences, and to use proof-theoretic methods to establish their properties. We prove a cut-elimination result for untyped lambda-calculus and describe how this generalises the usual confluence result. We give a notion of uniform proof for lambda-terms, and suggest how this can be viewed as a mixed logic-programming/functional programming framework with the ability to assume arbitrary reductions. Finally, we discuss related and future work.

Gentzen's Proof of Normalization for Natural Deduction

AbstractGentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933.

Coherence for star-autonomous categories

This paper presents a coherence theorem for star-autonomous categories exactly analogous to Kelly and Mac Lane’s coherence theorem for symmetric monoidal closed categories. The proof of this theorem is based on a categorial cut-elimination result, which is presented in some detail.

Cut Elimination in a Class of Sequent Calculi for Pure Type Systems

This paper present a new sequent calculus for Pure Type Systems (PTS). The calculus proposed is equiconsistent to the standard formulation (natural deduction like). The corresponding cut-free fragment makes it possible to introduce a notion of Cut Elimination. This property can be applied to develop proof-search strategies with dependent types.We prove that Cut Elimination holds in two important families of normalizing systems, including, in particular, three systems in the Barendregt's λ-cube: λ →, λ2, and λ ω . In addition, a cut elimination result is obtained for the minimal implicational second-order sequent calculus.

An Update on ’’Might‘‘

This paper is on the update semantics for ’’might‘‘ of Veltman (1996). Three consequence relations are introduced and studied in an abstract setting. Next we present sequent-style systems for each of the consequence relations. We show the logics to be complete and decidable. The paper ends with a syntactic cut elimination result.

Cut elimination for the unified logic

Vauzeilles, J., Cut elimination for the Unified Logic, Annals of Pure and Applied Logic 62 (1993) 1-16. In the paper entitled “On the Unity of Logic” Girard introduced and motivated the system LU. In Girard's article, the cut-elimination result for LU is stated and used as a key lemma, but not supported by any rigourous proof. In the present paper, we prove that LU enjoys cut elimination under minimal hypotheses: a notion of degree for a formula is introduced, which depends only on the number of exponentials of the formula; this refinement yields much better bounds for cut elimination and leaves open many possibilities as to non-well-foundedness of formulae.