A cut-elimination procedure designed for evaluating proofs as programs

2005 ◽  
pp. 316-325
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Cut Elimination ◽  
Elimination Procedure

scholarly journals A short proof of the strong normalization of classical natural deduction with disjunction

2003 ◽  
Vol 68 (4) ◽  
pp. 1277-1288 ◽  
Author(s):  
René David ◽  
Karim Nour
Keyword(s):  
Natural Deduction ◽  
Elementary Proof ◽  
Short Proof ◽  
Cut Elimination ◽  
Strong Normalization ◽  
Elimination Procedure

AbstractWe give a direct, purely arithmetical and elementary proof of the strong normalization of the cut-elimination procedure for full (i.e., in presence of all the usual connectives) classical natural deduction.

Multi-focused cut elimination

2017 ◽  
Vol 28 (5) ◽  
pp. 614-650
Author(s):  
TAUS BROCK-NANNESTAD ◽  
NICOLAS GUENOT
Keyword(s):  
Simple Proof ◽  
Linear Logic ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Elimination Procedure ◽  
The Impact

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

A Formally Verified Cut-Elimination Procedure for Linear Nested Sequents for Tense Logic

2021 ◽  
pp. 281-298
Author(s):  
Caitlin D’Abrera ◽  
Jeremy Dawson ◽  
Rajeev Goré
Keyword(s):  
Tense Logic ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Nested Sequents

scholarly journals A MALL geometry of interaction based on indexed linear logic

2021 ◽  
pp. 1-29
Author(s):  
Masahiro Hamano
Keyword(s):  
Monoidal Category ◽  
Linear Logic ◽  
Dynamic Modelling ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Dynamic Level

Abstract We construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

scholarly journals On permuting cut with contraction

2000 ◽  
Vol 10 (2) ◽  
pp. 99-136 ◽  
Author(s):  
MIRJANA BORISAVLJEVIĆ ◽  
KOSTA DOšEN ◽  
ZORAN PETRIĆ
Keyword(s):  
Propositional Logic ◽  
Cut Elimination ◽  
Intuitionistic Propositional Logic ◽  
Cut Rule ◽  
Elimination Procedure ◽  
Made In

This paper presents a cut-elimination procedure for intuitionistic propositional logic in which cut is eliminated directly, without introducing the multiple-cut rule mix, and in which pushing cut above contraction is one of the reduction steps. The presentation of this procedure is preceded by an analysis of Gentzen's mix-elimination procedure, made in the perspective of permuting cut with contraction. We also show that in the absence of implication, pushing cut above contraction does not pose problems for directly eliminating cut.

scholarly journals Sequent Calculi for the Propositional Logic of HYPE

Studia Logica ◽  
2021 ◽  
Author(s):  
Martin Fischer
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Cut Elimination ◽  
Philosophical Logic ◽  
Admissible Rule ◽  
Sequent Calculi ◽  
Complete Proof ◽  
Elimination Procedure ◽  
The One

AbstractIn this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple $$\mathbf{G1}$$ G 1 -system employing rules of contraposition. On the other hand we present a $$\mathbf{G3}$$ G 3 -system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

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