scholarly journals The New Normal: We Cannot Eliminate Cuts in Coinductive Calculi, But We Can Explore Them

2020 ◽  
Vol 20 (6) ◽  
pp. 990-1005
Author(s):  
Ekaterina Komendantskaya ◽  
Dmitry Rozplokhas ◽  
Henning Basold
Keyword(s):  
Cut Elimination ◽  
Horn Clause ◽  
Sequent Calculi ◽  
New Normal ◽  
Novel Method

AbstractIn sequent calculi, cut elimination is a property that guarantees that any provable formula can be proven analytically. For example, Gentzen’s classical and intuitionistic calculi LK and LJ enjoy cut elimination. The property is less studied in coinductive extensions of sequent calculi. In this paper, we use coinductive Horn clause theories to show that cut is not eliminable in a coinductive extension of LJ, a system we call CLJ. We derive two further practical results from this study. We show that CoLP by Gupta et al. gives rise to cut-free proofs in CLJ with fixpoint terms, and we formulate and implement a novel method of coinductive theory exploration that provides several heuristics for discovery of cut formulae in CLJ.

Gentzen sequent calculi for some intuitionistic modal logics

2019 ◽  
Vol 27 (4) ◽  
pp. 596-623
Author(s):  
Zhe Lin ◽  
Minghui Ma
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Modal Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Intuitionistic Propositional Logic ◽  
Propositional Language ◽  
Intuitionistic Modal Logics ◽  
Subformula Property ◽  
Gentzen Sequent Calculus

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

Frame definability, canonicity and cut elimination in common sense modal predicate logics

2020 ◽  
Author(s):  
Takahiro Sawasaki ◽  
Katsuhiko Sano
Keyword(s):  
Common Sense ◽  
Predicate Logic ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Axiom Schema ◽  
Modal Predicate Logics ◽  
Modal Predicate Logic

Abstract The paper presents semantically complete Hilbert-style systems for some variants of common sense modal predicate logic proposed by van Benthem and further developed by Seligman. The paper also investigates frame definability in the logics and shows what axiom schema is canonical in the logics. In addition to these semantic investigations on the logics, the paper provides the sequent calculi for some of the logics which enjoy cut elimination theorem.

scholarly journals Proof theory for quantified monotone modal logics

2019 ◽  
Vol 27 (4) ◽  
pp. 478-506
Author(s):  
Sara Negri ◽  
Eugenio Orlandelli
Keyword(s):  
Proof Theory ◽  
Modal Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Completeness Proof ◽  
First Order ◽  
Constant Domains ◽  
Normal Modal Logics ◽  
Order Extension

Abstract This paper provides a proof-theoretic study of quantified non-normal modal logics (NNML). It introduces labelled sequent calculi based on neighbourhood semantics for the first-order extension, with both varying and constant domains, of monotone NNML, and studies the role of the Barcan formulas in these calculi. It will be shown that the calculi introduced have good structural properties: invertibility of the rules, height-preserving admissibility of weakening and contraction and syntactic cut elimination. It will also be shown that each of the calculi introduced is sound and complete with respect to the appropriate class of neighbourhood frames. In particular, the completeness proof constructs a formal derivation for derivable sequents and a countermodel for non-derivable ones, and gives a semantic proof of the admissibility of cut.

scholarly journals GAME SEMANTICS AND THE GEOMETRY OF BACKTRACKING: A NEW COMPLEXITY ANALYSIS OF INTERACTION

2017 ◽  
Vol 82 (2) ◽  
pp. 672-708 ◽  
Author(s):  
FEDERICO ASCHIERI
Keyword(s):  
Complexity Analysis ◽  
Cut Elimination ◽  
Worst Case ◽  
First Order ◽  
Game Semantics ◽  
Case Complexity ◽  
Complexity Results ◽  
Worst Case Complexity ◽  
Novel Method

AbstractWe present abstract complexity results about Coquand and Hyland–Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.

Towards an algorithmic construction of cut-elimination procedures

2008 ◽  
Vol 18 (1) ◽  
pp. 81-105 ◽  
Author(s):  
AGATA CIABATTONI ◽  
ALEXANDER LEITSCH
Keyword(s):  
Substructural Logics ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Structural Rules ◽  
Algorithmic Construction ◽  
Logical Rules

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 generalised 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.

Modal Sequent Calculi Labelled with Truth Values: Cut Elimination

2005 ◽  
Vol 13 (2) ◽  
pp. 173-199 ◽  
Author(s):  
Paulo Mateus ◽  
João Rasga ◽  
Cristina Sernadas
Keyword(s):  
Cut Elimination ◽  
Sequent Calculi ◽  
Truth Values
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