scholarly journals A Characterisation of Open Bisimilarity using an Intuitionistic Modal Logic

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Ki Yung Ahn ◽  
Ross Horne ◽  
Alwen Tiu
Keyword(s):  
Modal Logic ◽  
Modal Logics ◽  
Completeness Proof ◽  
Intuitionistic Modal Logic ◽  
Free Variables ◽  
Open Process ◽  
Intuitionistic Modal Logics

Open bisimilarity is defined for open process terms in which free variables may appear. The insight is, in order to characterise open bisimilarity, we move to the setting of intuitionistic modal logics. The intuitionistic modal logic introduced, called $\mathcal{OM}$, is such that modalities are closed under substitutions, which induces a property known as intuitionistic hereditary. Intuitionistic hereditary reflects in logic the lazy instantiation of free variables performed when checking open bisimilarity. The soundness proof for open bisimilarity with respect to our intuitionistic modal logic is mechanised in Abella. The constructive content of the completeness proof provides an algorithm for generating distinguishing formulae, which we have implemented. We draw attention to the fact that there is a spectrum of bisimilarity congruences that can be characterised by intuitionistic modal logics.

Explicit Provability and Constructive Semantics

2001 ◽  
Vol 7 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Sergei N. Artemov
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
Modal Logics ◽  
Combinatory Logic ◽  
Underlying Structure ◽  
Intuitionistic Propositional Logic ◽  
Intuitionistic Modal Logics

AbstractIn 1933 Gödel introduced a calculus of provability (also known as modal logicS4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logicLPof propositions and proofs and show that Gödel's provability calculus is nothing but the forgetful projection ofLP. This also achieves Gödel's objective of defining intuitionistic propositional logicIntvia classical proofs and provides a Brouwer-Heyting-Kolmogorov style provability semantics forIntwhich resisted formalization since the early 1930s.LPmay be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and λ-calculus.

scholarly journals Constant Domain Quantified Modal Logics Without Boolean Negation

2005 ◽  
Vol 3 ◽  
Author(s):  
Greg Restall
Keyword(s):  
Modal Logic ◽  
Predicate Logic ◽  
Difficult Problem ◽  
Modal Logics ◽  
Modal Operator ◽  
Completeness Proof ◽  
Constant Domain ◽  
Relevant Logics ◽  
Modal Predicate Logic

his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 presentation of the completeness proof quite closely), but with an important twist, to do with the absence of Boolean negation.

A Modal Semantics of Negation in Logic Programming

1992 ◽  
Vol 16 (3-4) ◽  
pp. 231-262
Author(s):  
Philippe Balbiani
Keyword(s):  
Modal Logic ◽  
Logic Programming ◽  
Modal Logics ◽  
Kripke Models ◽  
Logic Programs ◽  
Modal Semantics ◽  
Soundness And Completeness ◽  
Modal Completion ◽  
Well Foundedness

The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

scholarly journals Canonical formulas via locally finite reducts and generalized dualities

10.29007/hgbj ◽  
2018 ◽  
Author(s):  
Nick Bezhanishvili
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Algebraic Approach ◽  
Modal Logics ◽  
Substructural Logic ◽  
Locally Finite ◽  
Canonical Formulas

The method of canonical formulas is a powerful tool for investigating intuitionistic and modal logics. In this talk I will discuss an algebraic approach to this method. I will mostly concentrate on the case of intuitionistic logic. But I will also review the case of modal logic and possible generalizations to substructural logic.

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.

PROOF SYSTEMS FOR VARIOUS FDE-BASED MODAL LOGICS

2019 ◽  
Vol 13 (4) ◽  
pp. 720-747
Author(s):  
SERGEY DROBYSHEVICH ◽  
HEINRICH WANSING
Keyword(s):  
Modal Logic ◽  
Sequent Calculus ◽  
Axiom System ◽  
Modal Logics ◽  
Cut Elimination ◽  
Proof Systems ◽  
Image Position ◽  
Axiomatic Extension

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.

scholarly journals Intuitionistic Modal Logic and Applications (IMLA 2008)

2011 ◽  
Vol 209 (12) ◽  
pp. 1435-1436
Author(s):  
Valeria de Paiva ◽  
Brigitte Pientka
Keyword(s):  
Modal Logic ◽  
Intuitionistic Modal Logic
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