Cut-free Sequent Research Articles

Lyndon’s interpolation property for the logic of strict implication

Abstract The main result proves Lyndon’s and Craig’s interpolation properties for the logic of strict implication ${\textsf{F}}$, with a purely syntactical method. A cut-free G3-style sequent calculus $ {\textsf{GF}} $ and its single-succedent variant $ \textsf{GF}_{\textsf{s}} $ are introduced. $ {\textsf{GF}} $ can be extended to a G3-variant of the sequent calculus GBPC3 for Visser’s basic logic. Also a simple syntactic proof of known embedding result of $ {\textsf{F}} $ into $ {\textsf{K}} $ is provided. An extension of $ {\textsf{F}} $, namely $ \textsf{FD}, $ is considered as well.

A derivation-loop method for temporal logic

Various types of calculi (Hilbert, Gentzen sequent, resolution calculi, tableaux) for propositional linear temporal logic (PLTL) have been considered in the literature. Cutfree Gentzen-type sequent calculi are convenient tools for backward proof-search search of formulas and sequents. In this paper we present a cut-free Gentzen type sequent calculus for PLTL with the operator

A proof-search system for the logic of likelihood

Abstract The cut-free Gentzen-type sequent calculus LLK for the logic of likelihood (LL) is introduced in the paper. It is proved that the calculus is sound and complete for LL. Using the introduced calculus LLK, a decision procedure for LL is presented.

GENERALITY AND EXISTENCE 1: QUANTIFICATION AND FREE LOGIC

AbstractIn this paper, I motivate a cut free sequent calculus for classical logic with first order quantification, allowing for singular terms free of existential import. Along the way, I motivate a criterion for rules designed to answer Prior’s question about what distinguishes rules for logical concepts, like conjunction from apparently similar rules for putative concepts like Prior’s tonk, and I show that the rules for the quantifiers—and the existence predicate—satisfy that condition.

Mathematical Text Processing in EA-style: a Sequent Aspect

The paper is devoted to the study of one of the aspects of the so-called Evidence Algorithm programme advanced by Academician V.M.~Glushkov and connected with the problem of automated theorem-proving search in the signature of first-order theories that can use different logics, such as classical, intuitionistic, and modal ones. An approach to the construction of such sequent logics (with or without equality) is described. It exploits the original notions of admissibility and compatibility, which permits to avoid preliminary skolemization being a forbidden operation for a number of non-classical logics in general. Following the approach, the cut-free sequent (modal) calculi avoiding the dependence of inference search on different orders of quantifier rules applications are described. Results on the coextensivity of sequent calculi are given. The research gives a way to the construction of computer-oriented quantifier-rule-free calculi for classical and intuitionistic logics and their modal extensions.

On the Computational Interpretation of CKn for Contextual Information Processing

We aim to establish the multi-modal logic CKn as a baseline for a constructive correspondence theory of constructive modal logics. Just like many classical multi-modal logics may be studied as theories of the basic system K obtained by model-theoretic specialisation, we envisage constructive modal logics to be derived as proof-theoretic enrichments of CKn. The system CKn would then act as a core system for constructive contextual reasoning with controlled information flow. In this paper, as a first step towards this goal, we study CKn as a type theory and introduce its computational λ-calculus, λCKn. Extending previous work on CKn, we present a cut-free contextual sequent system in the spirit of Masini's two-dimensional generalisation of natural deduction and Brunnler's nested sequents and give a computational interpretation for CKn following the Curry-Howard Correspondence. The associated modal type theory λCKn permits an interpretation for both the modalities □ and ◊ of CKn as type operators with simple and independent constructors and destructors, which has been missing in the literature. It is shown that the calculus satisfies subject reduction, strong normalisation and confluence. Since normal forms can be characterised by way of a Gentzen-style typing system with sub-formula property, λCKn is suitable for proof search in CKn. At the same time, λCKn enjoys natural deduction style typing which is important for programming applications. In contrast to most existing modal type theories, which are obtained as theories of the constructive modal logic S4, CKn is not bound to a particular contextual interpretation. Thus, λCKn constitutes the core of a functional language which provides static type checking of information processing to support safe contextual navigation in relational structures like those treated by description logics. We review some existing work on modal type theories and discuss their relation to λCKn.

A Connection-based Characterization of Bi-intuitionistic Validity

We give a connection-based characterization of validity in propositional bi-intuitionistic logic in terms of specific directed graphs called R-graphs. Such a characterization is well-suited for deriving labelled proof-systems with counter-model construction facilities. We first define the notion of bi-intuitionistic R-graphs from which we then obtain a connection-based characterization of propositional bi-intuitionistic validity and derive a sound and complete cut-free free-variable labelled sequent calculus that enjoys variable splitting.

The Properties of Determinative Disjunctive Normal Forms and Systems Based on Them

We study some structural and numerical properties of varieties for determinative disjunctive normal forms, introduced here for given propositional formula. We consider for classical and non-classical propositional logics some proof systems, which are constructed on the base of determinative disjunctive normal forms. We investigate also the relation between the proof complexities in some well-known classical and non-classical proof systems (Resolution, Cut-free sequent, Gentzen refutation, Cutting planes etc.) and numerical properties of varieties for determinative disjunctive normal forms for classical and non-classical tautologies.

The Power of Belnap: Sequent Systems for SIXTEEN 3

The trilattice \(\textit{SIXTEEN}_3\) is a natural generalization of the well-known bilattice \(\textit{FOUR}_2\). Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in \(\textit{SIXTEEN}_3\) are presented.

Displaying the modal logic of consistency

AbstractIt is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

Reasoning with first order nondeterministic specifications

The paper presents a variant of first order logic for specifying nondeterministic software. Models of the logics are multialgebras, i.e. multi-sorted algebras with set-valued operations, together with multi-sorted valuations of variables. We allow empty carrier sets but the valuations are kept total. Terms are interpreted as sets and the usual set of algebraic terms is extended by an additional \({\sf let}\) construct used for limiting nondeterminism. Atomic formulae are of the form \(t_1 \to t_2\) where \(\to\) is a rewrite operator, corresponding semantically to inclusion. For the above logic, we develop two complete deduction systems in the natural deduction style: a Rasiowa-Sikorski system for sequences of formulae, and a cut-free Gentzen-style sequent calculus. We also consider the issues of determinism and partiality, proposing alternate solutions to defining the respective predicates in our logic.