A representation of proper BC domains based on conjunctive sequent calculi

2019 ◽  
Vol 30 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Longchun Wang ◽  
Qingguo Li
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Continuous Functions ◽  
Logical System ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Syntactic Form ◽  
Scott Continuous ◽  
Continuous Domain

AbstractWe build a logical system named a conjunctive sequent calculus which is a conjunctive fragment of the classical propositional sequent calculus in the sense of proof theory. We prove that a special class of formulae of a consistent conjunctive sequent calculus forms a bounded complete continuous domain without greatest element (for short, a proper BC domain), and each proper BC domain can be obtained in this way. More generally, we present conjunctive consequence relations as morphisms between consistent conjunctive sequent calculi and build a category which is equivalent to that of proper BC domains with Scott-continuous functions. A logical characterization of purely syntactic form for proper BC domains is obtained.

Consistent disjunctive sequent calculi and Scott domains

2021 ◽  
pp. 1-24
Author(s):  
Longchun Wang ◽  
Qingguo Li
Keyword(s):  
Fixed Point ◽  
Propositional Logic ◽  
Continuous Functions ◽  
Consequence Relations ◽  
Sequent Calculi ◽  
Continuous Solutions ◽  
Syntactic Representation ◽  
Scott Continuous ◽  
Recursive Domain Equations ◽  
Standard Domain

Abstract Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

scholarly journals Continuous Domains in Formal Concept Analysis*

2021 ◽  
Vol 179 (3) ◽  
pp. 295-319
Author(s):  
Longchun Wang ◽  
Lankun Guo ◽  
Qingguo Li
Keyword(s):  
Formal Concept Analysis ◽  
Concept Analysis ◽  
Continuous Functions ◽  
Formal Context ◽  
Formal Concept ◽  
Contractive Mappings ◽  
Continuous Domains ◽  
Algebraic Domains ◽  
Scott Continuous ◽  
Continuous Domain

Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.

scholarly journals POSITIVE LOGIC WITH ADJOINT MODALITIES: PROOF THEORY, SEMANTICS, AND REASONING ABOUT INFORMATION

2010 ◽  
Vol 3 (3) ◽  
pp. 351-373 ◽  
Author(s):  
MEHRNOOSH SADRZADEH ◽  
ROY DYCKHOFF
Keyword(s):  
Modal Logic ◽  
Multiagent Systems ◽  
Epistemic Logic ◽  
Proof Theory ◽  
Sequent Calculus ◽  
Dynamic Epistemic Logic ◽  
Algebraic Semantics ◽  
Sequent Calculi ◽  
Closure Operators ◽  
Positive Logic

We consider a simple modal logic whose nonmodal part has conjunction and disjunction as connectives and whose modalities come in adjoint pairs, but are not in general closure operators. Despite absence of negation and implication, and of axioms corresponding to the characteristic axioms of (e.g.) T, S4, and S5, such logics are useful, as shown in previous work by Baltag, Coecke, and the first author, for encoding and reasoning about information and misinformation in multiagent systems. For the propositional-only fragment of such a dynamic epistemic logic, we present an algebraic semantics, using lattices with agent-indexed families of adjoint pairs of operators, and a cut-free sequent calculus. The calculus exploits operators on sequents, in the style of “nested” or “tree-sequent” calculi; cut-admissibility is shown by constructive syntactic methods. The applicability of the logic is illustrated by reasoning about the muddy children puzzle, for which the calculus is augmented with extra rules to express the facts of the muddy children scenario.

scholarly journals A contribution to automated-oriented reasoning about permutability of sequent calculi rules

2013 ◽  
Vol 10 (3) ◽  
pp. 1185-1210 ◽  
Author(s):  
Tatjana Lutovac ◽  
James Harland
Keyword(s):  
Proof Theory ◽  
Sequent Calculus ◽  
Sufficient Conditions ◽  
General Treatment ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Different Parts ◽  
Sequence Rules

Many important results in proof theory for sequent calculus (cut-elimination, completeness and other properties of search strategies, etc) are proved using permutations of sequent rules. The focus of this paper is on the development of systematic and automated-oriented techniques for the analysis of permutability in some sequent calculi. A representation of sequent calculi rules is discussed, which involves greater precision than previous approaches, and allows for correspondingly more precise and more general treatment of permutations. We define necessary and sufficient conditions for the permutation of sequence rules. These conditions are specified as constraints between the multisets that constitute different parts of the sequent rules. The authors extend their previous work in this direction to include some special cases of permutations.

Sequent Calculi for Global Modal Consequence Relations

Studia Logica ◽  
2018 ◽  
Vol 107 (4) ◽  
pp. 613-637
Author(s):  
Minghui Ma ◽  
Jinsheng Chen
Keyword(s):  
Consequence Relations ◽  
Sequent Calculi

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

scholarly journals A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
Juan Carlos Ferrando
Keyword(s):  
Continuous Functions ◽  
Compact Open Topology ◽  
Tychonoff Space ◽  
Bounded Sets

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

scholarly journals The Method of Socratic Proofs Meets Correspondence Analysis

2019 ◽  
Vol 48 (2) ◽  
pp. 99-116
Author(s):  
Dorota Leszczyńska-Jasion ◽  
Yaroslav Petrukhin ◽  
Vasilyi Shangin
Keyword(s):  
Correspondence Analysis ◽  
Boolean Functions ◽  
Propositional Logic ◽  
Sequent Calculus ◽  
Sequent Calculi ◽  
Classical Propositional Logic ◽  
Logic Of Questions ◽  
Proof Systems

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for designing proof systems. In this paper it is used to consider sequent calculi with non-branching (the only exception being the rule of cut), invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions.

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