Soundness and Completeness Results for LEA and Probability Semantics

2021 ◽  
pp. 198-216
Author(s):  
Eoin Moore
Keyword(s):  
Soundness And Completeness ◽  
Probability Semantics

scholarly journals Kripke Semantics for Intersection Formulas

2021 ◽  
Vol 22 (3) ◽  
pp. 1-16
Author(s):  
Andrej Dudenhefner ◽  
Paweł Urzyczyn
Keyword(s):  
Kripke Semantics ◽  
Type Assignment ◽  
Soundness And Completeness

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

An Automata-Theoretic Decision Procedure for Propositional Temporal Logic with Since and Until1

1992 ◽  
Vol 17 (3) ◽  
pp. 271-282
Author(s):  
Y.S. Ramakrishna ◽  
L.E. Moser ◽  
L.K. Dillon ◽  
P.M. Melliar-Smith ◽  
G. Kutty
Keyword(s):  
Temporal Logic ◽  
Decision Procedure ◽  
Linear Time ◽  
Computer Systems ◽  
Hierarchical Abstraction ◽  
Soundness And Completeness

We present an automata-theoretic decision procedure for Since/Until Temporal Logic (SUTL), a linear-time propositional temporal logic with strong non-strict since and until operators. The logic, which is intended for specifying and reasoning about computer systems, employs neither next nor previous operators. Such operators obstruct the use of hierarchical abstraction and refinement and make reasoning about concurrency difficult. A proof of the soundness and completeness of the decision procedure is given, and its complexity is analyzed.

scholarly journals Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 115 ◽  
Author(s):  
Joanna Golińska-Pilarek ◽  
Magdalena Welle
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Classical Propositional Logic ◽  
Tableau System ◽  
Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

scholarly journals On Polymorphic Sessions and Functions

2021 ◽  
Vol 43 (2) ◽  
pp. 1-55
Author(s):  
Bernardo Toninho ◽  
Nobuko Yoshida
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Linear Formulation ◽  
Session Types ◽  
Logical Foundation ◽  
Π Calculus ◽  
Soundness And Completeness ◽  
System F ◽  
Algebraic Representations

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation.

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.

Modulated fibring and the collapsing problem

2002 ◽  
Vol 67 (4) ◽  
pp. 1541-1569 ◽  
Author(s):  
Cristina Sernadas ◽  
João Rasga ◽  
Walter A. Carnielli
Keyword(s):  
Mathematical Logic ◽  
Soundness And Completeness ◽  
The Given ◽  
Combining Logics

AbstractFibring is recognized as one of the main mechanisms in combining logics, with great significance in the theory and applications of mathematical logic. However, an open challenge to fibring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that fibring imposes unwanted interconnections between the given logics. Modulated fibring allows a finer control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with fibring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem.

The Boolean Algebra Logic: The Soundness and Completeness Theorem

2017 ◽  
Author(s):  
Chen Bo ◽  
Zhang Xingyou ◽  
Zhang Pengfei ◽  
Cao Cong ◽  
Liu Wenxue ◽  
...  
Keyword(s):  
Boolean Algebra ◽  
Completeness Theorem ◽  
Algebra Logic ◽  
Soundness And Completeness

Multiplicative conjunction and an algebraic meaning of contraction and weakening

1998 ◽  
Vol 63 (3) ◽  
pp. 831-859 ◽  
Author(s):  
A. Avron
Keyword(s):  
Classical Logic ◽  
Linear Logic ◽  
Substructural Logics ◽  
Interesting Property ◽  
Relevance Logic ◽  
Proper Subset ◽  
Algebraic Structures ◽  
Elimination Rule ◽  
First Case ◽  
Soundness And Completeness

AbstractWe show that the elimination rule for the multiplicative (or intensional) conjunction Λ is admissible in many important multiplicative substructural logics. These include LLm (the multiplicative fragment of Linear Logic) and RMIm (the system obtained from LLm by adding the contraction axiom and its converse, the mingle axiom.) An exception is Rm (the intensional fragment of the relevance logic R, which is LLm together with the contraction axiom). Let SLLm and SRm be, respectively, the systems which are obtained from LLm and Rm by adding this rule as a new rule of inference. The set of theorems of SRm is a proper extension of that of Rm, but a proper subset of the set of theorems of RMIm. Hence it still has the variable-sharing property. SRm has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLLm relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLLm corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCKm, in the second - the system SRm). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts.

Update Semantics in Communicated Information System

2011 ◽  
Vol 403-408 ◽  
pp. 1460-1465
Author(s):  
Guang Ming Chen ◽  
Xiao Wu Li
Keyword(s):  
Information Systems ◽  
Propositional Logic ◽  
General Purpose ◽  
Main Task ◽  
Information Communication ◽  
Classical Propositional Logic ◽  
Update Semantics ◽  
Essential Form ◽  
Soundness And Completeness ◽  
Multi Agents

An approach, which is called Communicated Information Systems, is introduced to describe the information available in a number of agents and specify the information communication among the agents. The systems are extensions of classical propositional logic in multi-agents context, providing with us a way by which not only the agent’s own information, but the information from other agents may be applied to agent’s reasoning as well. Communication rules, which are defined in the most essential form, can be regarded as the base to characterize some interesting cognitive proporties of agents. Since the corresponding communication rules can be chosen for different applications, the approach is general purpose one. The other main task is that the soundness and completeness of the Communicated Information Systems for the update semantics have been proved in the paper.

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