Natural deduction proof theory for logic programming

1993 ◽  
pp. 265-281 ◽  
Author(s):  
Seppo Keronen
Keyword(s):  
Logic Programming ◽  
Proof Theory ◽  
Natural Deduction

Some Fixpoint Techniques in Algebraic Structures and Applications to Computer Science

1987 ◽  
Vol 10 (4) ◽  
pp. 387-413
Author(s):  
Irène Guessarian
Keyword(s):  
Logic Programming ◽  
Computer Science ◽  
Proof Theory ◽  
Algebraic Structures ◽  
Data Bases

This paper recalls some fixpoint theorems in ordered algebraic structures and surveys some ways in which these theorems are applied in computer science. We describe via examples three main types of applications: in semantics and proof theory, in logic programming and in deductive data bases.

scholarly journals Hybrid Deduction–Refutation Systems

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Valentin Goranko
Keyword(s):  
Propositional Logic ◽  
Proof Theory ◽  
Natural Deduction ◽  
Basic Theory ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Deductive Systems ◽  
Refutation Systems ◽  
Soundness And Completeness

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which I show soundness and completeness for both deductions and refutations.

Logic as Programming

1992 ◽  
Vol 17 (4) ◽  
pp. 285-317
Author(s):  
Johan Van Benthem
Keyword(s):  
Fixed Point ◽  
Dynamic Analysis ◽  
Logic Programming ◽  
Proof Theory ◽  
Type Inference ◽  
Horn Clause ◽  
Structural Rules ◽  
Inference Engines ◽  
Fixed Point Semantics ◽  
General Dynamic

Starting from a general dynamic analysis of reasoning and programming, we develop two main dynamic perspectives upon logic programming. First, the standard fixed point semantics for Horn clause programs naturally supports imperative programming styles. Next, we provide axiomatizations for Prolog-type inference engines using calculi of sequents employing modified versions of standard structural rules such as monotonicity or permutation. Finally, we discuss the implications of all this for a broader enterprise of ‘abstract proof theory’.

On the proof theory of the intermediate logic MH

1986 ◽  
Vol 51 (3) ◽  
pp. 626-647 ◽  
Author(s):  
Jonathan P. Seldin
Keyword(s):  
Proof Theory ◽  
Natural Deduction ◽  
Interpolation Theorem ◽  
Intermediate Logic

AbstractA natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

scholarly journals Constraint Logic Programming with Hereditary Harrop formulas

2001 ◽  
Vol 1 (4) ◽  
pp. 409-445 ◽  
Author(s):  
JAVIER LEACH ◽  
SUSANA NIEVA ◽  
MARIO RODRÍGUEZ-ARTALEJO
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Proof Theory ◽  
Completeness Theorem ◽  
Constraint Logic Programming ◽  
Strong Completeness ◽  
Logic Programming Language ◽  
Constraint System ◽  
Soundness And Completeness ◽  
Abstract Formulation

Constraint Logic Programming (CLP) and Hereditary Harrop formulas (HH) are two well known ways to enhance the expressivity of Horn clauses. In this paper, we present a novel combination of these two approaches. We show how to enrich the syntax and proof theory of HH with the help of a given constraint system, in such a way that the key property of HH as a logic programming language (namely, the existence of uniform proofs) is preserved. We also present a procedure for goal solving, showing its soundness and completeness for computing answer constraints. As a consequence of this result, we obtain a new strong completeness theorem for CLP that avoids the need to build disjunctions of computed answers, as well as a more abstract formulation of a known completeness theorem for HH.

scholarly journals Proof Search and Certificates for Evidential Transactions

2021 ◽  
pp. 234-251
Author(s):  
Vivek Nigam ◽  
Giselle Reis ◽  
Samar Rahmouni ◽  
Harald Ruess
Keyword(s):  
Logic Programming ◽  
Proof Theory ◽  
Knowledge Bases ◽  
Foreign Country ◽  
Proof System ◽  
Complete Proof ◽  
Proof Search ◽  
Logical Data

AbstractAttestation logics have been used for specifying systems with policies involving different principals. Cyberlogic is an attestation logic used for the specification of Evidential Transactions (ETs). In such transactions, evidence has to be provided supporting its validity with respect to given policies. For example, visa applicants may be required to demonstrate that they have sufficient funds to visit a foreign country. Such evidence can be expressed as a Cyberlogic proof, possibly combined with non-logical data (e.g., a digitally signed document). A key issue is how to construct and communicate such evidence/proofs. It turns out that attestation modalities are challenging to use established proof-theoretic methods such as focusing. Our first contribution is the refinement of Cyberlogic proof theory with knowledge operators which can be used to represent knowledge bases local to one or more principals. Our second contribution is the identification of an executable fragment of Cyberlogic, called Cyberlogic programs, enabling the specification of ETs. Our third contribution is a sound and complete proof system for Cyberlogic programs enabling proof search similar to search in logic programming. Our final contribution is a proof certificate format for Cyberlogic programs inspired by Foundational Proof Certificates as a means to communicate evidence and check its validity.

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