Proof-theory for extensions of Logic Programming

1996 ◽  
pp. 161-175
Author(s):  
Catholijn M. Jonker
Keyword(s):  
Logic Programming ◽  
Proof Theory

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.

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’.

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.

scholarly journals A Survey of the Proof-Theoretic Foundations of Logic Programming

2021 ◽  
pp. 1-46
Author(s):  
DALE MILLER
Keyword(s):  
Logic Programming ◽  
Programming Languages ◽  
Proof Theory ◽  
Linear Logic ◽  
Higher Order ◽  
Data Types ◽  
Abstract Data ◽  
Logic Programming Languages ◽  
Extended Logic Programming ◽  
High Level

Abstract Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use of structural proof theory as an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.

Resolution and the Origins of Structural Reasoning: Early Proof-Theoretic Ideas of Hertz and Gentzen

2002 ◽  
Vol 8 (2) ◽  
pp. 246-265 ◽  
Author(s):  
Peter Schroeder-Heister
Keyword(s):  
Normal Form ◽  
Logic Programming ◽  
Proof Theory ◽  
Structural Rules ◽  
General Proof ◽  
Early Proof ◽  
Sld Resolution ◽  
Structural Reasoning ◽  
Structural Proof Theory

AbstractIn the 1920s, Paul Hertz (1881–1940) developed certain calculi based on structural rules only and established normal form results for proofs. It is shown that he anticipated important techniques and results of general proof theory as well as of resolution theory, if the latter is regarded as a part of structural proof theory. Furthermore, it is shown that Gentzen, in his first paper of 1933, which heavily draws on Hertz, proves a normal form result which corresponds to the completeness of propositional SLD-resolution in logic programming.

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