Some Proof Theory of First Order Logic Programming

1997 ◽  
pp. 201-228
Author(s):  
Gerhard Jäger
Keyword(s):  
Logic Programming ◽  
Proof Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

Uniqueness, definability and interpolation

1988 ◽  
Vol 53 (2) ◽  
pp. 554-570 ◽  
Author(s):  
Kosta Došen ◽  
Peter Schroeder-Heister
Keyword(s):  
Proof Theory ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
General Notion ◽  
Structural Part ◽  
First Order ◽  
First Case ◽  
Additional Constant ◽  
Special Case

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, Tα a set of sentences, α* an additional constant of the same syntactical category as α and Tα, a copy of Tα with α* instead of α, then for implicit definability of α in Tα one has, in the case of predicate constants, to derive α(x1,…,xn) ↔ α*(x1,…,xn) from Tα ∪ Tα*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata Sα and derivability from instances of Sα ∪ Sα* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

Logics for the Semantic Web

2011 ◽  
pp. 24-43
Author(s):  
J. Bruijn
Keyword(s):  
Semantic Web ◽  
Logic Programming ◽  
Description Logic ◽  
Description Logics ◽  
Order Logic ◽  
First Order Logic ◽  
Horn Logic ◽  
First Order ◽  
Logical Language ◽  
Rule Languages

This chapter introduces a number of formal logical languages which form the backbone of the Semantic Web. They are used for the representation of both ontologies and rules. The basis for all languages presented in this chapter is the classical first-order logic. Description logics is a family of languages which represent subsets of first-order logic. Expressive description logic languages form the basis for popular ontology languages on the Semantic Web. Logic programming is based on a subset of first-order logic, namely Horn logic, but uses a slightly different semantics and can be extended with non-monotonic negation. Many Semantic Web reasoners are based on logic programming principles and rule languages for the Semantic Web based on logic programming are an ongoing discussion. Frame Logic allows object-oriented style (frame-based) modeling in a logical language. RuleML is an XML-based syntax consisting of different sublanguages for the exchange of specifications in different logical languages over the Web.

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

2011 ◽  
Vol 76 (2) ◽  
pp. 673-699 ◽  
Author(s):  
Michael Gabbay
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Order Logic ◽  
First Order Logic ◽  
Cut Elimination ◽  
Logic Programming Language ◽  
First Order ◽  
Functional Programming Language ◽  
Sequent System

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

scholarly journals GEOMETRISATION OF FIRST-ORDER LOGIC

2015 ◽  
Vol 21 (2) ◽  
pp. 123-163 ◽  
Author(s):  
ROY DYCKHOFF ◽  
SARA NEGRI
Keyword(s):  
Proof Theory ◽  
Normal Forms ◽  
Order Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Conservative Extension ◽  
First Order ◽  
First Order Theory ◽  
Intermediate Logics ◽  
Normal Form Theorem

AbstractThat every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem’s argument from 1920 for his “Normal Form” theorem. “Geometric” being the infinitary version of “coherent”, it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms.

scholarly journals Compositional Neural Logic Programming

2021 ◽  
Author(s):  
Son N. Tran
Keyword(s):  
Neural Networks ◽  
Logic Programming ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Forward Chaining ◽  
Symbolic Reasoning ◽  
New Knowledge ◽  
Neural Logic

This paper introduces Compositional Neural Logic Programming (CNLP), a framework that integrates neural networks and logic programming for symbolic and sub-symbolic reasoning. We adopt the idea of compositional neural networks to represent first-order logic predicates and rules. A voting backward-forward chaining algorithm is proposed for inference with both symbolic and sub-symbolic variables in an argument-retrieval style. The framework is highly flexible in that it can be constructed incrementally with new knowledge, and it also supports batch reasoning in certain cases. In the experiments, we demonstrate the advantages of CNLP in discriminative tasks and generative tasks.

A First Course in Logic

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Proof Theory ◽  
Order Logic ◽  
First Order Logic ◽  
University Of Maryland ◽  
First Order ◽  
Science Philosophy ◽  
The University ◽  
Second Order Logic ◽  
Theory And Model

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.

Visual And Diagrammatic Languages

2011 ◽  
pp. 22-50
Author(s):  
Bernd Meyer ◽  
Paolo Bottoni
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Linear Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Visual Languages ◽  
New Approach ◽  
Logic Programming Language ◽  
First Order ◽  
Complex Forms

In this paper we investigate a new approach to formalizing interpretation of and reasoning with visual languages based on linear logic. We argue that an approach based on logic makes it possible to deal with different computational tasks in the usage of visual notations, from parsing and animation to reasoning about diagrams. However, classical first order logic, being monotonic, is not a suitable basis for such an approach. The paper therefore explores linear logic as an alternative. We demonstrate how parsing corresponds to linear proofs and prove the soundness and correctness of this mapping. As our mapping of grammars is into a subset of a linear logic programming language, we also demonstrate how multi-dimensional parsing can be understood as automated linear deduction. We proceed to discuss how the same framework can be used as the foundation of more complex forms of reasoning with and about diagrams.

On the Proof-Theory of two Formalisations of Modal First-Order Logic

Studia Logica ◽  
2010 ◽  
Vol 96 (3) ◽  
pp. 349-373
Author(s):  
Yehuda Schwartz ◽  
George Tourlakis
Keyword(s):  
Proof Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

Proof Procedures for Logic Programming

1998 ◽  
Author(s):  
Donald W. Loveland ◽  
Gopalan Nadathur
Keyword(s):  
Logic Programming ◽  
Theorem Proving ◽  
Automated Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Proof Procedure ◽  
Sld Resolution ◽  
Work Done ◽  
Resolution Proof

A proof procedure is an algorithm (technically, a semi-decision procedure) which identifies a formula as valid (or unsatisfiable) when appropriate, and may not terminate when the formula is invalid (satisfiable). Since a proof procedure concerns a logic the procedure takes a special form, superimposing a search strategy on an inference calculus. We will consider a certain collection of proof procedures in the light of an inference calculus format that abstracts the concept of logic programming. This formulation allows us to look beyond SLD-resolution, the proof procedure that underlies Prolog, to generalizations and extensions that retain an essence of logic programming structure. The inference structure used in the formulation of the logic programming concept and first realization, Prolog, evolved from the work done in the subdiscipline called automated theorem proving. While many proof procedures have been developed within this subdiscipline, some of which appear in Volume 1 of this handbook, we will present a narrow selection, namely the proof procedures which are clearly ancestors of the first proof procedure associated with logic programming, SLD-resolution. Extensive treatment of proof procedures for automated theorem proving appear in Bibel [Bibel, 1982], Chang and Lee [Chang and Lee, 1973] and Loveland [Loveland, 1978]. Although the consideration of proof procedures for automated theorem proving began about 1958 we begin our overview with the introduction of the resolution proof procedure by Robinson in 1965. We then review the linear resolution procedures, model elimination and SL-resolution procedures. Our exclusion of other proof procedures from consideration here is due to our focus, not because other procedures are less important historically or for general use within automated or semi-automated theorem process. After a review of the general resolution proof procedure, we consider the linear refinement for resolution and then further restrict the procedure format to linear input resolution. Here we are no longer capable of treating full first-order logic, but have forced ourselves to address a smaller domain, in essence the renameable Horn clause formulas. By leaving the resolution format, indeed leaving traditional formula representation, we see there exists a linear input procedure for all of first-order logic.

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