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A Compositional Typed Higher-Order Logic with Definitions

Authors Ingmar Dasseville, Matthias van der Hallen, Bart Bogaerts, Gerda Janssens, Marc Denecker

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Ingmar Dasseville
Matthias van der Hallen
Bart Bogaerts
Gerda Janssens
Marc Denecker

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Ingmar Dasseville, Matthias van der Hallen, Bart Bogaerts, Gerda Janssens, and Marc Denecker. A Compositional Typed Higher-Order Logic with Definitions. In Technical Communications of the 32nd International Conference on Logic Programming (ICLP 2016). Open Access Series in Informatics (OASIcs), Volume 52, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/OASIcs.ICLP.2016.14

Abstract

Expressive KR languages are built by integrating different language constructs, or extending a language with new language constructs. This process is difficult if non-truth-functional or non-monotonic constructs are involved. What is needed is a compositional principle.

This paper presents a compositional principle for defining logics by modular composition of logical constructs, and applies it to build a higher order logic integrating typed lambda calculus and rule sets under a well-founded or stable semantics. Logical constructs are formalized as triples of a syntactical rule, a semantical rule, and a typing rule. The paper describes how syntax, typing and semantics of the logic are composed from the set of its language constructs. The base semantical concept is the infon: mappings from structures to values in these structures. Semantical operators of language constructs operate on infons and allow to construct the infons of compound expressions from the infons of its subexpressions. This conforms to Frege's principle of compositionality.

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Keywords
  • Logic
  • Semantics
  • Compositionality

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