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Syntax and semantics of multi-adjoint normal logic programming

M. Eugenia Cornejo, David Lobo, Jesús Medina

TL;DR

The paper extends multi-adjoint logic programming with a negation operator to form multi-adjoint normal logic programming (MANLP) and develops its syntax and semantics. It defines an immediate consequence operator and a stable-model semantics via reducts, then proves existence of stable models for MANLPs on convex compact carriers using Schauder fixed-point theory, provided the connectives, negation, and aggregators are continuous. It further establishes a uniqueness result for MANLPs defined on the interval lattice C([0,1]) under a contractivity condition, obtainable by Banach’s fixed-point theorem. Together, these results broaden non-classical logic programming to include default negation with guarantees of solvability and determinism in relevant semantic settings, enabling application to a range of generalized annotated, fuzzy, and possibilistic frameworks.

Abstract

Multi-adjoint logic programming is a general framework with interesting features, which involves other positive logic programming frameworks such as monotonic and residuated logic programming, generalized annotated logic programs, fuzzy logic programming and possibilistic logic programming. One of the most interesting extensions of this framework is the possibility of considering a negation operator in the logic programs, which will improve its flexibility and the range of real applications. This paper introduces multi-adjoint normal logic programming, which is an extension of multi-adjoint logic programming including a negation operator in the underlying lattice. Beside the introduction of the syntax and semantics of this paradigm, we will provide sufficient conditions for the existence of stable models defined on a convex compact set of a euclidean space. Finally, we will consider a particular algebraic structure in which sufficient conditions can be given in order to ensure the unicity of stable models of multi-adjoint normal logic programs.

Syntax and semantics of multi-adjoint normal logic programming

TL;DR

The paper extends multi-adjoint logic programming with a negation operator to form multi-adjoint normal logic programming (MANLP) and develops its syntax and semantics. It defines an immediate consequence operator and a stable-model semantics via reducts, then proves existence of stable models for MANLPs on convex compact carriers using Schauder fixed-point theory, provided the connectives, negation, and aggregators are continuous. It further establishes a uniqueness result for MANLPs defined on the interval lattice C([0,1]) under a contractivity condition, obtainable by Banach’s fixed-point theorem. Together, these results broaden non-classical logic programming to include default negation with guarantees of solvability and determinism in relevant semantic settings, enabling application to a range of generalized annotated, fuzzy, and possibilistic frameworks.

Abstract

Multi-adjoint logic programming is a general framework with interesting features, which involves other positive logic programming frameworks such as monotonic and residuated logic programming, generalized annotated logic programs, fuzzy logic programming and possibilistic logic programming. One of the most interesting extensions of this framework is the possibility of considering a negation operator in the logic programs, which will improve its flexibility and the range of real applications. This paper introduces multi-adjoint normal logic programming, which is an extension of multi-adjoint logic programming including a negation operator in the underlying lattice. Beside the introduction of the syntax and semantics of this paradigm, we will provide sufficient conditions for the existence of stable models defined on a convex compact set of a euclidean space. Finally, we will consider a particular algebraic structure in which sufficient conditions can be given in order to ensure the unicity of stable models of multi-adjoint normal logic programs.
Paper Structure (13 sections, 12 theorems, 68 equations)

This paper contains 13 sections, 12 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Syntax of the propositional language
  4. Semantics of the propositional language
  5. Multi-adjoint logic programs
  6. Some notions of algebraic topology
  7. On the syntax and semantics of multi-adjoint normal logic programs
  8. Immediate consequence operator
  9. Stable models
  10. On the existence and unicity of stable models
  11. Existence of stable models in convex compact sets
  12. Unicity of stable models in MANLPs defined on $\mathcal{C}([0,1])$
  13. Conclusions and future work

Key Result

Proposition 1

If $(L,\preceq)$ is a complete lattice, then $(\cal{I}_\mathfrak{L},\sqsubseteq)$ is a complete lattice where the least interpretation $\Delta$ applies every propositional symbol to the bottom element of $L$, and the greatest interpretation $\nabla$ applies every propositional symbol to the top elem

Theorems & Definitions (37)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • definition 7
  • Proposition 1: lpnmr01
  • definition 8
  • definition 9
  • ...and 27 more