Extensions of Simple Conceptual Graphs: the Complexity of Rules and Constraints

TL;DR

This paper explores extensions of simple conceptual graphs with rules and constraints, analyzing their semantics, relationships with first-order logic, and the complexity and decidability of reasoning tasks, including conditions for polynomial hierarchy inclusion.

Contribution

It introduces formal models of extended conceptual graphs with rules and constraints, analyzing their semantics, complexity, and decidability, and identifies conditions for polynomial hierarchy classification.

Findings

Reasoning with rules is generally undecidable.

Certain restrictions on rules and constraints place reasoning within the polynomial hierarchy.

The paper extends previous results with new complexity analyses and formal definitions.

Abstract

Simple conceptual graphs are considered as the kernel of most knowledge representation formalisms built upon Sowa's model. Reasoning in this model can be expressed by a graph homomorphism called projection, whose semantics is usually given in terms of positive, conjunctive, existential FOL. We present here a family of extensions of this model, based on rules and constraints, keeping graph homomorphism as the basic operation. We focus on the formal definitions of the different models obtained, including their operational semantics and relationships with FOL, and we analyze the decidability and complexity of the associated problems (consistency and deduction). As soon as rules are involved in reasonings, these problems are not decidable, but we exhibit a condition under which they fall in the polynomial hierarchy. These results extend and complete the ones already published by the authors. Moreover we systematically study the complexity of some particular cases obtained by restricting the form of constraints and/or rules.