Space Efficiency of Propositional Knowledge Representation Formalisms

TL;DR

This paper analyzes the space efficiency of propositional knowledge representation formalisms, introducing measures and classes to compare their ability to compactly represent models and theorems, especially in nonmonotonic reasoning.

Contribution

It introduces formal measures and classes for comparing the space efficiency of PKR formalisms, linking these to their relative ability to represent knowledge compactly.

Findings

Formalisms with the same time complexity can differ in space efficiency.

New compactness measures effectively classify PKR formalisms.

Nonmonotonic reasoning formalisms vary significantly in space efficiency.

Abstract

We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency of a formalism F in representing a certain piece of knowledge A, is the size of the shortest formula of F that represents A. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space efficiency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms with the same time complexity do not necessarily belong to the same space efficiency class.