Axiomatic Aspects of Default Inference

TL;DR

This paper explores axioms for nonmonotonic reasoning based on default structures, establishing representation theorems that connect axiomatic systems with extension-based semantics, and demonstrating the sufficiency of shallow extensions.

Contribution

It provides a formal characterization of nonmonotonic consequence relations via axioms and shows their equivalence to default-structure-based reasoning, with a focus on shallow extensions.

Findings

Axioms for nonmonotonic consequence relations are characterized and linked to default structures.

Representation theorems establish a one-to-one correspondence between axiomatic systems and default-structure semantics.

Shallow extensions, requiring at most three iterations, are sufficient for constructing extensions.

Abstract

This paper studies axioms for nonmonotonic consequences from a semantics-based point of view, focusing on a class of mathematical structures for reasoning about partial information without a predefined syntax/logic. This structure is called a default structure. We study axioms for the nonmonotonic consequence relation derived from extensions as in Reiter's default logic, using skeptical reasoning, but extensions are now used for the construction of possible worlds in a default information structure. In previous work we showed that skeptical reasoning arising from default-extensions obeys a well-behaved set of axioms including the axiom of cautious cut. We show here that, remarkably, the converse is also true: any consequence relation obeying this set of axioms can be represented as one constructed from skeptical reasoning. We provide representation theorems to relate axioms for nonmonotonic consequence relation and properties about extensions, and provide one-to-one correspondence between nonmonotonic systems which satisfies the law of cautious monotony and default structures with unique extensions. Our results give a theoretical justification for a set of basic rules governing the update of nonmonotonic knowledge bases, demonstrating the derivation of them from the more concrete and primitive construction of extensions. It is also striking to note that proofs of the representation theorems show that only shallow extensions are necessary, in the sense that the number of iterations needed to achieve an extension is at most three. All of these developments are made possible by taking a more liberal view of consistency: consistency is a user defined predicate, satisfying some basic properties.