Outlier detection in default logics: the tractability/intractability frontier

TL;DR

This paper explores the computational boundaries of detecting outliers in default logics, identifying tractable cases, and providing algorithms within specific logical fragments, thereby advancing understanding of complexity and expressive power.

Contribution

It characterizes the tractability frontier for outlier detection in default logics, introduces a polynomial algorithm for strong outliers, and demonstrates that simple logic fragments can express all NL queries.

Findings

Polynomial time algorithm for enumerating strong outliers

Identification of tractable cases in default logic fragments

Normal unary theories can express all NL queries

Abstract

In default theories, outliers denote sets of literals featuring unexpected properties. In previous papers, we have defined outliers in default logics and investigated their formal properties. Specifically, we have looked into the computational complexity of outlier detection problems and proved that while they are generally intractable, interesting tractable cases can be singled out. Following those results, we study here the tractability frontier in outlier detection problems, by analyzing it with respect to (i) the considered outlier detection problem, (ii) the reference default logic fragment, and (iii) the adopted notion of outlier. As for point (i), we shall consider three problems of increasing complexity, called Outlier-Witness Recognition, Outlier Recognition and Outlier Existence, respectively. As for point (ii), as we look for conditions under which outlier detection can be done efficiently, attention will be limited to subsets of Disjunction-free propositional default theories. As for point (iii), we shall refer to both the notion of outlier of [ABP08] and a new and more restrictive one, called strong outlier. After complexity results, we present a polynomial time algorithm for enumerating all strong outliers of bounded size in an quasi-acyclic normal unary default theory. Some of our tractability results rely on the Incremental Lemma that provides conditions for a deafult logic fragment to have a monotonic behavior. Finally, in order to show that the simple fragments of DL we deal with are still rich enough to solve interesting problems and, therefore, the tractability results that we prove are interesting not only on the mere theoretical side, insights into the expressive capabilities of these fragments are provided, by showing that normal unary theories express all NL queries, hereby indirectly answering a question raised by Kautz and Selman.