Tractability of Theory Patching

TL;DR

This paper investigates the computational complexity of theory patching, identifying conditions under which revising parts of a logical theory to fit training data is computationally feasible.

Contribution

It introduces the concept of stability in theories and characterizes when theory patching is tractable based on monotonicity and independence conditions.

Findings

Theory patching is tractable under certain conditions.

Stability of theory components determines patching complexity.

Methods to assess soundness and completeness of patching algorithms.

Abstract

In this paper we consider the problem of `theory patching', in which we are given a domain theory, some of whose components are indicated to be possibly flawed, and a set of labeled training examples for the domain concept. The theory patching problem is to revise only the indicated components of the theory, such that the resulting theory correctly classifies all the training examples. Theory patching is thus a type of theory revision in which revisions are made to individual components of the theory. Our concern in this paper is to determine for which classes of logical domain theories the theory patching problem is tractable. We consider both propositional and first-order domain theories, and show that the theory patching problem is equivalent to that of determining what information contained in a theory is `stable' regardless of what revisions might be performed to the theory. We show that determining stability is tractable if the input theory satisfies two conditions: that revisions to each theory component have monotonic effects on the classification of examples, and that theory components act independently in the classification of examples in the theory. We also show how the concepts introduced can be used to determine the soundness and completeness of particular theory patching algorithms.