On the Size Complexity and Decidability of First-Order Progression

TL;DR

This paper analyzes the size and decidability of first-order progression in knowledge bases, showing polynomial growth and preservation of decidability in certain logical fragments.

Contribution

It provides a systematic analysis of the size complexity of first-order progression for specific action classes within the Situation Calculus framework.

Findings

First-order progression grows polynomially under reasonable assumptions.

Progression remains within decidable fragments like two-variable logic.

Ensures practical applicability of reasoning about actions.

Abstract

Progression, the task of updating a knowledge base to reflect action effects, generally requires second-order logic. Identifying first-order special cases, by restricting either the knowledge base or action effects, has long been a central topic in reasoning about actions. It is known that local-effect, normal, and acyclic actions, three increasingly expressive classes, admit first-order progression. However, a systematic analysis of the size of such progressions, crucial for practical applications, has been missing. In this paper, using the framework of Situation Calculus, we show that under reasonable assumptions, first-order progression for these action classes grows only polynomially. Moreover, we show that when the KB belongs to decidable fragments such as two-variable first-order logic or universal theories with constants, the progression remains within the same fragment, ensuring decidability and practical applicability.