On the Size Complexity and Decidability of First-Order Progression

2026-05-12

AI Summary

This study addresses the problem of updating knowledge bases after actions in first-order logic, focusing on three classes of actions: those with local effects, normal actions, and acyclic actions. Within the situation calculus framework, the work provides the first systematic proof that the first-order progression of such actions exhibits only polynomial growth in size. It further establishes the closure of this progression within decidable fragments of first-order logic, including binary first-order logic and universal theories with constants. By delineating the precise complexity boundaries of progression, this research not only clarifies fundamental theoretical limits but also offers crucial guarantees and efficiency justifications for the practical deployment of knowledge representation and reasoning systems.

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.

Problem

• first-order progression
• size complexity
• decidability
• Situation Calculus
• knowledge base update

Innovation

• first-order progression
• size complexity
• decidability
• Situation Calculus
• knowledge base update