On the Size Complexity and Decidability of First-Order Progression
Provides theoretical guarantees for the practical applicability of first-order progression in reasoning about actions, addressing a long-standing gap in size complexity analysis.
The paper proves that first-order progression for three action classes (local-effect, normal, acyclic) grows polynomially and remains within decidable fragments like two-variable FO logic or universal theories with constants, ensuring tractability and decidability.
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.