On the Progression of Situation Calculus Universal Theories with Constants

The progression of action theories is an important problem in knowledge representation. Progression is second-order definable and known to be first-order definable and effectively computable for restricted classes of theories. Motivated by the fact that universal theories with constants (UTCs) are expressive and natural theories whose satisfiability is decidable, in this paper we provide a thorough study of the progression of situation calculus UTCs. First, we prove that progression of a (possibly infinite) UTC is always first-order definable and results in a UTC. Though first-order definable, we show that the progression of a UTC may be infeasible, that is, it may result in an infinite UTC that is not equivalent to any finite set of first-order sentences. We then show that deciding whether %or not there is a feasible progression of a UTC is undecidable. Moreover, we show that deciding whether %or not a sentence (in an expressive fragment of first-order logic) is in the progression of a UTC is CONEXPTIME-complete, and that there exists a family of UTCs for which the size of every feasible progression grows exponentially. Finally, we discuss resolution-based approaches to compute the progression of a UTC. This comprehensive analysis contributes to a better understanding of progression in action theories, both in terms of feasibility and difficulty.