Synthesising Recursive Functions for First-Order Model Counting: Challenges, Progress, and Conjectures
This work addresses a bottleneck in exact FOMC for computational logic and AI, representing a novel method rather than an incremental improvement.
The paper tackles the limitation of first-order model counting (FOMC) algorithms in expressing recursive computations by generalizing circuits to directed graphs with cycles, adapting the ForcLift algorithm to handle such graphs and introducing new compilation rules for recursive function calls, enabling efficient solutions to previously unsolvable counting problems.
First-order model counting (FOMC) is a computational problem that asks to count the models of a sentence in finite-domain first-order logic. In this paper, we argue that the capabilities of FOMC algorithms to date are limited by their inability to express many types of recursive computations. To enable such computations, we relax the restrictions that typically accompany domain recursion and generalise the circuits used to express a solution to an FOMC problem to directed graphs that may contain cycles. To this end, we adapt the most well-established (weighted) FOMC algorithm ForcLift to work with such graphs and introduce new compilation rules that can create cycle-inducing edges that encode recursive function calls. These improvements allow the algorithm to find efficient solutions to counting problems that were previously beyond its reach, including those that cannot be solved efficiently by any other exact FOMC algorithm. We end with a few conjectures on what classes of instances could be domain-liftable as a result.