A Distribution Semantics for Probabilistic Term Rewriting

TL;DR

This paper introduces a distribution semantics for probabilistic term rewrite systems (PTRS) that parallels distribution semantics in probabilistic logic programming. It defines worlds as selections of probabilistic rule choices and proves how to compute the probability that a term $s$ reduces to a term $t$ by marginalizing over these worlds, $P(s\to^* t)$. To address combinatorial explosion, it proposes explanations (covering sets of consistent atomic choices) and leverages knowledge compilation (e.g., BDDs) for exact inference and iterative/Monte Carlo methods for approximation. The framework accommodates ground and (via grounding or measures) variable probabilistic rules and extends to conditional rewriting, enabling modeling tasks such as Bayesian networks within the PTRS paradigm. Overall, it provides a rigorous, extensible approach to integrating uncertainty into term rewriting with practical inference strategies and illustrative examples.

Abstract

Probabilistic programming is becoming increasingly popular thanks to its ability to specify problems with a certain degree of uncertainty. In this work, we focus on term rewriting, a well-known computational formalism. In particular, we consider systems that combine traditional rewriting rules with probabilities. Then, we define a novel "distribution semantics" for such systems that can be used to model the probability of reducing a term to some value. We also show how to compute a set of "explanations" for a given reduction, which can be used to compute its probability in a more efficient way. Finally, we illustrate our approach with several examples and outline a couple of extensions that may prove useful to improve the expressive power of probabilistic rewrite systems.