Probabilistic Rewriting: Normalization, Termination, and Unique Normal Forms

Abstract

While a mature body of work supports the study of rewriting systems, even infinitary ones, abstract tools for Probabilistic Rewriting are still limited. Here, we investigate questions such as uniqueness of the result (unique limit distribution) and normalizing strategies (is there a strategy to find a result with *greatest probability* ?). The goal is to have tools to analyze the operational properties of probabilistic calculi such as probabilistic lambda-calculi, whose evaluation is also non-deterministic, where non-determinism arises from a choice between several redexes. We investigate how the asymptotic behavior of different rewrite sequences starting from the same term compare w.r.t. normal forms, propose a robust analogue of the notion of "unique normal form", and we develop methods to study and compare strategies. Our approach is that of Abstract Rewrite Systems, i.e. we search for general properties of probabilistic rewriting, which hold independently of the specific nature of the objects.