On Probabilistic Term Rewriting

Abstract

We study the termination problem for probabilistic term rewrite systems. We prove that the interpretation method is sound and complete for a strengthening of positive almost sure termination, when abstract reduction systems and term rewrite systems are considered. Two instances of the interpretation method-polynomial and matrix interpretations-are analyzed and shown to capture interesting and nontrivial examples when automated. We capture probabilistic computation in a novel way by means of multidistribution reduction sequences, thus accounting for both the nondeterminism in the choice of the redex and the probabilism intrinsic in firing each rule.

On randomised strategies in the λ-calculus

“…The first preliminary concept we need is that of a distribution. Please notice that if (S, →) is an ARS and P is a randomised reduction strategy for it, then (S, P) can be seen as a fully probabilistic abstract reduction system (FPARS), namely a probabilistic abstract reduction system [2] whose dynamics is purely probabilistic, without any nondeterminism. In the following, we will study randomised strategies as FPARS.…”

“…they are (partial) functions on (possibly shared representations of) terms. There is however some work on probabilistic term rewriting systems [4,10,2], in particular regarding termination, and about randomised strategies in the abstract [5]. What would happen if the redexes to reduce were picked according to some probability distribution?…”

Type-Based Complexity Analysis of Probabilistic Functional Programs

“…To reason about expected runtimes, our type system assigns to every typeable term t a weight. Slightly simplifying the exposition, typing judgments have the form Φ; Γ R t : µ , denoting that under the logical context Φ and typing context Γ, the term t receives the DDT µ and a weight of R. Inspired by Lyapunov or probabilistic ranking functions [15], also referred to as ranking supermartingales [37], the type system ensures that weights reduce in expectation along reductions, and consequently, relate tightly to expected runtimes. With this intuition in mind the majority of the typing rules are fairly standard.…”

“…The programming language research community has indeed devoted quite some effort to this research problem, with many interesting results coming out in recent years, ranging from ranking-function based methodologies [12], to Hoare-Logic based ones [13], through amortized analysis [14] and the interpretation method [15]. Remarkably, higher-order functional programming languages have been left out of the picture, despite type-based techniques for them are wellknown to be useful in the complexity analysis of deterministic programs [16], [17], but also of a problem deeply related to complexity analysis, namely that of termination analysis [18], [19].…”

“…. This motivates the following definition, compare [15] for further justification of this definition.…”