Intersection Types and (Positive) Almost-Sure Termination

TL;DR

This paper demonstrates that intersection types can precisely characterize both the probability of convergence and the expected evaluation time in probabilistic lambda calculus, providing tight, optimal approximations for termination analysis.

Contribution

It introduces a type system using non-idempotent intersection types that captures both probabilistic convergence and expected evaluation steps in a unified framework.

Findings

Type derivations underapproximate convergence probability

Type weights provide lower bounds on expected evaluation steps

Both approximations are proven to be tight and optimal

Abstract

Randomized higher-order computation can be seen as being captured by a lambda calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any lambda-term can be underapproximated by its type, while the underlying derivation's weight gives a lower bound to the term's expected number of steps to normal form. Noticeably, both approximations are tight -- not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.