Intersection types and (positive) almost-sure termination

Abstract

Randomized higher-order computation can be seen as being captured by a λ-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 λ-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.

On continuation-passing transformations and expected cost analysis

“…The recent works on expected cost analysis by way of linear dependent types [Avanzini et al 2019a] and intersection types [Dal Lago et al 2021] aim at giving very expressive type systems in which bounds on the expected cost of the typed programs can be derived. In the first case, we are talking about a system obtained by generalizing Dal Lago and Gaboardi's ideas [Dal Lago and Gaboardi 2011] to a probabilistic -calculus, obtaining a very expressive, although not relatively complete, methodology.…”

On probabilistic termination of functional programs with continuous distributions

“…We conjecture these ideas to also be applicable to the system of [8]. Independent of us, [35] designed an intersection type system that is also able to reason about the expected time to termination. Their approach is, however restricted to a language with discrete samples and it is not obvious whether the approach extends to continuous samples.…”

The Theory of Call-by-Value Solvability (long version)

“…In particular, for head reduction, which in CbN is the reduction characterizing solvability. De Carvalho's seminal work has been extended to many notions of reduction and formalisms.A first wave was inspired directly from his original work [21,34,35,47,61], and a second wave [6,7,10,11,16,23,31,[55][56][57] started after the revisitation of de Carvalho's technique by Accattoli et al [8].…”