Type-Based Complexity Analysis of Probabilistic Functional Programs

Abstract

We show that complexity analysis of probabilistic higher-order functional programs can be carried out compositionally by way of a type system. The introduced type system is a significant extension of refinement types. On the one hand, the presence of probabilistic effects requires adopting a form of dynamic distribution type, subject to a coupling-based subtyping discipline. On the other hand, recursive definitions are proved terminating by way of Lyapunov ranking functions. We prove not only that the obtained type system, called RPCF, provides a sound methodology for average case complexity analysis, but also that it is extensionally complete, in the sense that any average case polytime Turing machines can be encoded as a term typable in RPCF. ∀l : L K J (l).∀n : n ≤ K ∧ notElem(n, l). Nat(n) ⊗ List(l) List(g | L K CntLe(l,n) (g)) ⊗ List(g | L K CntGt(l,n) (g)) (a) Type of partition. ∀l : L K J+1 (l). List(l) 1 J+1 : List(g | L K CntLe(l,Nth(l,i)) (g)) ⊗ Nat(b | b ≤ K) ⊗ List(g | L K CntGt(l,Nth(l,i)) (g)) | i ≤ J (b) Type of ppartition. List(l | L K J+1 (l)). 1 J+1 : List(g | L K c (g)) ⊗ Nat(b | b ≤ K) ⊗ List(g | L K J−c (g)) | c ≤ J (c) Subtype of type ppartition.

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.…”

“…Some of the approaches used in the deterministic setting extend to the probabilistic setting, namely type systems and program logics [Kaminski and Katoen 2017;. In a higher-order setting, the only technique which has been studied is the one based on types, which have indeed been proved to be applicable to higher-order randomized languages [Avanzini et al 2019a;Dal Lago and Grellois 2017;Wang et al 2020]. The main advantage of adopting type systems is the inherent compositionality of the obtained methodology.…”

Intersection types and (positive) almost-sure termination

“…Systems of types for probabilistic programs exist in the literature. In particular, sized types [Hughes et al 1996], and linear dependent types [Dal Lago and Gaboardi 2011] have been generalised to probabilistic programming languages, and have been proved to be sound methodologies for checking almost-sure termination [Dal Lago and Grellois 2019] and positive almost-sure termination [Avanzini et al 2019] in an higher-order setting. None of such systems is complete, however.…”

A modular cost analysis for probabilistic programs

“…functions [Bournez and Garnier 2005]; model checking [Katoen 2016]; program logics McIver et al 2018;Ngo et al 2018;Wang et al 2018]; proof assistants [Barthe et al 2009]; recurrence relations [Sedgewick and Flajolet 1996]; methods based on program analysis [Celiku and McIver 2005;Katoen et al 2010;Kozen 1985]; or symbolic inference [Gehr et al 2016]; and finally type systems [Avanzini et al 2019;Breuvart and Dal Lago 2018].…”