Lambda Calculus and Probabilistic Computation

Abstract

We introduce two extensions of the λ-calculus with a probabilistic choice operator, Λ cbv ⊕ and Λ cbn ⊕ , modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic, Λ ! ⊕ , which allows us to develop a unified, modular approach.

A Relational Theory of Monadic Rewriting Systems, Part I

“…2) The second one [14]- [16] analyses the global behaviour of probabilistic systems by lifting the underlying rewriting relation R : A + → DA to a relation acting on a variation of distributions, viz. multi-distributions, this way obtaining an ARS whose objects are themselves multi-distributions.…”

“…To the best of the authors' knowledge, only specific computational effects have been studied in the setting of rewriting theory (the main example being the recent theory of probabilistic rewriting [12]- [16]) and nothing has been done at the general level of monadic or algebraic effects. This is rather unsatisfactory, as a proper analysis of effectful programming languages and computations requires a general theory of effectful rewriting.…”

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic

“…This approach has so far mainly be exploited semantically [21,10,11,15,9,7], but can be used it also to study operational properties [15,30,13]. In this paper, we push forward this operational direction.…”

Decomposing Probabilistic Lambda-Calculi

“…Probabilistic lambda-calculi [18,17,14,8,15,6,12] extend the lambda-calculus with a probabilistic choice operator N ⊕ p M , which chooses N with probability p and M with probability 1 − p (throughout this paper, we let p = 0.5 and will omit it). Duplication of N ⊕ M , as is wont to happen in lambda-calculus, raises a fundamental question about its semantics: do the duplicate occurrences represent the same probabilistic event, or different ones with the same probability?…”

“…Second, a probabilistic lambdacalculus must derive its semantics from a prescribed reduction strategy, and its terms only have meaning in the context of that strategy. Third, combining different kinds of probabilities becomes highly involved [12], as it would require specialized reduction strategies. These issues present themselves even in a more general setting, namely that of commutative (algebraic) effects, which in general do not commute with copying.…”