On coinductive equivalences for higher-order probabilistic functional programs

Abstract

We study bisimulation and context equivalence in a probabilistic λ-calculus. The contributions of this paper are threefold. Firstly we show a technique for proving congruence of probabilistic applicative bisimilarity. While the technique follows Howe's method, some of the technicalities are quite different, relying on non-trivial "disentangling" properties for sets of real numbers. Secondly we show that, while bisimilarity is in general strictly finer than context equivalence, coincidence between the two relations is attained on pure λ-terms. The resulting equality is that induced by Levy-Longo trees, generally accepted as the finest extensional equivalence on pure λ-terms under a lazy regime. Finally, we derive a coinductive characterisation of context equivalence on the whole probabilistic language, via an extension in which terms akin to distributions may appear in redex position. Another motivation for the extension is that its operational semantics allows us to experiment with a different congruence technique, namely that of logical bisimilarity.

Effectful applicative bisimilarity: Monads, relators, and Howe's method

“…Applicative bisimilarity has been generalised to effectful λ-calculi of various kinds, and in particular to untyped probabilistic λ-calculi [10]. It is known to be not only a congruence (thus sound for contextual equivalence) but also fully abstract, at least for call-by-value evaluation [8].…”

“…As an example, nondeterministic extensions of the λ-calculus can be dealt with by looking at terms as a labelled transition system, while probabilistic extensions of the λ-calculus require a different definition akin to Larsen and Skou's probabilistic bisimulation [10]. What kind of transition system do we need when, e.g., dealing with the example programs from Figure 2?…”

“…There have been many attempts to study effectful λ-calculi [34], [30] by way of denotational semantics [21], [14], [12], logical relations [6], and applicative bisimilarity [25], [10], [8]. But while the denotational and logical relation semantics of effectful calculi have been studied in the abstract [18], [20], the same cannot be said about applicative bisimilarity and related coinductive techniques.…”

“…But while the denotational and logical relation semantics of effectful calculi have been studied in the abstract [18], [20], the same cannot be said about applicative bisimilarity and related coinductive techniques. There is a growing body of literature on applicative bisimilarity for calculi with, e.g., nondeterministic [25], and probabilistic effects [10], but each notion of an effect has been studied independently, often getting different results. Distinct proofs of congruence for applicative bisimilarity, even if done through a common methodology, namely the so-called Howe's method [19], do not at all have the same difficulty in each of the cases cited above.…”

Practical probabilistic programming with monads

“…Some recent works develop operational equivalence for probabilistic λ-calculus but without considering continuous distributions, or constructs for conditioning. Dal Lago et al (2014) develop a bisimulation theory for an untyped calculus, while Bizjak and Birkedal (2015) investigate operational equivalence and stepindexed logical relations for a typed probabilistic λ-calculus. Gretz et al (2015) consider the semantics of conditioning in probabilistic while-programs, together with transformations to eliminate conditioning, but without higher-order functions.…”