Probabilistic operational semantics for the lambda calculus

Abstract

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin's CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.

Probabilistic coherence spaces are fully abstract for probabilistic PCF

“…We have chosen a quite standard one, rand, implementing CAML function Random.int. In [5,21] it is also considered the more basic Coin ∆ = (rand) 2. Another possibility is to allow an arbitrary superposition of two terms M ⊕κ N , for κ ∈ [0, 1], evaluating to M with probability κ and to N with probability 1 − κ, see e.g.…”

On Applicative Similarity, Sequentiality, and Full Abstraction

“…The proof is by induction on the structure of derivations for M ⇓ D. We only consider two cases, since the others are the same as in [6]:…”

“…We first define an approximation semantics, which attributes finite probability distributions to terms, and only later define the actual semantics, which is the least upper bound of all distributions obtained through the approximation semantics. Big-step semantics is given by means of a binary relation ⇓ between This evaluation relation is the natural extension to Λ ⊕or of the evaluation relation given in [6] for the untyped probabilistic λ-calculus. Since the calculus has a call-by-value evaluation strategy, function arguments are evaluated before being passed to functions.…”

“…Various notions of bisimilarity have been proved adequate and, in some cases, fully abstract, for deterministic and nondeterministic computation [1,17,15]. A recent paper [5] generalizes Abramsky's applicative bisimulation [1] to a call-by-name, untyped λ-calculus endowed with binary, fair, probabilistic choice [6]. Probabilistic applicative bisimulation is shown to be a congruence, thus included in context equivalence.…”

Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs

“…In our examples, we use Church [13], a probabilistic programming language based on the stochastic lambda calculus. This calculus is universal in the sense that it can be used to define any computable discrete probability distribution [6] (and indeed, continuous distributions when encoded via rational approximation).…”