Distributing probability over non-determinism

Abstract

We study the combination of probability and non-determinism from a categorical point of view. In category theory, non-determinism and probability are represented by suitable monads. However, these two monads do not combine well as they are. To overcome this problem, we introduce the notion of indexed valuations. This notion is used to define a new monad that can be combined with the usual non-deterministic monad via a categorical distributive law. We give an equational characterisation of our construction. We discuss the computational meaning of indexed valuations, and we show how they can be used by giving a denotational semantics of a simple imperative language.

Relating strong behavioral equivalences for processes with nondeterminism and probabilities

“…Indeed, as motivated in [41] and [40], mutual distributivity of these operators has a key role in the definition of denotational models of nondeterministic and probabilistic systems. Thus, it would be interesting to see which of the many equivalences that we have considered guarantees distributivity.…”

Towards a Coalgebraic Chomsky Hierarchy

“…-If R is the interval [0, +∞) then T R is the monad of finite valuations used for modelling probabilistic computations [38]. Two other well-known monads of finite distributions and finite subdistributions serving the same purpose embed into T R : the formal sums (3.1) for them are requested to satisfy the additional constraints r 1 + · · · + r n = 1 and r 1 + · · · + r n ≤ 1, respectively.…”

Events, Causality and Symmetry

“…Combining probability and nondeterminism is problematic because the two forms of powerdomain together do not satisfy a distributive law (their combination forces extra laws to be imposed [67]). However, if one works with the indexed probabilistic powerdomain where the probability distribution is carried by the ways in which values are computed, one recovers a distributive law [95,96]. Nondeterministic dataflow: While, as Kahn was early to show [57], deterministic dataflow is a shining application of simple domain theory, nondeterministic dataflow is beyond its scope.…”

“…The output event structure (to the right of the span) would now stand for a type of probabilistic processes. In special cases the idea relates to, again largely-unexplored, categorical versions of the indexed-probability powerdomains [95,96]. Each construction gives a category of random variables on a category, and mimics the corresponding construction on domains.…”