The Theory of Traces for Systems with Nondeterminism and Probability

Abstract

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories. Monads and Algebraic TheoriesIn this paper, on the algebraic side, we deal with Eilenberg-Moore algebras of a monad on the category Sets of sets and functions, for which we also give presentations in terms of operations and equations, i.e., algebraic theories. MonadsA monad on Sets is a functor M : Sets → Sets together with two natural transformations: a unit η : Id ⇒ M and multiplication µ :We next introduce several monads on Sets, relevant to this paper. Each monad can be seen as giving side-effects.Nondeterminism. The finite powerset monad P maps a set X to its finite powerset PX = {U | U ⊆ X, U is finite} and a function f :The unit η of P is given by singleton, i.e., η(x) = {x} and the multiplication µ is given by union, i.e., µ(S) = U∈S U for S ∈ PPX. Of particular interest to us in this paper is the submonad P ne of non-empty finite subsets, that acts on functions just like the (finite) powerset monad, and has the same unit and multiplication. We rarely mention the unrestricted (not necessarily finite) powerset monad, which we denote by P u . We sometimes write f for P u f in this paper.Probability. The finitely supported probability distribution monad D is defined, for a set X and a function f : X → Y , asThe support set of a distribution ϕ ∈ DX is supp(ϕ) = {x ∈ X | ϕ(x) = 0}. The unit of D is given by a Dirac distribution η(x) = δ x = (x → 1) for x ∈ X and the multiplication by µ(Φ)(x) = ϕ∈supp(Φ) Φ(ϕ) · ϕ(x) for Φ ∈ DDX. We sometimes write i∈I p i x i for a distribution ϕ with supp(ϕ) = {x i | i ∈ I} and ϕ(x i ) = p i . 1. Personal communication with Gordon Plotkin.

Combining probabilistic and non-deterministic choice via weak distributive laws

“…In domain theory there has been a lot of work in this direction by Tix, Kleimel and Plotkin [16,29], Mislove [21] and Varacca and Winskel [30,31]. More recently, the same notorious difficulties appear in the coalgebraic semantics of Segala systems [6,7].…”

“…A second application concerns trace semantics of systems combining probabilistic and non-deterministic features -as studied in [7] -using tools akin to the generalized determinization. The lack of a proper distributive law means that the authors of [7] had to build from scratch an algebraic presentation for the algebras of the composite monad P D and to mend the mechanism of the generalized determinization accordingly.…”

Combining Semilattices and Semimodules

“…In the next Proposition we show that also the µ S X -convex closure is superfluous, due to the fact that Φ ∈ SP a c X (and not simply SPX), thus obtaining (10). Proposition 28.…”

“…By (5), λ • αa A = α a (Φ) where Φ = (A → λ). By (10), α a (Φ) = {a(ϕ) | ϕ ∈ c(Φ)}. Following the definition of c(Φ) given in (7), one has to consider functions u : supp Φ → X such that u(B) ∈ B for all B ∈ supp Φ: if λ = 0, then supp Φ = {A} and thus, for each x ∈ A, there is exactly one function u x : supp Φ → X mapping A into x.…”

A trustful monad for axiomatic reasoning with probability and nondeterminism

“…Now that we have completed our formalization, it should be easier to extend it with finitely generated convex sets. Indeed, looking at Bonchi et al (2020b), we recognize several technical results that we have already formalized (e.g., parts of Lemma 4.4). Concretely, the approach would start by defining the data structure for the non-empty finitely generated convex sets by adding an axiom for the existence of a finite generator to the type necset, and then by replaying and fixing the proofs (the category part of our framework should stay unchanged).…”