Trace Semantics via Determinization

Abstract

Abstract. This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category. This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.

Combining probabilistic and non-deterministic choice via weak distributive laws

“…Whenever the functor can be lifted to a functor on the Eilenberg-Moore category of T, i.e., in the presence of a distributive law ⇒ of the functor over the monad T, then one can transform -coalgebras with carrier into -coalgebras with carrier . This was nicely summarized in [13] by stating that the free functor T : C → EM(T) can be lifted to a functor between the categories of coalgebras. We thus have the following commuting diagram, where the vertical arrows denote the forgetful functors mapping an -coalgebra, respectively an -coalgebra, to its carrier set, respectively T-algebra.…”

Long-Term Values in Markov Decision Processes, (Co)Algebraically

“…We also recall (cf. [3,14]) that such a distributive law induces an operation (−) : Coalg C (F T ) → Coalg EM(T ) (F λ ), which is often referred to as an abstract form of determinization (cf. [26,14]).…”

“….). The distributive law given by χ is an EM-law in the terminology of [14], where determinization was studied for the purpose of obtaining trace semantics. The trace semantics of m σ : S → R × ∆S is the function that maps a state s to the stream of expected rewards (r σ 0 (s), r σ 1 (s), r σ 2 (s), .…”

“…This leads us to introduce the notions of b-categories and b-corecursive algebras (bcas). Combining these with well-known techniques from coinductive specification and trace sematics [3,14], we obtain the desired universal maps.…”

“…The algebras E : ∆R → R and α γ , from(14), both preserve B.Next, we formulate definitions of functors and natural transformations which incorporate b-structures. The main motivation for the definitions below are the examples which follow and also the properties that we shall see at the end of this section, in Proposition 5 and Example 8.…”