Abstract
Abstract In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, the Kleisli category of a monad or its Eilenberg–Moore category. This paper elaborates two new unifying ideas: (i) coalgebraic,draftrules trace semantics is naturally presented in terms of corecursive algebras, and (ii) all three approaches arise as instances of the same abstract setting. Our perspective puts the different approaches under a common roof and allows to derive conditions under which some of them coincide.
Highlights
Traces are used in the semantics of state-based systems as a way of recording the consecutive behaviour of a state in terms of sequences of observable actions
We extend the above treatment of trace semantics of BT-coalgebras via Eilenberg–Moore categories, to cover coalgebras for a composite functor TA as well, where A is another endofunctor on the base category C
By Theorem 2.6, we obtain two corecursive algebras by applying these liftings to the inverse of the initial algebra, i.e. the final coalgebra in Dop: (12) These corecursive algebras define trace semantics for any TB-coalgebra (X, c) and BT-coalgebra (Y, d): (13) It is instructive to characterize this trace semantics in terms of the transpose and the step-induced coalgebra liftings Fτ δ and Fδ τ, showing how they arise as unique maps from an initial algebra: (14) In the remainder of this section, we show two classes of examples of the logical approach to trace semantics
Summary
To illustrate the versatility of our framework, we show that it underpins a trace example quite different from the previous ones, one that arises in programming language semantics and involves both input and output actions [5]. – In the automata literature and the previous sections, a ‘trace’ ends in acceptance. Semanticists would call this a ‘complete trace’. – By contrast, in the semantics literature [5, 24, 32, 33, 36, 41] and this section, a ‘trace’ need not end in acceptance. The second (Section 6.3) characterizes the poset of all strategies as a final coalgebra. This is a result that appeared in [5]
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