## Coalgebraic semantics for timed processes (2006)

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Venue: | Inf. & Comp |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Kick06coalgebraicsemantics,

author = {Marco Kick and John Power and Alex Simpson},

title = {Coalgebraic semantics for timed processes},

journal = {Inf. & Comp},

year = {2006},

volume = {204},

pages = {2006}

}

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### Abstract

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws 1

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Citation Context ...we refer to the development of Chapter 7 of the first author’s thesis [11]. Constructions 19sof cofree comonads on endofunctors abound in the coalgebraic literature, for instance in [23] but see also =-=[8,18]-=-. In particular, if C is locally presentable and B and D are accessible, the cofree comonads B ∞ and (BD) ∞ exist [5], and so the theorem holds. More specifically still, recall that in the cases of di... |

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Citation Context ...the distributive law λ (2) the category of coalgebras for the comonad DT on T -Alg. (3) the category of algebras for the monad TD on D-Coalg Now recall the corresponding situation for coalgebras (see =-=[2]-=- for the dual): Proposition 4.4 The following are equivalent: (1) a distributive law of a comonad D over a comonad D ′ (2) a lifting of the comonad D to a comonad DD ′ on D′ -Coalg. Given these equiva... |

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Citation Context ...However, in the cases of primary interest to us, one of the comonads, that given by the action behaviour, is the cofree comonad on an endofunctor. And in that case, the dual of a result for monads in =-=[3]-=- does give us a reasonable construction as follows. Theorem 5.9 For any category C and any endofunctor B and comonad D for which the cofree comonads B ∞ and (BD) ∞ on B and BD exist, the product B ∞ ×... |

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Citation Context ...oduct of comonads is then the cofree comonad on the product of endofunctors, which in turn is given pointwise. In Section 6, following the work of Turi, Plotkin and later authors on distributive laws =-=[14,15,18,19,21]-=-, we study the combination of operational semantics generated by two sorts of behaviour, our leading class of examples having one sort of behaviour generated by time with the other sort of behaviour g... |

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Citation Context ...t observe that the functor T ⇀ (−) is accessible as it is isomorphic to (1 + (−)) T and accessible functors on Set are closed under limits in general, hence products in particular (see, for instance, =-=[1]-=-). Next, obtain ET as the equaliser of natural transformations ρ, ρ ′ : (T ⇀ (−)) ⇒ 2 × 2 T ×T , where we write 2 for the set {true, false} of truth values, defined by ρX(e) = (e(0)↓, λ(t, u). (e(t + ... |

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Citation Context ...transition relation, i.e., it satisfies axioms of determinacy, zero-delay, and continuity. The concept of timed transition system was at the heart of the first author’s thesis [11], was summarised in =-=[10]-=-, and was synthesised from various accounts of time in the literature, such as [7,17,22]. The central result of Section 2 is that a timed transition system amounts exactly to a coalgebra for what we c... |

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Citation Context ...inuity. The concept of timed transition system was at the heart of the first author’s thesis [11], was summarised in [10], and was synthesised from various accounts of time in the literature, such as =-=[7,17,22]-=-. The central result of Section 2 is that a timed transition system amounts exactly to a coalgebra for what we call the evolution comonad ET on Set generated by the time domain T . The evolution comon... |

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Citation Context ..., λ(t, u). (e(t + u)↓)) ρ ′ X(e) = (true, λ(t, u). (e(t + u)↓ ∧ e(t)↓) . This exhibits ET as a limit of accessible endofunctors, hence accessible. ✷ We remark that it follows from the above, see e.g. =-=[5]-=-, that pAct T is a topos. In fact it can be shown to be a presheaf topos. 3 Delay operators In this paper, we use coalgebra to provide a principled treatment of operations on timed processes. Most int... |

5 | Modularity of behaviours for mathematical operational semantics - Kick, Power - 2004 |