MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

84 Robert GoldblattThis abstracts the relation of observational equivalence of processes discussed insection 7.2.Another fundamental notion is that of a final, or terminal, coalgebra, categoricallydual to the notion of initial algebra discussed for modal algebras above. AT -coalgebra (F, τ F ) is called final if for each T -coalgebra (A, τ A ) there is a uniquef Acoalgebraic morphism (A, τ A ) −→ (F, τ F ). In the process algebra context thestates of a final coalgebra are thought of as representing all possible “observablebehaviours” of processes, because observationally equivalent processes are identifiedby the unique morphism to a final coalgebra. More precisely, for any states xand y of coalgebra (A, τ A ), if x ∼ y then f A (x) = f A (y), and the converse is alsotrue under a mild restriction on T [Rutten and Turi, 1993, Corollary 2.9].It is a well known observation of Joachim Lambek that the transition structureτ F of a final T -coalgebra is an isomorphism between F and T F . So it followsfrom Cantor’s Theorem that there cannot exist any final P-coalgebra, since thereis no bijection from any set A onto its powerset PA. Thus the category of modalframes and p-morphisms has no final object. More generally there is no finalcoalgebra for the functor (P−) I whose coalgebras are non-deterministic transitionsystems with input set I. On the other hand, we can model finitely branchingnon-determinism by using the finitary powerset functor P ω , where P ω A is the setof all finite subsets of A. A (P ω −) I -coalgebra is an image-finite transition systemin the sense, described in section 7.2, that the set {y : x −→ i y} of possible nextstates is finite for each state x and each input i. There does exist a final (P ω −) I -coalgebra: this follows from general results about the existence of final coalgebras[Aczel and Mendler, 1989; Barr, 1993; Kawahara and Mori, 2000; Rutten, 2000].In particular, a final T -coalgebra exists whenever T is bounded, which means thatthere is some cardinal number κ such that any state of a T -coalgebra belongs tosome subcoalgebra with no more than κ states. The functor P ω is bounded withκ = ℵ 0 , and for each set I, (P ω −) I is bounded with κ = max{ℵ 0 , card I}.Devising a suitable syntax and semantics for T -coalgebras is a matter thatdepends on the nature of the functor T involved. A natural desideratum is asatisfaction relation τ A , x |= α, expressing “formula α is true/satisfied at statex in coalgebra τ A ”, that provides a logical characterisation of bisimilarity in thefollowing form:x ∼ AB y iff for all formulas α, τ A , x |= α iff τ B , y |= α.If this holds we will say that the logic, or the functor T , has the Hennessy–Milner(HM) property (see (∗) in section 7.2).The first explicit coalgebraic logic with this property was introduced by LawrenceMoss [1999] for a broad class of functors that have final coalgebras. The languageinvolved was infinitary, allowing formation of the conjunction of any set of formulas.For certain functors it was shown that this language has sufficient expressivepower to characterise each state of the final coalgebra uniquely by a single formula.