MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 85Finitary modal languages with the HM-property were developed by AlexanderKurz [1998; 2001], Martin Rößiger [1998; 2001] and Bart Jacobs [2000] forcoalgebras of polynomial functors. A functor is polynomial if it can be inductivelyconstructed from the identity functor A ↦→ A and functors A ↦→ C withsome constant value C, by forming products A ↦→ T 1 A × T 2 A, disjoint unionsA ↦→ T 1 A + T 2 A, and “exponential” functors A ↦→ (T A) I with fixed exponentI. The value C of a constant functor can be thought of as a set of “outputs” or“observable values” and an exponent I as an “input” set. For example, considerthe functor having T A = (C × A) I with fixed sets C and I. The correspondingmodal language has a modality [i] for each i ∈ I. Given a state x in a T -coalgebra(A, τ A ), and an “input” i ∈ I, we obtain a pair τ A (x)(i) ∈ C × A whose secondprojection π 2 (τ A (x)(i)) is a new state from A. We declare a modal formula [i]α tobe true at x when α is true at this next state:τ A , x |= [i]α iff τ A , π 2 (τ A (x)(i)) |= α.Note that the first projection π 1 (τ A (x)(i)) here is an output value from C. Thelanguage for T -coalgebras in this case has formulas (i)c for each c ∈ C with thesemanticsτ A , x |= (i)c iff π 1 (τ A (x)(i)) = c.Similarly, the logic for a general polynomial functor T has modal formulas [p]α and“observational” formulas (p)c built from certain path expressions p that syntacticallyreflect the internal structure and inductive formation of T . The Lemmon–Scott canonical model construction (section 5.1) can be adapted to such logics,and Kurz and Rößiger proved that the canonical model is a final T -coalgebra inthe case that the constant sets C occurring in the definition of T are all finite.Jacobs showed that under this same restriction a contravariant duality of the kindconsidered in section 6.5 can be constructed between the category of T -coalgebrasand a certain category of Boolean algebras with operators corresponding to thepath-modalities [p].Another approach to polynomial coalgebraic logic was introduced in [Goldblatt,2001b; Goldblatt, 2003b] by working with terms for algebraic expressions, likeπ 1 (τ A (x)(i)), that have a single state-valued variable x. Boolean combinations ofequations between observable-valued terms were shown to give a class of formulasthat has the Hennessy–Milner property. Bisimilar states were also characterisedas those that assign the same values to all observable-valued terms. Equationswith the same semantics as the above formulas [p]α and (p)c can be defined in thislanguage.Of course the idea of a formula or term having a single state-valued variableis an implicitly modal one, and goes all the way back to Meredith’s U-calculusinterpretation of propositional modal formulas as formulas of first-order logic thathave a single free variable (Sections 4.4 and 6.3). At the same time this equationalapproach is closer to classical universal algebra and model theory, and leads to naturalcoalgebraic constructions of ultraproducts [Goldblatt, 2003d] and ultrafilter