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The bibliography lists many articles of Robert Goldblatt and Bart Jacobs on modal logic.
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An important difference of the coalgebraic approach compared to the algebraic one is that coalgebras generalize rather than dualize the model theory of modal logic. One may generalize the concept of modal logic from Kripke frames to arbitrary coalgebras. In fact the link between modal logic and coalgebra is so strong that one may claim that modal logic is the natural logic for coalgebras (just like equational logic is that for algebra)
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The notion of coalgebra was formed in the sixties but the topic attracted a wider attention only when was realized that coalgebras can be seen as a general uniform theory of dynamic systems.
We list some applications (or examples) of the theory of coalgebras.
The main examples for coalgebras are Kripke frames.
Aczel in [non-well founded sets] models transition systems and non-well fondet sets as coalgebras.
Barwise-Moss in [Vicious Circles] discuss notions of circularity and self-reference (and many (other) applications).
Rutten [Automata and coinduction] on coalgebras and deterministic automata.
(Definition 146, p.392) A polynomial functor is a $Set$-valued functor inductively defined by
where $I$ denotes the identity functor, $C$ denotes the constant functor on an object $C$, $K_0 +K_1$ denotes the coproduct functor of two polynomial functors, and likewise for the product functor, $K^D$ denotes the exponent functor $X\mapsto K(X)^D$.
A Kripke polynomial functor is inductively defined by
where $P K$ denotes the composition of a polynomial functor with the power set functor.
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The initial ideas in this section are attributed to L. Moss:
Coalgebraic logic. Annals of Pure and Applied Logic, 96:277–317, 1999. (Erratum published Ann.P.Appl.Log. 99:241–259, 1999).
J. Barwise and L. Moss. Vicious Circles, volume 60 of CSLI Lecture Notes. CSLI Publications, 1996.
where $\diamond \phi:=\{\diamond \varphi|\varphi\in \phi\}$ and likewise $\Box\phi$. Then we have
where $\diamond \phi:=\{\diamond \varphi|\varphi\in \phi\}$ and likewise $\Box\phi$. Then we have
Let $\mathbb{S}:=\langle S,\lambda,R[\cdot]\rangle$ be a modal model in coalgebraic shape(§9 Example 143). Then $\mathbb{S},s\Vdash\nabla \phi$ iff $(R[s],\phi)$ belongs to the relation lifting (§11) $\overline{\mathcal{P}}(\Vdash_{\mathbb{S}})$ of the satisfaction relation $\Vdash_{\mathbb{S}}\subseteq S\times \phi$: Every $\varphi\in \Phi$ must hold at some successor $t\in R[s]$, and at every successor $t$ of some $\varphi\in \phi$ must hold. The previous lines are the founding insights to coalgebraic logic in which the same principle is applied to an arbitrary set functor $\Omega$ and the basic idea is to have
Let $\mathbb{S}:=\langle S,\lambda,R[\cdot]\rangle$ be a modal model in coalgebraic shape(§9 Example 143). Then $\mathbb{S},s\Vdash\nabla \phi$ iff $(R[s],\phi)$ belongs to the relation lifting (§11) $\overline{\mathcal{P}}(\Vdash_{\mathbb{S}})$ of the satisfaction relation $\Vdash_{\mathbb{S}}\subseteq S\times \phi$: Every $\varphi\in \Phi$ must hold at some successor $t\in R[s]$, and at every successor $t$ of some $\varphi\in \phi$ must hold.
The previous lines are the founding insights to coalgebraic logic in which the same principle is applied to an arbitrary set functor $\Omega$ and the basic idea is to have
In this situation, the satisfaction relation is much like a bisimulation between a language and a coalgebra; for this see
For the existence of the language of coalgebraic formulas for $\Omega$ it is necessary to allow class based algebras. This may be accomplished by letting $\Omega^+$ be the unique extension of $\Omega$ to a set based functor on the category $SET$ having classes as objects and set-continuous (standard) set functors as morphisms. In examples these functors are often polynomial functors; e.g. Kripke functors are of this kind.
The initial algebra of the functor $(P+\Omega)^+$ is called the language of coalgebraic formulas for $\Omega$ and is denoted by
Unwinding this definition yields the following equivalent characterization.
Let $\Omega:Set\to Set$ be (for convenience) be a set functor sending inlusions to inclusions (call this a standard set functor). Then the language of coalgebraic formulas denoted by $\mathcal{L}_\Omega$ is defined to be the least class $C$ such that
(1) $\phi\in \mathcal{L}_\Omega$ implies $\bigwedge\phi\in \mathcal{L}_\Omega$.
(2) $P\in \Omega^+(\mathcal{L}_\Omega)$ implies $\nabla P\in \mathcal{L}_\Omega$.Let $\Omega^+$ is the unique extension of $\Omega$ to a set based functor on the category $SET$ having classes as objects and set-continuous (standard) set functors as morphisms.
Let $\Omega:\Set\to Set$ be a functor, let $\nu:\Omega\to Prop$ be some natural transformation, let $\bigwedge$ be some collection of natural transformations $\Omega\to \mathcal{P}$. then $\mathcal{L}_{\nu,\bigwedge}$ is the standard modal language obtained by taking $Prop$ as the collection off propositional variables, and $\tau_\bigwedge:=\{\diamond_\lambda|\lambda\in \bigwedge\}$ as the modal similarity type.
Predicate liftings (from $\mathcal{P} S\to \mathcal{P}\Omega S$ can be used to obtain modal operators.
A predicate lifting for a set functor $\Omega$ is a natural transformation $\mu:\check{\mathcal{P}}\to \check{\mathcal{P}} \Omega$ where $\check{P}$ denotes the contravariant power set functor. Which each predicate lifting we can associate a modal operator $\diamond_\mu$ with the following semantics:
For polynomial functors see also the references at polynomial functor. In particular
(Joachim) Kock, Joyal, Batanin, Mascari, Polynomial functors and opetopes, arXiv:0706.1033
Joachim Kock. Polynomial functors and trees. Preprint, arXiv:0807.2874
Nicola Gambino and Joachim Kock (2009); Polynomial functors and polynomial monads; arXiv.