Suppose we have a poset, which is a simple sort of category. The example he already mentioned is very nice. I think David and other people can give you piles of comonads with a computer science or logic flavor to them. Jim Dolan made me solve this puzzle, and it made me stronger, so I feel good about inflicting it on everyone else.Īnyway, this is why nobody cares about comonoids in Set Set (or any other category with finite products). Nobody tell Mike the answer! If he gets stuck, he can find it buried in an old n n-Café article featuring similar puzzles. However, comonoids in Set Set are very dull, and it’s a great exercise to work out what they are. The endofunctor on the semantic side, i.e., on N N is We use the presheaf category N N of functors from finite sets (of channel names) with injective renamings to a certain category of domains (representing observable behaviour in the presence of recursion). We can even consider the π \pi-calculus as a modal logic: The resulting logic is Probabilistic modal logic. Coalgebras for this functor are Markov chains. On the semantic side of this operator, we will need to revist our old friend the Giry monad which sends a set to the set of subprobability distributions it supports. It is at least p % p \% probable that at the next step ϕ \phi. In the case of probabilistic modalities, we might want modal operators such as Likelihood’) and temporal knowledge (‘weekends’) under a quantitative regime (‘smaller’). Smaller on weekends’, one implicitly makes use of default logics (‘normally’), probabilistic reasoning (‘the In a seemingly simple piece of knowledge such as ‘Normally, the likelihood of road congestion is 1)Īll of these are needed to analyse typical statements: ‘necessarily’, ‘in the future’, ‘everywhere’, ‘probably’, ‘as everyone knows’, or ‘normally’. One large class of logics is the vast and growing family of modal logics, which are characterised by having operators that qualify formulas as holding in a certain way, e.g. Properties of the modalities, such as □ P ⊢ □ □ P \square P \vdash \square \square P, correspond to properties of the relation, here transitivity.īut there are many other modalities we might want to consider, as explained here: This accessibility relation is thus a relation R R between worlds, or in other words a coalgebra for the powerset functor: So, □ P \square P (it is necessarily the case that P P) holds in my world if P P holds in all worlds accessible from my world. To relate all this to something perhaps familiar, when you introduce the necessity operator □ \square into a propositional theory, semantically you start thinking about the accessibility of one world from another. As I haven’t you can see what’s going on on page 2 of Coalgebras and Their Logics.
MODAL LOGIC PLAYGROUND HOW TO
It would be handy at this point if I’d learned how to draw suitable diagrams.
![modal logic playground modal logic playground](https://www.jubalstore.com/wp-content/uploads/2021/01/DP1-768x321.jpg)
If all goes well, we end up lifting the Stone adjunction to one between coalgebras on one side and algebras on the other.
![modal logic playground modal logic playground](https://bestof.nyc3.digitaloceanspaces.com/react/react-react-designer.jpeg)
Meanwhile, on the semantic side, we have a endofunctor whose coalgebras provide the semantics for the modal propositional theories. Modalities are then introduced on the syntactic side in the shape of an endofunctor, where the new modal propositional theories are algebras for this functor. Typically this is a Stone-style duality between (Lindenbaum algebras of) propositions and spaces of models or valuations for a propositional theory. The key idea is to start from a dual adjunction between the semantics and syntax of propositional theories, as explained on page 3 of Exemplaric Expressivity of Modal Logics. Recently I’ve been looking into the coalgebra community’s take on modal logic, which in a phrase states simply that