Monads have been recognized by Moggi as an elegant device for dealing with stateful computation in functional programming languages. In previous work, we have introduced a Hoare calculus for partial correctness of monadic programs. All this has been done in an entirely monad-independent way. Here, we extend this to a monad-independent dynamic logic (assuming a moderate amount of additional infr...

This paper considers a generalisation of selection functions over an arbitrary strong monad T , as functionals of type J RX = (X → R) → TX. It is assumed throughout that R is a T -algebra. We show that J R is also a strong monad, and that it embeds into the continuation monad KRX = (X → R) → R. We use this to derive that the explicitly controlled product of T -selection functions is definable f...

Iterative monads, introduced by Calvin Elgot in the 1970’s, are those ideal monads in which every guarded system of recursive equations has a unique solution. For every ideal monad M we prove that an iterative reflection, i.e., an embedding M ↪−→ M̂ into an iterative monad with the expected universal property, exists. We also introduce the concept of iterativity for algebras for the monad M, fol...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C “generated” by solutions of recursive ...

Solutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad C “generated” by solutions of recursive ...

Techniques for monad compositions can be used to provide a basis for categorical unification in the framework of generalised terms. In [4], we gave results for the many-valued sets of terms, and showed that this composition of set functors can be extended to a monad. In this work we introduce new sufficient conditions for two monads being composable, and show that the construction in [4] also s...

This paper analyses the notion of a positivity relation of Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finall...

We present here a novel approach to the analysis of common knowledge based on category theory. In particular, we model global epistemic state for given set agents through hierarchy beliefs represented by presheaf construction. Then, employing properties categorical monad, prove existence state, obtained in an iterative fashion, which all acquire some underlying statement. order guarantee fixed ...

Many categories of semantic domains can be considered from an order-theoretic point of view and from a topological point of view via the Scott topology. The topological point of view is particularly fruitful for considerations of computability in classical spaces such as the Euclidean real line. When one embeds topological spaces into domains, one requires that the Scott continuous maps between...

Abstract In the theory of coalgebras, trace semantics can be defined in various distinct ways, including through algebraic logics, Kleisli category a monad or its Eilenberg–Moore category. This paper elaborates two new unifying ideas: (i) coalgebraic,draftrules is naturally presented terms corecursive algebras, and (ii) all three approaches arise as instances same abstract setting. Our perspect...