The notion of a monad cannot be expressed within higher-order logic (HOL) due to type system restrictions. We show that if a monad is used with values of only one type, this notion can be formalised in HOL. Based on this idea, we develop a library of effect specifications and implementations of monads and monad transformers. Hence, we can abstract over the concrete monad in HOL definitions and ...

We consider the theory of “extended subsitutions” involving both angelic and demonic choice. For other related formal theories describing program semantics the implicit model of computation is based on a combination of monads by a distributive law. We show how the model of computation underlying extended subsitutions is based on a monad which, while not being a compound monad, has strong simila...

The category theoretic structures of monads and comonads can be used as an abstraction mechanism for simplifying both language semantics and programs. Monads have been used to structure impure computations, whilst comonads have been used to structure context-dependent computations. Interestingly, the class of computations structured by monads and the class of computations structured by comonads...

The paper investigates non-deterministic, probabilistic and quantum walks, from the perspective of coalgebras and monads. Nondeterministic and probabilistic walks are coalgebras of a monad (powerset and distribution), in an obvious manner. It is shown that also quantum walks are coalgebras of a new monad, involving additional control structure. This new monad is also used to describe Turing mac...

In 2010, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.Since any 2-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijection...

This paper identifies several key properties of a monad that allow us to formulate the basics of conditional probability theory, using states for distributions/measures and predicates for events/probability density functions (pdf’s). The distribution monad for discrete probability and the Giry monad for continuous probability are leading examples. Our categorical description handles discrete an...

We give a logical reconstruction of all-solution predicates in terms of list comprehensions in Prolog's and we describe a variety of logic programming constructs in terms of monads and monad morphisms. Novel monad structures are described for lazy function lists, clause unfoldings and a monad morphism based embedding of Prolog in Prolog is given.