For the semantics of probabilistic features in programming mainly two approaches are used for building models. One is the Giry monad of Borel probability measures over metric spaces, and the other is Jones’ probabilistic powerdomain monad [6] over dcpos (directed complete partial orders). This paper places itself in the second domain theoretical tradition. The probabilistic powerdomain monad is...

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational of a system, by means distributive law between its syntax (encoded as monad) dynamics (an endofunctor). This setup instrumental in showing that semantic specification (a coalgebra) compositional. In this work, we use derive well-behaved structural string diagrams, graphical increasingly used study int...

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...

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...

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...

Abstract We develop the Scott model of programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for non-termination PCF, use lifting monad (also known as partial map classifier monad) from topos theory, which has been extended to theory by Escardó Knapp. Our results show that is a viable approach partiality can be constructed predicative ...

Every finitary monad T on the category of sets is described by an algebraic theory whose n-ary operations are the elements of the free algebra Tn generated by n letters. This canonical presentation of the monad (called its Lawvere theory) offers a precious guideline in the search for an intuitive presentation of the monad by generators and relations. Hence, much work has been devoted to extend ...

For programmers, Monads are a well-known way to represent an abstract data structure, but without knowing its true nature. Admittedly, the definition of Monads in Category Theory is too subtle and obscure so makes people to avoid from trying to understand them. In this paper, we attempt to explore popular Monads in a purely functional programming language, Haskell, with a Category-theoretic min...

Probability is often counter-intuitive, and it always involves a great deal of math. This is unfortunate, because many applications in robotics and AI increasingly rely on probability theory. We introduce a modular toolkit for constructing probability monads, and show that it can be used for everything from discrete distributions to weighted particle filtering. This modular approach allows us t...

For a quantaloid Q, considered as a bicategory, Walters introduced categories enriched in Q. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad T over the discrete presheaf monad of the small quantaloid Q the primary data of the theory, rather than the l...