We use the theorem prover Isabelle to formalise and machinecheck results of the theory of generalised substitutions given by Dunne and used in the B method. We describe the model of computation implicit in this theory and show how this is based on a compound monad, and we contrast this model of computation and monad with those implicit in Dunne’s theory of abstract commands. Subject to a qualif...

Article history: Received 7 June 2014 Available online 16 October 2014 Communicated by G. Rosolini MSC: 03B15; 03G30 We give an account of the basic combinatorial structure underlying the notion of type dependency. We do so by considering the category of generalised algebraic theories in the sense of Cartmell, and exhibiting it as the category of algebras for a monad on a presheaf category. The...

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces ...

The theory of monads on categories equipped with a dagger (a contravariant identity-on-objects involutive endofunctor) works best when all structure respects the dagger: the monad and adjunctions should preserve the dagger, and the monad and its algebras should satisfy the so-called Frobenius law. Then any monad resolves as an adjunction, with extremal solutions given by the categories of Kleis...

We propose an intermediate form based on monad-algebra comprehensions (to represent queries), folds (to represent computation), and setoids over polynomial datatypes (to represent data), suitable for use in collection processing. Such an intermediate form captures, in a uniform way, large fragments of many recent largescale collection processing languages such as MapReduce, PIG, DryadLINQ, and ...

We continue the study of Capretta’s delay monad as a means of introducing non-termination from iteration into Martin-Löf type theory. In particular, we explain in what sense this monad provides a canonical solution. We discuss a class of monads that we call ω-complete pointed classifying monads. These are monads whose Kleisli category is an ωcomplete pointed restriction category where pure maps...

The delay datatype was introduced by Capretta [2] as a means to incorporate general recursion to Martin-Löf type theory and it is useful in this setting for modeling non-terminating behaviours. This datatype is a (strong) monad and constitutes a constructive alternative to the maybe monad. For a given set X, each element of DX is a possibly infinite computation that returns a value of X, if it ...

In the theory of programming languages, the use of proof assistants has become mainstream. It is considered good form to provide a formal connection between a language and its semantics. Currently, the main tools for this are based on either higher order logic, or on type theory. Here we will focus on Coq, the biggest system of the latter class. The ALEA [2] Coq library formalizes discrete meas...

Synopsis Domain-theoretic categories are axiomatised by means of categorical non-order-theoretic requirements on a cartesian closed category equipped with a commutative monad. We prove an enrichment theorem showing that every axiomatic domain-theoretic category can be endowed with an intensional notion of approximation, the path relation, with respect to which the category Cpo-enriches. Subsequ...