In the semantics of programming, nite data types such as finite lists, have traditionally been modelled by initial algebras. Later final coalgebras were used in order to deal with in finite data types. Coalgebras, which are the dual of algebras, turned out to be suited, moreover, as models for certain types of automata and more generally, for (transition and dynamical) systems. An important property of initial algebras is that they satisfy the familiar principle of induction. Such a principle was missing for coalgebras until the work of Aczel (Non-Well-Founded sets, CSLI Leethre Notes, Vol. 14, center for the study of Languages and information, Stanford, 1988) on a theory of non-wellfounded sets, in which he introduced a proof principle nowadays called coinduction. It was formulated in terms of bisimulation, a notion originally stemming from the world of concurrent programming languages. Using the notion of coalgebra homomorphism, the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras. Thus, the three basic notions of universal algebra: algebra, homomorphism of algebras, and congruence, turn out to correspond to coalgebra, homomorphism of coalgebras, and bisimulation, respectively. In this paper, the latter are taken

ing from syntax. Conventionally an equation for algebra ' is just a pair of terms built from variables, the constituent operations of ' , and some fixed standard operations. An equation is valid if the two terms are equal for all values of the variables. We are going to model a syntactic term as a morphism that has the values of the variables as source. For example, the two terms ` x ' and ` x join x ' (with variable x of type tree ) are modeled by morphisms id and id \Delta id ; join of type tree ! tree . So, an equation for ' is modeled by a pair of terms (T '; T 0 ') , T and T 0 being mappings of morphisms which we call `transformers'. This faces us with the following problem: what properties must we require of an arbitrary mapping T in order that it model a classical syntactic Datatype Laws without Signatures 7 term? Or, rather, what properties of classical syntactic terms are semantically essential, and how can we formalise these as properties of a transformer T ? Of course...

ion Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : 82 6.3 The Beautiful Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 6.4 The Rolling Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 89 6.5 The Square Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 6.6 The Exchange Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 6.7 The Diagonal Rule : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.1 One Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 100 6.7.2 The Other Half : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 103 6.8 Mutual Recursion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 106 7 Monads 111 7.1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 111 7.2 Monads and Adjunctions : : : : : : : : : : : : : : : : : : : : : : : : : : : : 112 7.3 Basic Adjunction : : : : : : : : : : : : ...

This paper concerns the introduction of an iterator into the refinement calculus. The construct is based on concepts from functional programming, and the work gives an interesting example of cross-fertilisation between the functional and imperative programming worlds. Specifically, the iterator construct it..ti uses the idea of a catamorphism — the unique homomorphism from an initial algebra. The datatype for which the iterator is to be defined is considered as an initial algebra of an appropriate functor. The it..ti construct is formally defined as a recursive procedure, and it is shown that, if the value to be obtained by an iteration can be expressed as a catamorphism, then the it..ti construct provides a very natural implementation. Examples are given to show typical uses of the new construct. 1

Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]-probabilities. It will be shown that there are translations back-and-forth, in the form of an adjunction, between effect monoids and “convex ” monads. This convexity property is formalised, both for monads and for categories. In the end this leads to “triangles of adjunctions ” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1