We develop an equational de nition of exception monads that characterizes Moggi's exception monad transformer. This axiomatization is then used to de ne an extension of previously described monadindependent computational logics by abnormal termination. Instantiating this generic formalism with the Java monad used in the LOOP project yields in particular the known Hoare calculi with abnormal termination and JML's method speci cations; this opens up the possibility of extending these formalisms by hitherto missing computational features such as I/O and nondeterminism.

Abstract. We present an algebraic treatment of exception handlers and, more generally, introduce handlers for other computational effects representable by an algebraic theory. These include nondeterminism, interactive input/output, concurrency, state, time, and their combinations; in all cases the computation monad is the free-model monad of the theory. Each such handler corresponds to a model of the theory for the effects at hand. The handling construct, which applies a handler to a computation, is based on the one introduced by Benton and Kennedy, and is interpreted using the homomorphism induced by the universal property of the free model. This general construct can be used to describe previously unrelated concepts from both theory and practice. 1

A Freyd-category is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in call-by-value programming languages, such as the computational λ-calculus, with computational effects. We develop the theory of Freyd-categories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freyd-category. We then give canonical, universal embeddings of Freyd-categories into closed Freyd-categories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λc-calculus. Our leading examples of signatures arise from side-effects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism. Key words: Freyd-category, enriched Yoneda embedding, conical colimit completion, canonical model

We provide a syntactic analysis of contextual preorder and equivalence for a polymorphic programming language with effects. Our approach applies uniformly to arbitrary algebraic effects, and thus incorporates, as instances: errors, input/output, global state, nondeterminism, probabilistic choice, and combinations thereof. Our approach is to extend Plotkin and Power’s structural operational semantics for algebraic effects (FoSSaCS 2001) with a primitive “basic preorder ” on ground type computation trees. The basic preorder is used to derive notions of contextual preorder and equivalence on program terms. Under mild assumptions on this relation, we prove fundamental properties of contextual preorder (hence equivalence) including extensionality properties, a characterisation via applicative contexts, and machinery for reasoning about polymorphism using relational parametricity. 1.

Nondeterminism is modelled in domain theory by the notion of a powerdomain, while probability is modelled by that of the probabilistic powerdomain. Some problems arise when we want to combine them in order to model computation in which both nondeterminism and probability are present. In particular there is no categorical distributive law between them. We introduce the powerdomain of indexed valuations which modifies the usual probabilistic powerdomain to take more detailed account of where probabilistic choices are made. We show the existence of a distributive law between the powerdomain of indexed valuations and the nondeterministic powerdomain. By means of an equational theory we give an alternative characterisation of indexed valuations and the distributive law. We study the relation between valuations and indexed valuations. Finally we use indexed valuations to give a semantics to a programming language. This semantics reveals the computational intuition lying behind the mathematics. In the second part of the thesis we provide an operational reading of continuous valuations on certain domains (the distributive concrete domains of Kahn and Plotkin) through the model of probabilistic event structures. Event structures are a model for concurrent computation that account for causal relations between events. We propose a way of adding probabilities to confusion free event structures, defining the notion of probabilistic event structure. This leads to various ideas of a run for probabilistic event structures. We show a confluence theorem for such runs. Configurations of a confusion free event structure form a distributive concrete domain. We give a representation theorem which characterises completely the powerdomain of valuations of such concrete domains in terms of prob...

This paper is part of a line of work on using the logical techniques of polarity and focusing to design a dependent programming language, with particular emphasis on programming with deductive systems such as programming languages and proof theories. Polarity emphasizes the distinction between positive types, which classify data, and negative types, which classify computation. In previous work, we showed how to use Zeilberger’s higher-order formulation of focusing to integrate a positive function space for representing variable binding, an essential tool for specifying logical systems, with a standard negative computational function space. However, our previous work considers only a simply-typed language. The central technical contribution of the present paper is to extend higher-order focusing with a form of dependency that we call positively dependent types: We allow dependency on positive data, but not negative computation, and we present the syntax of dependent pair and function types using an iterated inductive definition, mapping positive data to types, which gives an account of type-level computation. We construct our language inside the dependently typed programming language Agda 2, making essential use of coinductive types and induction-recursion.

Abstract We outline a possible logic that will allow us to give a unified approach to reasoning about computational effects. The logic is given by extending Moggi's computational *-calculus by basic types and a signature, the latter given by constant symbols, function symbols, and operation symbols, and by including a _ operator. We give both syntax and semantics for the logic except for _. We consider a number of sound and complete classes of models, all given in category-theoretic terms. We illustrate the ideas with some of our leading examples of computational effects, and we observe that operations give rise to natural modalities.

One of the appeals of pure functional programming is that it is so amenable to equational reasoning. One of the problems of pure functional programming is that it rules out computational effects. Moggi and Wadler showed how to get round this problem by using monads to encapsulate the effects, leading in essence to a phase distinction—a pure functional evaluation yielding an impure imperative computation. Still, it has not been clear how to reconcile that phase distinction with the continuing appeal of functional programming; does the impure imperative part become inaccessible to equational reasoning? We think not; and to back that up, we present a simple axiomatic approach to reasoning about programs with computational effects.

Abstract. Every algebraic theory gives rise to a monad, and monads allow a meta-language which is a basic programming language with sideeffects. Equations in the algebraic theory give rise to equations between programs in the meta-language. An interesting question is this: to what extent can we put equational reasoning for programs into the algebraic theory for the monad? In this paper I focus on local state, where programs can allocate, update and read the store. Plotkin and Power (FoSSaCS’02) have proposed an algebraic theory of local state, and they conjectured that the theory is complete, in the sense that every consistent equation is already derivable. The central contribution of this paper is to confirm this conjecture. To establish the completeness theorem, it is necessary to reformulate the informal theory of Plotkin and Power as an enriched algebraic theory in the sense of Kelly and Power (JPAA, 89:163–179). The new presentation can be read as 14 program assertions about three effects. The completeness theorem for local state is dependent on certain conditions on the type of storable values. When the set of storable values is finite, there is a subtle additional axiom regarding quotient types. 1

The incremental approach to modular monadic semantics constructs complex monads by using monad transformers to add computational features to a preexisting monad. A complication of this approach is that the operations associated to the pre-existing monad need to be lifted to the new monad. In a companion paper by Jaskelioff, the lifting problem has been addressed in the setting of system F ω. Here, we recast and extend those results in a category-theoretic setting. We abstract and generalize from monads to monoids (in a monoidal category), and from monad transformers to monoid transformers. The generalization brings more simplicity and clarity, and opens the way for lifting of operations with applicability beyond monads. Key words: Monad, Monoid, Monoidal Category