We present a natural semantics that models the untyped, normal order -calculus plus McCarthy's amb in the context of call-by-need parameter passing. This results in a singular semantics for amb. Previous work on singular choice has concentrated on erratic choice, a less interesting nondeterministic choice operator, and only in relation to callby -value parameter passing, or call-by-name restricted to deterministic terms. The natural semantics contains rules for both convergent and divergent behaviour, allowing it to distinguish programs that dier only in their divergent behaviour. As a result, it is more discriminating than current domain-theoretic models. This, and the fact that it models singular amb, makes the natural semantics suitable for reasoning about lazy, functional languages containing McCarthy's amb. 1 Introduction The need for non-determinism in functional programming is apparent. There are parallel algorithms that are inherently non-deterministic, deterministic para...

Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.

Domain theory is a branch of classical computer science. It has proven to be a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. In this paper, we study the extension of domain theory to the quantum setting. By defining a quantum domain we introduce a rigourous definition of quantum computability for quantum states and operators. Furthermore we show that the denotational semantics of quantum computation has the same structure as the denotational semantics of classical probabilistic computation introduced by Kozen [23]. Finally, we briefly review a recent result on the application of quantum domain theory to quantum information processing. 1

An R-module M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable R-modules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are non-observable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains. 1 Introduction A power domain construction maps every domain X into a so-called power domain over X whose points represent sets of points of the ground domain. Power domain constructions were originally proposed to model the semantics of non-deterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the sema...

We present a natural semantics for the untyped lazy -calculus plus McCarthy's amb, a nondeterministic choice operator. The natural semantics includes rules for both convergent behaviour (dened inductively) and divergent behaviour (dened co-inductively). This semantics is equivalent to a small step reduction semantics that corresponds closely to our operational intuitions about McCarthy's amb. We present equivalences for convergent and divergent behaviour based on the natural semantics and prove a Context Lemma for the convergence equivalence. We then give a -theory l 8 , based on the equivalences for convergent and divergent behaviour. Since it is able to distinguish between programs that dier only in their divergent behaviour, the -theory is more discriminating than equational theories based on current domain-theoretic models. It is therefore more suitable for reasoning about functional programs containing McCarthy's amb. Contents 1 Introduction 2 2 Related Work 3 3 ...

This paper is a summary of the following six publications: (1) Stable Power Domains [Hec94d] (2) Product Operations in Strong Monads [Hec93b] (3) Power Domains Supporting Recursion and Failure [Hec92] (4) Lower Bag Domains [Hec94a] (5) Probabilistic Domains [Hec94b] (6) Probabilistic Power Domains, Information Systems, and Locales [Hec94c] After a general introduction in Section 0, the main results of these six publications are summarized in Sections 1 through 6. 0 Introduction In this section, we provide a common framework for the summarized papers. In Subsection 0.1, Moggi's approach to specify denotational semantics by means of strong monads is introduced. In Subsection 0.2, we specialize this approach to languages with a binary choice construct. Strong monads can be obtained in at least two ways: as free constructions w.r.t. algebraic theories (Subsection 0.3), and by using second order functions (Subsection 0.4). Finally, formal definitions of those concepts which are used in all...

We present two probabilistic powerdomain constructions in topological domain theory. The first is given by a free ”convex space ” construction, fitting into the theory of modelling computational effects via free algebras for equational theories, as proposed by Plotkin and Power. The second is given by an observationally induced approach, following Schröder and Simpson. We show the two constructions coincide when restricted to ω-continuous dcppos, in which case they yield the space of (continuous) probability valuations equipped with the Scott topology. Thus either construction generalises the classical domain-theoretic probabilistic powerdomain. On more general spaces, the constructions differ, and the second seems preferable. Indeed, for countably-based spaces, we characterise the observationally induced powerdomain as the space of probability valuations with weak topology. However, we show that such a characterisation does not extend to non countablybased spaces. 1

Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as P oset, ωCpo, or functor categories such as [Inj, Set] or [Inj, ωCpo]. That involves allowing the arities of Lawvere theories to be extended to being size-restricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabriel-Ulmer duality.