On the Herbrand-Kleene universe for nondeterministic computations

AbstractFor nondeterministic recursive equations over an arbitrary signature of function symbols including the nondeterministic choice operator “or” the interpretation is factorized according to the techniques developed by the present author (1982). It is shown that one can either associate an infinite tree with the equations, then interpret the function symbol “or” as a nondeterministic choice operator and so mapping the tree onto a set of infinite trees and then interpret these trees. Or one can interpret the recursive equation directly yielding a set-valued function. Both possibilities lead to the same result, i.e., one obtains a commuting diagram. However, one has to use more refined techniques than just powerdomains. This explains and solves a problem posed by Nivat (1980). Basically, the construction gives a generalization of the powerdomain approach applicable to arbitrary nonflat (nondiscrete) algebraic domains

A representable approach to finite nondeterminism

AbstractWe reformulate denotational semantics for nondeterminism, taking a nondeterministic operation V on programs, and sequential composition, as primitive. This gives rise to binary trees. We analyse semantics for both type and program constructors such as products and exponential types, conditionals and recursion, in this setting. In doing so, we define new category-theoretic structures, in particular premonoidal categories. We also account for equivalences of programs such as those induced by associativity, symmetry and idempotence of V, and we study finite approximation by enrichment over the category of ω-cpos with least element. We also show how to recover the classical powerdomains, especially the convex powerdomain, as three instances of a general, computationally natural, construction

Slice Nondeterminism

This paper studies a technique for describing and formalising nondeterministic functions, using slice categories. Results of a nondeterministic function are modelled by an object of the slice category over the codomain of the function, which is an indexed family over the codomain. Two such families denote the same set of results if slice morphisms exist between them in both directions. We formulate the category of nondeterministic functions by expressing a set of possible results as an equivalence class of objects. If we allow families to use any indexing set, this category will be equivalent to the category of relations. When we limit ourselves to a smaller universe of indexing sets, we get a subcategory which more closely resembles nondeterministic programs. We compare this category with other representations of the category of relations, and see how many properties can be carried over, such as its product, coproduct and other monoidal structures. We can describe inductive nondeterministic structures by lifting free monads from the category of sets. Moreover, due to the intensional nature of the slice representation, nondeterministic processes are easily represented, such as interleaving concurrency and labelled transition systems. This paper has been formalised in Agda

Computable concurrent processes

AbstractWe study relative computability for processes and process transformations, in general, and in particular the non-deterministic and concurrent processes which can be specified in terms of various fair merge constructs. The main result is a normal form theorem for these (relatively) computable process functions which implies that although they can be very complex when viewed as classical set-functions, they are all “loosely implementable” in the sense of Park (1980). The precise results are about the player model of concurrency introduced in Moschovakis (1991), which supports both fairness constructs and full recursion

Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

Reasoning about orchestrations of web services using partial correctness

Abstract A service is a remote computational facility which is made available for general use by means of a wide-area network. Several types of service arise in practice: stateless services, shared state services and services with states which are customised for individual users. A service-based orchestration is a multi-threaded computation which invokes remote services in order to deliver results back to a user (publication). In this paper a means of specifying services and reasoning about the correctness of orchestrations over stateless services is presented. As web services are potentially unreliable the termination of even finite orchestrations cannot be guaranteed. For this reason a partial-correctness powerdomain approach is proposed to capture the semantics of recursive orchestrations. </jats:p

Convex powerdomains II

AbstractThe study of powerdomains defined as completions via Frink ideals is continued. It is shown how to represent directed ideals ofP(D)by certain compact subsets of the original domainD, and arbitrary Frink ideals by sets of such subsets. The operations union and big union are defined and their properties studied. Finally, some results on the relationship of this powerdomain to the classical Plotkin powerdomain are presented

Denotational semantics for unguarded recursion: the demonic case

We show that the technique to prove equivalence of operational and denotational cpo based semantics using retractions, as introduced in de Bruin & Vink [1989] for a sequential backtracking language, can be applied to parallel languages as well. We prove equivalence for a uniform language in which procedure calls need not be guarded. The unguardedness is taken care of by giving a semantics in which the nondeterminism is demonic