Constraint Logic Programming (CLP) is a merger of two declarative paradigms: constraint solving and logic programming. Although a relatively new field, CLP has progressed in several quite different directions. In particular, the early fundamental concepts have been adapted to better serve in different areas of applications. In this survey of CLP, a primary goal is to give a systematic description of the major trends in terms of common fundamental concepts. The three main parts cover the theory, implementation issues, and programming for applications.

We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's "Introduction to Functional Programming" can be expressed using these operators. 1 Introduction Among the many styles and methodologies for the construction of computer programs the Squiggol style in our opinion deserves attention from the functional programming community. The overall goal of Squiggol is to calculate programs from their specification in the way a mathematician calculates solutions to differential equations, or uses arithmetic to solve numerical problems. It is not hard to state, prove and use laws for well-known operations such as addition, multiplication and ---at the function level--- composition. It is, however, quite hard to state, prove and use laws for arbitrarily recursively ...

The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higher-order processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higher-order processes. For this it is useful to generalise event structures to allow events which “persist.”

Concurrent constraint programming [Sar89,SR90] is a sim-ple and powerful model of concurrent computation based on the notions of store-as-constraint and process as information transducer. The store-as-valuation conception of von Neu-mann computing is replaced by the notion that the store is a constraint (a finite representation of a possibly infinite set of valuations) which provides partial information about the possible values that variables can take. Instead of “reading” and “writing ” the values of variables, processes may now ask (check if a constraint is entailed by the store) and tell (augment the store with a new constraint). This is a very general paradigm which subsumes (among others) nonde-terminate data-flow and the (concurrent) (constraint) logic programming languages. This paper develops the basic ideas involved in giving a coherent semantic account of these languages. Our first con-tribution is to give a simple and general formulation of the notion that a constraint system is a system of partial infor-mation (a la the information systems of Scott). Parameter passing and hiding is handled by borrowing ideas from the cylindric algebras of Henkin, Monk and Tarski to introduce diagonal elements and “cylindrification ” operations (which mimic the projection of information induced by existential quantifiers). The se;ond contribution is to introduce the notion of determinate concurrent constraint programming languages. The combinators treated are ask, tell, parallel composition, hiding and recursion. We present a simple model for this language based on the specification-oriented methodology of [OH86]. The crucial insight is to focus on observing the resting points of a process—those stores in which the pro-cess quiesces without producing more information. It turns out that for the determinate language, the set of resting points of a process completely characterizes its behavior on all inputs, since each process can be identified with a closure operator over the underlying constraint system. Very nat-ural definitions of parallel composition, communication and hiding are given. For example, the parallel composition of two agents can be characterized by just the intersection of the sets of constraints associated with them. We also give a complete axiomatization of equality in this model, present

this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of higher-type functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as well---and that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the so-called untyped -calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary way---avoiding heavy prerequisites (such as category theory) but still maintaining some level of abstraction---with the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...

Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.

We present the clp(FD) system: a Constraint Logic Programming language with finite domain constraints...

Contexts have been proposed as a means of performing strictness analysis on non-flat domains. Roughly speaking, a context describes how much a sub-expression will be evaluated by the surrounding program. This paper shows how contexts can be represented using the notion of projection from domain theory. This is clearer than the previous explanation of contexts in terms of continuations. In addition, this paper describes finite domains of contexts over the non-flat list domain. This means that recursive context equations can be solved using standard fixpoint techniques, instead of the algebraic manipulation previously used. Praises of lazy functional languages have been widely sung, and so have some curses. One reason for praise is that laziness supports programming styles that are inconvenient or impossible otherwise [Joh87, Hug84, Wad85a]. One reason for cursing is that laziness hinders efficient implementation. Still, acceptable efficiency for lazy languages is at last being achieved...

We propose a formal model of trust informed by the Global Computing scenario and focusing on the aspects of trust formation, evolution, and propagation. The model is based on a novel notion of trust structures which, building on concepts from trust management and domain theory, feature at the same time a trust and an information partial order.

Constraint logic programming languages such asCHIP[26,5] havedemonstrated the practicality ofdeclarative languages supporting consistency techniques and nondeterminism. Nevertheless they su er from the black-box e ect: the programmer must work with a monolithic, unmodi able, inextensible constraint-solver. This problem can be overcome within the logically and computationally richer concurrent constraint (cc) programming paradigm [17]. We showthat some basic constraint-operations currently hardwired into constraint-solvers can be abstracted and made available as combinators in the programming language. This allows complex constraint-solvers to be decomposed into logically clean and e ciently implementable cc programs over a much simpler constraint system. In particular, we show that the CHIP constraint-solver can be simply programmed in cc(FD), acc language with an extremely simple built-in constraint solver for nite domains. 1