This paper presents for the first time the semantic foundations of CLP in a self-contained and complete package. The main contributions are threefold. First, we extend the original conference paper by presenting definitions and basic semantic constructs from first principles, giving new and complete proofs for the main lemmas. Importantly, we clarify which theorems depend on conditions such as solution compactness, satisfaction completeness and independence of constraints. Second, we generalize the original results to allow for incompleteness of the constraint solver. This is important since almost all CLP systems use an incomplete solver. Third, we give conditions on the (possibly incomplete) solver which ensure that the operational semantics is confluent, that is, has independence of literal scheduling.

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.

This paper presents a new and very rich class of (con-current) programming languages, based on the notion of comput.ing with parhal information, and the con-commitant notions of consistency and entailment. ’ In this framework, computation emerges from the inter-action of concurrently executing agents that communi-cate by placing, checking and instantiating constraints on shared variables. Such a view of computation is in-teresting in the context of programming languages be-cause of the ability to represent and manipulate partial information about the domain of discourse, in the con-text of concurrency because of the use of constraints for communication and control, and in the context of AI because of the availability of simple yet powerful mechanisms for controlling inference, and the promise that very rich representational/programming languages, sharing the same set of abstract properties, may be pos-sible. To reflect this view of computation, [Sar89] develops the cc family of languages. We present here one mem-ber of the family, CC(.L,+) (pronounced “cc with Ask and Choose”) which provides the basic operations of blocking Ask and atomic Tell and an algebra of be-haviors closed under prefixing, indeterministic choice, interleaving, and hiding, and provides a mutual recur-sion operator. cc(.L,-t) is (intentionally!) very similar to Milner’s CCS, but for the radically different under-lying concept of communication, which, in fact, pro-’ The class is founded on the notion of “constraint logic pro-gramming ” [JL87,Mah87], fundamentally generalizes concurrent logic programming, and is the subject of the first author’s disser-tation [Sar89], on which this paper is substantially based.

Constraint Handling Rules (CHR) are our proposal to allow more flexibility and application-oriented customization of constraint systems. CHR are a declarative language extension especially designed for writing user-defined constraints. CHR are essentially a committed-choice language consisting of multi-headed guarded rules that rewrite constraints into simpler ones until they are solved. In this broad survey we aim at covering all aspects of CHR as they currently present themselves. Going from theory to practice, we will define syntax and semantics for CHR, introduce an important decidable property, confluence, of CHR programs and define a tight integration of CHR with constraint logic programming languages. This survey then describes implementations of the language before we review several constraint solvers - both traditional and non- standard ones - written in the CHR language. Finally we introduce two innovative applications that benefited from using CHR.

The Oz Programming Model (OPM) is a concurrent programming model subsuming higher-order functional and object-oriented programming as facets of a general model. This is particularly interesting for concurrent object-oriented programming, for which no comprehensive formal model existed until now. The model

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 paper describes the design, implementation, and applications of the constraint logic language cc(FD). cc(FD) is a declarative nondeterministic constraint logic language over finite domains based on the cc framework [33], an extension of the CLP scheme [21]. Its constraint solver includes (non-linear) arithmetic constraints over natural numbers which are approximated using domain and interval consistency. The main novelty of cc(FD) is the inclusion of a number of general-purpose combinators, in particular cardinality, constructive disjunction, and blocking implication, in conjunction with new constraint operations such as constraint entailment and generalization. These combinators signi cantly improve the operational expressiveness, extensibility, and flexibility of CLP languages and allow issues such as the definition of non-primitive constraints and disjunctions to be tackled at the language level. The implementation of cc(FD) (about 40,000 lines of C) includes a WAM-based engine [44], optimal arc-consistency algorithms based on AC-5 [40], and incremental implementation of the combinators. Results on numerous problems, including scheduling, resource allocation, sequencing, packing, and hamiltonian paths are reported and indicate that cc(FD) comes close to procedural languages on a number of combinatorial problems. In addition, a small cc(FD) program was able to nd the optimal solution and prove optimality to a famous 10/10 disjunctive scheduling problem [29], which was left open for more than 20 years and nally solved in 1986.

Constraints allow programmers and users to state declaratively a relation that should be maintained, rather than requiring them to write procedures to maintain the relation themselves. They are thus useful in such applications as programming languages, user interface toolkits, and simulation packages. In many situations, it is desirable to be able to state both required and preferential constraints. The required constraints must hold. Since the other constraints are merely preferences, the system should try to satisfy them if possible, but no error condition arises if it cannot. A constraint hierarchy consists of a set of constraints, each labeled as either required or preferred at some strength. An arbitrary number of different strengths is allowed. In the discussion of a theory of constraint hierarchies, we present alternate ways of selecting among competing possible solutions, and prove a number of propositions about the relations among these alternatives. We then outline algorit...

Almost all constraint logic programming systems include negation, yet nowhere has a sound operational model for negation in CLP been discussed. The SLDNF approach of only allowing ground negative subgoals to execute is very restrictive in constraint logic programming where most variables appearing in a derivation never become ground. By describing a scheme for constructive negation in constraint logic programming we give a sound and complete operational model for negation in these languages. Constructive negation was first formulated for logic programming in the Herbrand Universe and involves introducing disequality constraints. Constraint logic programming thus provides a much more natural framework for describing constructive negation. In this paper we describe a framework for constructive negation for constraint logic programming over arbitrary structures which is sound and complete with respect to the three-valued consequences of the completion of a program. Through this descriptio...

Constraint programming is newly flowering in industry. Several companies have recently started up to exploit the technology, and the number of industrial applications is now growing very quickly. This survey will seek, by examples,