Magic squares, chess-like problems, cryptarithmetic puzzles, and similar classes of problems have been extensively used to challenge human reasoning capabilities. Lo Shu magic square can be traced back to 650 B.C., the eight-queens problem was proposed in 1848 by the chess player Max Bazzel TWO × TWO = THREE puzzle appeared in Strand Magazine in 1924. These puzzles are nowadays widely used in constraint programming courses. The first programming language provided with constraint modelling primitives (Sketchpad) has been proposed by the Turing award winner Ivan Sutherland in his PhD thesis (1963). Logemann and Loveland, when implementing the Davis–Putnam procedure (Davis and Putnam 1960) for testing the satisfiability of a propositional formula (SAT), devised an algorithm (Davis–Putnam–Logemann–Loveland (DPLL)) that has become the core of all SAT/and Answer Set Programming solvers (50 years later). It consists in choosing an un-assigned variable, assigning it a value 0 or 1, propagating the chosen value (unit propagation), and proceeding with the alternative value, if the original assignment leads to a contradiction (backtracking). Some years later Waltz (1975) introduced the notion of domain filtering (arc-consistency-based constraint propagation). With this idea the same DPLL scheme can be used for verifying the satisfiability of a constraint satisfaction problem, where the assignment is no longer 0/1 and the unit propagation is replaced by constraint propagation. For a detailed history of these early years achievements, we refer the reader to the works by Loveland et al. (2017), Jaffar and Maher (1994), and Freuder and Mackworth (2006).

Open forms of global constraints allow the addition of new variables to an argument during the execution of a constraint program. Such forms are needed for difficult constraint programming problems, where problem construction and problem solving are interleaved, and fit naturally within constraint logic programming. However, in general, filtering that is sound for a global constraint can be unsound when the constraint is open. This paper provides a simple characterization, called contractibility, of the constraints, where filtering remains sound when the constraint is open. With this characterization, we can easily determine whether a constraint has this property or not. In the latter case, we can use it to derive a contractible approximation to the constraint. We demonstrate this work on both hard and soft constraints. In the process, we formulate two general classes of soft constraints.

We present the third generation of the constraint answer set system clingcon, combining Answer Set Programming (ASP) with finite domain constraint processing (CP). While its predecessors rely on a black-box approach to hybrid solving by integrating the CP solver gecode, the new clingcon system pursues a lazy approach using dedicated constraint propagators to extend propagation in the underlying ASP solver clasp. No extension is needed for parsing and grounding clingcon's hybrid modeling language since both can be accommodated by the new generic theory handling capabilities of the ASP grounder gringo. As a whole, clingcon 3 is thus an extension of the ASP system clingo 5, which itself relies on the grounder gringo and the solver clasp. The new approach of clingcon offers a seamless integration of CP propagation into ASP solving that benefits from the whole spectrum of clasp's reasoning modes, including, for instance, multi-shot solving and advanced optimization techniques. This is accomplished by a lazy approach that unfolds the representation of constraints and adds it to that of the logic program only when needed. Although the unfolding is usually dictated by the constraint propagators during solving, it can already be partially (or even totally) done during preprocessing. Moreover, clingcon's constraint preprocessing and propagation incorporate several well-established CP techniques that greatly improve its performance. We demonstrate this via an extensive empirical evaluation contrasting, first, the various techniques in the context of CSP solving and, second, the new clingcon system with other hybrid ASP systems.

Researchers in answer set programming and constraint programming have spent significant efforts in the development of hybrid languages and solving algorithms combining the strengths of these traditionally separate fields. These efforts resulted in a new research area: constraint answer set programming. Constraint answer set programming languages and systems proved to be successful at providing declarative, yet efficient solutions to problems involving hybrid reasoning tasks. One of the main contributions of this paper is the first comprehensive account of the constraint answer set language and solver ezcsp, a mainstream representative of this research area that has been used in various successful applications. We also develop an extension of the transition systems proposed by Nieuwenhuis et al. in 2006 to capture Boolean satisfiability solvers. We use this extension to describe the ezcsp algorithm and prove formal claims about it. The design and algorithmic details behind ezcsp clearly demonstrate that the development of the hybrid systems of this kind is challenging. Many questions arise when one faces various design choices in an attempt to maximize system's benefits. One of the key decisions that a developer of a hybrid solver makes is settling on a particular integration schema within its implementation. Thus, another important contribution of this paper is a thorough case study based on ezcsp, focused on the various integration schemas that it provides.

Satisfiability modulo theories (SMT) solvers have been used successfully as reasoning engines for automated verification and other applications based on automated reasoning. Current techniques for dealing with quantified formulas in SMT are generally incomplete, forcing SMT solvers to report “unknown” when they fail to prove the unsatisfiability of a formula with quantifiers. This inability to return counter models limits their usefulness in applications that produce queries involving quantified formulas. In this paper, we reduce these limitations by integrating finite model finding techniques based on constraint solving into the architecture used by modern SMT solvers. This approach is made possible by a novel solver for cardinality constraints, as well as techniques for on-demand instantiation of quantified formulas. Experiments show that our approach is competitive with the state of the art in SMT, and orthogonal to approaches in automated theorem proving.

Constraint answer set programming is a promising research direction that integrates answer set programming with constraint processing. It is often informally related to the field of satisfiability modulo theories. Yet, the exact formal link is obscured as the terminology and concepts used in these two research areas differ. In this paper, we connect these two research areas by uncovering the precise formal relation between them. We believe that this work will boost the cross-fertilization of the theoretical foundations and the existing solving methods in both areas. As a step in this direction, we provide a translation from constraint answer set programs with integer linear constraints to satisfiability modulo linear integer arithmetic that paves the way to utilizing modern satisfiability modulo theories solvers for computing answer sets of constraint answer set programs.

Constraint answer set programming (CASP) is an extension of answer set programming that allows for numerical constraints to be added in the rules. PDDL+ is an extension of the PDDL standard language of automated planning for modeling mixed discrete-continuous dynamics. In this paper, we present CASP solutions for dealing with PDDL+ problems, i.e., encoding from PDDL+ to CASP, and extensions to the algorithm of the ezcsp CASP solver in order to solve CASP programs arising from PDDL+ domains. An experimental analysis, performed on well-known linear and non-linear variants of PDDL+ domains, involving various configurations of the ezcsp solver, other CASP solvers, and PDDL+ planners, shows the viability of our solution.

This paper explores the use of Answer Set Programming (ASP) in solving Distributed Constraint Optimization Problems (DCOPs). The paper provides the following novel contributions: (1) it shows how one can formulate DCOPs as logic programs; (2) it introduces ASP-DPOP, the first DCOP algorithm that is based on logic programming; (3) it experimentally shows that ASP-DPOP can be up to two orders of magnitude faster than DPOP (its imperative programming counterpart) as well as solve some problems that DPOP fails to solve, due to memory limitations; and (4) it demonstrates the applicability of ASP in a wide array of multi-agent problems currently modeled as DCOPs.