Adapting Real Quantifier Elimination Methods for Conflict Set Computation

The satisfiability problem in real closed fields is decidable. In the context of satisfiability modulo theories, the problem restricted to conjunctive sets of literals, that is, sets of polynomial constraints, is of particular importance. One of the central problems is the computation of good explanations of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the input constraints whose conjunction is already unsatisfiable. We adapt two commonly used real quantifier elimination methods, cylindrical algebraic decomposition and virtual substitution, to provide such conflict sets and demonstrate the performance of our method in practice

An introduction to interval-based constraint processing.

Constraint programming is often associated with solving problems over finite domains. Many applications in engineering, CAD and design, however, require solving problems over continuous (real-valued) domains. While simple constraint solvers can solve linear constraints with the inaccuracy of floating-point arithmetic, methods based on interval arithmetic allow exact (interval) solutions over a much wider range of problems. Applications of interval-based programming extend the range of solvable problems from non-linear polynomials up to those involving ordinary differential equations. In this text, we give an introduction to current approaches, methods and implementations of interval-based constraint programming and solving. Special care is taken to provide a uniform and consistent notation, since the literature in this field employs many seemingly different, but yet conceptually related, notations and terminology

How to take into account general and contextual knowledge for interactive aiding design: Towards the coupling of CSP and CBR approaches

The goal of this paper is to show how it is possible to support design decisions with two different tools relying on two kinds of knowledge: case-based reasoning operating with contextual knowledge embodied in past cases and constraint filtering that operates with general knowledge formalized using constraints. Our goals are, firstly to make an overview of existing works that analyses the various ways to associate these two kinds of aiding tools essentially in a sequential way. Secondly, we propose an approach that allows us to use them simultaneously in order to assist design decisions with these two kinds of knowledge. The paper is organized as follows. In the first section, we define the goal of the paper and recall the background of case-based reasoning and constraint filtering. In the second section, the industrial problem which led us to consider these two kinds of knowledge is presented. In the third section, an overview of the various possibilities of using these two aiding decision tools in a sequential way is drawn up. In the fourth section, we propose an approach that allows us to use both aiding decision tools in a simultaneous and iterative way according to the availability of knowledge. An example dealing with helicopter maintenance illustrates our proposals

Modular Constraint Solver Cooperation via Abstract Interpretation

Cooperation among constraint solvers is difficult because different solving paradigms have different theoretical foundations. Recent works have shown that abstract interpretation can provide a unifying theory for various constraint solvers. In particular, it relies on abstract domains which capture constraint languages as ordered structures. The key insight of this paper is viewing cooperation schemes as abstract domains combinations. We propose a modular framework in which solvers and cooperation schemes can be seamlessly added and combined. This differs from existing approaches such as SMT where the cooperation scheme is usually fixed (e.g., Nelson-Oppen). We contribute to two new cooperation schemes: (i) interval propagators completion that allows abstract domains to exchange bound constraints, and (ii) delayed product which exchanges over-approximations of constraints between two abstract domains. Moreover, the delayed product is based on delayed goal of logic programming, and it shows that abstract domains can also capture control aspects of constraint solving. Finally, to achieve modularity, we propose the shared product to combine abstract domains and cooperation schemes. Our approach has been fully implemented, and we provide various examples on the flexible job shop scheduling problem. Under consideration for acceptance in TPLP.Comment: Paper presented at the 36th International Conference on Logic Programming (ICLP 2020), University Of Calabria, Rende (CS), Italy, September 2020, 17 pages. v2: Fix an example in Section 3.2 (improved closure

TOY: A System for Experimenting with Cooperation of Constraint Domains

AbstractThis paper presents, from a user point-of-view, the mechanism of cooperation between constraint domains that is currently part of the system TOY, an implementation of a constraint functional logic programming scheme. This implementation follows a cooperative goal solving calculus based on lazy narrowing. It manages the invocation of solvers for each domain, and projection operations for converting constraints into mate domains via mediatorial constraints. We implemented the cooperation among Herbrand, real arithmetic (R), finite domain (FD) and set (S) domains. We provide two mediatorial constraints: The first one relates the numeric domains FD and R, and the second one relates FD and S