A Proof Theoretic View of Constraint Programming

We provide here a proof theoretic account of constraint programming that attempts to capture the essential ingredients of this programming style. We exemplify it by presenting proof rules for linear constraints over interval domains, and illustrate their use by analyzing the constraint propagation process for the {\tt SEND + MORE = MONEY} puzzle. We also show how this approach allows one to build new constraint solvers.Comment: 25 page

Towards declarative diagnosis of constraint programs over finite domains

The paper proposes a theoretical approach of the debugging of constraint programs based on a notion of explanation tree. The proposed approach is an attempt to adapt algorithmic debugging to constraint programming. In this theoretical framework for domain reduction, explanations are proof trees explaining value removals. These proof trees are defined by inductive definitions which express the removals of values as consequences of other value removals. Explanations may be considered as the essence of constraint programming. They are a declarative view of the computation trace. The diagnosis consists in locating an error in an explanation rooted by a symptom.Comment: In M. Ronsse, K. De Bosschere (eds), proceedings of the Fifth International Workshop on Automated Debugging (AADEBUG 2003), September 2003, Ghent. cs.SE/030902

Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556