We discuss the construction of proper widening operators on several weakly-relational numeric abstractions. Time and space complexities of the resolution algorithms are proved to be polynomial.
The paper provides the first graph-based proofs of the 4-CSP tractability and elaborates algorithms for 4-CSP resolution based on the positive linear dependence theory, the hypergraph closure and the constraint propagation technique. In this paper, we deal with a CSP subclass which we call 4-CSP and whose constraint network infers relations of the form: $\ $. It consists in explicitly excluding values or combination of values for some variables whenever they make a given subset of constraints unsatisfied. One of the most widely used techniques is the Constraint Propagation. Even though the problem is in general NP-complete, there are some approximation and practical techniques to tackle its intractability. Once the problem is expressed as a finite set of constraints, the goal is to find the variables' values satisfying them. Constraint Satisfaction Problem (CSP) is a framework for modeling and solving a variety of real-world problems.
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