By Marie Pelleau

ISBN-10: 1785480103

ISBN-13: 9781785480102

Constraint Programming goals at fixing demanding combinatorial difficulties, with a computation time expanding in perform exponentially. The equipment are at the present time effective sufficient to unravel huge commercial difficulties, in a popular framework. even if, solvers are devoted to a unmarried variable kind: integer or genuine. fixing combined difficulties will depend on advert hoc changes. In one other box, summary Interpretation bargains instruments to turn out software houses, by way of learning an abstraction in their concrete semantics, that's, the set of attainable values of the variables in the course of an execution. numerous representations for those abstractions were proposed. they're known as summary domain names. summary domain names can combine any form of variables, or even signify kin among the variables.

In this paintings, we outline summary domain names for Constraint Programming, with the intention to construct a customary fixing procedure, facing either integer and genuine variables. We additionally examine the octagons summary area, already outlined in summary Interpretation. Guiding the quest through the octagonal kin, we receive strong effects on a continual benchmark. We additionally outline our fixing procedure utilizing summary Interpretation concepts, so that it will comprise present summary domain names. Our solver, AbSolute, is ready to clear up combined difficulties and use relational domains.

- Exploits the over-approximation the right way to combine AI instruments within the equipment of CP
- Exploits the relationships captured to resolve non-stop difficulties extra effectively
- Learn from the builders of a solver able to dealing with virtually all summary domains

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**Extra resources for Abstract Domains in Constraint Programming**

**Sample text**

Sn ) ∈ D State of the Art 1, p , Ci (s1 . . sn )}. For a constraint C, we ˆ | C(s1 . . sn )} as the solution set for C. SC = {(s1 . . – Let us consider the following 4 × 4 Sudoku grid: 3 1 4 1 2 1 A possible model is to associate to each cell a variable as follows: v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12 v13 v14 v15 v16 Each variable can take a value between 1 and 4. Thus, we have ˆ1 = D ˆ2 = · · · = D ˆ 16 = 1, 4 . To specify that a cell has a ﬁxed D ˆ 3 = {1} or add the constraint value, we can either modify its domain D v3 = 1.

Note that even if ρ is not an optimal abstraction of ρ , Yδ may be signiﬁcantly more precise than ρ (X ). A relevant application is the analysis of complex test conjunction C1 ∧ · · · ∧ Cp , where each atomic test Ci is modeled in the abstract as ρi . Generally, ρ = ρ1 ◦ · · · ◦ ρp is not optimal, even when each ρi is. Lower closure operators may be reformulated as a ﬁxpoint. This uniﬁes the use of narrowings and brings out the similarities in the iterative computations. Given an element X, ρ computes the greatest ﬁxpoint smaller than X, that is ρ(X) = gfpX ρ.

Indeed, by replacing v1 and v2 by their domains, we have [0, 2]+[0, 2] = [0, 4], and so [0, 4] ≤ 6 is true since any point of the interval [0, 4] is less than or equal to 6. The second constraint C2 answers false, indeed [0, 2] − [0, 2] = [−2, 2] and the condition [−2, 2] ≥ 4 is always false. As for the last constraint, it answers maybe: we have [−2, 2] = 0, the only possible deduction is that there are perhaps values of v1 and v2 for which the constraint is satisﬁed. 3. Solutions and approximations Given a problem, we try to solve it, that is, to ﬁnd a solution.

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