By Marie Pelleau

Constraint Programming goals at fixing demanding combinatorial difficulties, with a computation time expanding in perform exponentially. The tools are this present day effective sufficient to resolve huge commercial difficulties, in a time-honored framework. although, solvers are devoted to a unmarried variable variety: integer or genuine. fixing combined difficulties depends upon advert hoc variations. In one other box, summary Interpretation bargains instruments to turn out application homes, 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. a variety of representations for those abstractions were proposed. they're referred to as summary domain names. summary domain names can combine any kind of variables, or even symbolize family members among the variables.

In this paintings, we outline summary domain names for Constraint Programming, so that it will construct a customary fixing approach, facing either integer and actual variables. We additionally learn the octagons summary area, already outlined in summary Interpretation. Guiding the hunt by means of the octagonal family, we receive strong effects on a continuing benchmark. We additionally outline our fixing process utilizing summary Interpretation recommendations, with a view to comprise present summary domain names. Our solver, AbSolute, is ready to clear up combined difficulties and use relational domains.

  • Exploits the over-approximation how 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 in a position to dealing with virtually all summary domains

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Note that these two trees have the same area. In the case of failure, the cut in the search tree is greater when the first-fail strategy is used. 1. A discret solver using integer Cartesian product to represent the domains. zip int j ← 0 /* j indicates the depth in the search tree */ int op[] /* at depth j, stores the index of the variable on which the hypothesis is made, uniformly initialized at 0 */ int width[] /* at depth j, stores the width of the tree already explored, uniformly initialized at 0 */ integer Cartesian product e ∈ S list of integer Cartesian product sols /* stores the solutions */ e←D /* initialization with the initial domains */ repeat e ← generalized arc-consistency(e) width[j]+ + if e is a solution then /* success */ sols ← sols ∪ e end if if e = ∅ or e is a solution then /* back to the last backtrackable point */ while j ≥ 0 et width[j] ≥ |Dop[j] | do width[j] ← 0 j– – end while else /* new hypothesis */ choose a variable vi to instantiate op[j] ← i j+ + end if if j ≥ 0 then assign vop[j−1] to the (width[j] + 1)-th possible value /* backtrackable point */ end if until j < 0 Following this idea, the heuristic proposed in [BRÉ 79] chooses the variable with the smallest domain (dom) and appearing in the biggest number of constraints deg.

Some major domains, such as polyhedra, do not feature any. This difference of interest between the widening and the narrowing may be explained by three facts: first, narrowings are not necessary to achieve soundness unlike widenings which are mandatory. Indeed, the widening allows an approximation of the least fixpoint to be computed even if the partially ordered set has an infinite increasing chain. So, if the result obtained by the widening meets the given specifications, then the considered program is sound.

2. It uses the HC. In these examples, the resolution stops when: – all solutions are found, that is, when: - all solutions have been listed in the discrete case; - all computed boxes only contain solutions or are smaller than a certain accuracy in the continuous case. – it has been proved that there is no solution, in other words, the search space has been fully explored without success. Note that in both the discrete and continuous cases, the resolution process can be modified in order to stop as soon as the first solution is found.

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