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...and a narrowing algorithm, become the instances of our CIIQ algorithm and CII algorithm. Other two attempts to provide a general setting for constraint propagation algorithms can be found in Benhamou =-=[4]-=- and Telerman and Ushakov [27]. In these papers instead of ⊔-po’s specific families of subsets of the considered domain are taken with the inverse subset order. In Benhamou [4] they are called approxi...

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...ox C(φ, B) such that 214sEfficient Solving of Complex-Structured Arithmetic Constraint Systems C(φ, B) ⊆ B and such that C(φ, B) contains all solutions of φ in B (cf. the notion of narrowing operator =-=[5, 3]-=-). There are several methods for implementing such contractors. The most basic method [7, 6] decomposes all atomic constraints (i.e., constraints of the form t ≥ 0 or t = 0, where t is a term) into co...

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...loating-point box B, they compute another floating-point box N(φ, B) such that N(φ, B) ⊆ B (contractance), and such that N(φ, B) contains all solutions of φ in B (cf. the notion of narrowing operator =-=[5, 4]-=-, sometimes also called contractor). There are several methods for implementing such a pruning algorithm. The most basic method [13, 11, 6] decomposes all atomic constraints (i.e., constraints of the ...

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...′ in F − Q is at fixpoint at the start of the loop. The incremental propagation solver isolv with this definition of new (assuming Fo = ∅) is more or less equivalent to the propagation algorithms in [=-=Benhamou 1996-=-] and [Apt 2003, page 267]. Example 3.2 Incremental propagation. Consider the problem with constraints ACM Transactions on Programming Languages and Systems, Vol. V, No. N, Month 20YY.s8 · C. Schulte ...

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... X 3 =e -> X 4 N 0 : X 1, X 2 N 1 : X 1 =b -> X 3 N 2 : X 2 =c -> X 4 N 3 : X 3 =e -> X 4 N 3 : X 4 =g -> X 3 V I = {X 1 ,X 2 } X 4 =g -> X 3 (a) X 2 =c -> X 4 X 1 =b, X 3 (c) V I = {X 1 ,X 2 } N 1 : =-=[1]-=- N 3 : [2] X2 =c, X4 X2 <> c X2 =c, X X 4 2 <> c X 3 =e, X 4X 3 <> e X 1 <> b inconsistent ! X4 =g, X3 X4 <> g N 0 : X 1, X 2 N 0 : [0] X 1 =b -> X 3 X3 =e -> X 4 N 2 : X 2 =c -> X 4 N 1 : X 1 =b -> X...

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...rithms in a uniform way. Using such a single framework we can easier verify, compare, modify, parallelize or combine these algorithms. The last point has already been made to large extent in Benhamou =-=[2]-=-. Additionally, we clarified the role played by the notions of commutativity and semi-commutativity. The line of research presented here could be extended in a number of ways. First, it would be inter...

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...[Moore 1966; Alefeld and Herzberger 1983] to compute verified approximate solutions of real constraint systems. The reader will find in various works [Older and Vellino 1990; Benhamou and Older 1997; =-=Benhamou 1996-=-; Van Hentenryck et al. 1997; Benhamou et al. 1994; Van Hentenryck et al. 1997; Benhamou et al. 1999] a thorough presentation of the interval constraint solving framework and of the associated state-o...