Constraints in Term Algebras (Short Survey)

Uniication, which consists in solving equations in the (free) term algebra, is known to be a fundamental operation in many areas of computer science and, in particular, in logic programming. Disuniication, which consists in solving more complex formulae in the (free) term algebra, also revealed to be a fundamental operation (see 29, 13] for surveys on uniication and disuniication respectively). Recently, these computations have been seen as constraint solving in term algebras and this point of view is actually fruitful. A constraint system is deened by a logical language C (which is in practice a fragment of a rst-order language), a structure M in which the formulae of C are interpreted and an algorithm which decides, for every 2 C, whether is satissable in M or not. There are many examples: C can be a full rst-order language, in which case, the third condition implies the decidability of the ((rst-order) theory of M. For example, the constraint system could correspond to Presburger arithmetic or the theory of real numbers. It could also be the theory of nite trees, since this theory has been shown decidable 35, 34, 18]. Many other examples will be given later. Now, constraints can be (and have been) studied for their own mathematical interest. But, they can also be used to constrain other formulae. More precisely, given a logical language L, a class of structures S and a satisfaction relation j= on the one hand, and a constraint system (C; M;) on the other hand and given in addition, for each structure S in S, a mapping H S from the domain D of M into the domain D S of the structure S, the constrained logic consists of the language of pairs of formulae (called constrained formulae) jC where 2 L and C 2 C a satisfaction relation deened as follows. Given an assignment of the free variables of C into D and an assignment of the free variables of into D S ,

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