Combination of Constraint Systems II: Rational Amalgamation

38 and the rich amount of interesting relationships between them, are now well-understood (e.g., Co83, Ma88]). We believe that free amalgamation, rational amalgamation and a further construction called \innnite amalgamation" (still to be investigated) reeect this role on the higher level of amalgamation constructions. Many of the results that we have obtained for free and rational amalgamation can be interpreted in this sense: The universality-property of the free amalgamated product (see BS95]) reeects the status of the free term algebra as the absolutely free-algebra. We have seen that the free amalgamated product is always a substructure of the rational amalgamated product. This reeects the fact that the free term algebra is always a substructure of the algebra of rational trees. It is well-known that the uniication algorithm for the algebra of rational trees can be considered as the variant of the uniication algorithm for the free term algebra where we omit the occur-check. Similarly, the decomposition scheme for rational amalgamation as given here is|essentially|the decomposition scheme for free amalgamation where we omit the \inter-structural" occur-check that is provided by the choice of a linear ordering in the latter scheme. We would not be surprised if much more principles, techniques and theorems, well-known on the level of tree constructions, could be lifted to the level of combining structures. Our experience with rational amalgamation seems to indicate that this is a diicult, but promising line of research if we want to understand the scale of possibilities, and the limitations for combining solution domains and constraint solvers. \A feature-based constraint system for logic programming with entailment," Theoretical Comp. 37 where the sequence ~ y 0 is obtained from ~ y by removing y i. Since the elements in the sequence ~ y 0 are distinct atoms it follows as above that B j = 0. In this second case we have seen that we can construct a new output pair (0 ; 0) of Algorithm 2 such that A j = 0 and B j = 0. Moreover, the number of variables in (0 ; 0) is strictly smaller than the number of variables in (;). We may now use the same subcase analysis as above, replacing (;) by (0 ; 0), and iterate this contraction of formulae, if necessary. After a nite number of steps we reach an output pair that satisses all the assumptions that we made for (;) in the …