Overview of Existing Recurrent Schematizations

Innnite sets of objects with structural similarities appear frequently at diierent places in the theory of equational reasoning, automatic deduction, logic programming, and databases. The objects in such innnite families can have several forms. They are represented most frequently by terms. The formalisms to deal with innnite families are fairly limited. As long as only the representation is concerned, such formalisms exist and are well-known. The rst used formalisms were regular and context-free grammars, later extended by tree automata GS84]. Recently, this arsenal was enlarged by lazy evaluation and the concept of constraints. The problems appear if we wish to unify innnite families represented by these formalisms. In addition to the drawback that context-free grammars describe only sets of strings, it is undecidable if the intersection of two context-free languages is empty. Hence the uniication of context free languages is undecidable since strings in context-free languages are ground terms. Regular grammars and regular tree automata often lack a necessary schematization power. Innnite families often display patterns that cannot be captured by them. Lazy evaluation cannot be combined with uniication in a satisfactory way. Constraints represent innnite families implicitly , by restrictions on domains. Therefore, uniication of constraint expressions results in constraint solving that can be as problematic as the direct uniication of the innnite families itself. Three phenomena are usually observed: (1) the constraint solving is decidable, but the constraint formalism lacks schematization power, (2) the constraint formalism is powerful enough to represent the innnite families but the uniication problem for these constraints is undecidable, and (3) the constraint formalism is powerful enough and the uniication problem is decidable, but the corresponding constraint solving uniication algorithm produces an innnite family of constraints. In practice, the manipulated sets must be nite, unless there is a possibility to use constraints and implicit representations. The previous considerations clearly indicate the need for a formalism that allows to express explicitly innnite families by nite means, that has a decidable uniication problem, and that has a terminating uniication algorithm yielding a nite representation of the results. Moreover, such formalism should have a semantics 1