The Formal and Computational Theory of Complex Constraint Solution

The framework presented in this chapter assumes that any grammar is a particular first order theory with equality. We will turn our attention to those first order theories admitting complete models and extend them to recursively defined relations. Logic programming is an ideal paradigm for such a framework in that it supports a direct mapping between grammars, seen as first order logic theories, and the first order theory of their implementation and at the same time provide a formally sound and efficient computational scheme. It is also particularly well suited to simultaneously satisfy the main requirements put on computational grammar formalisms, namely expressivity, formal soundness and computational tractability. As we will see, the framework presented in this chapter is not only compatible with the unification grammar tradition but it also constitutes a simple framework for extending the notion of unification to complex constraint resolution. At the same time a high degree of declarativeness is achieved by avoiding any reference to an operation like unification. The motivations for the choice of the particular family of first order theories will become apparent in the rest of this chapter. The main reasons derives from the fact that restricting the formal analysis to the static properties of formalisms does not do justice to the computational complexity of modern linguistic frameworks. A finer grained analysis of the formal and computational properties of formalisms than decidability, formal complexity and model theoretic properties, sheds a different light on