The concept of "strong generative capacity" (SGC) of a linguistic formalism was introduced by Chomsky in the early sixties in order to characterize descriptive capacity. However, the original definition proposed by Chomsky turned out to be unuseable, especially when one wished to compare the SGC of different types of formalisms. This book provides for the first time a rigorous and useful characterization of SGC, defining it as the model theoretic semantics of linguistic formalism. Specifically, abstract interpretation domains are defined in theory-neutral set-theoretical terms, and the SGC of a theory with respect to a given interpretation domain is characterized as the range of a specific interpretation function mapping structural descriptions of that theory into elements of that domain. Interpretation domains are defined for such notions as labeled constituency, dependency, endocentricity and linking and applied to the analysis of a range of linguistic formalisms, among which context-free grammars, dependency grammars, X-bar grammars, tree-adjoing grammars, transformational grammars and categorial grammars.