In setting up a formal system to specify a grammar formalism, the conventional (mathematical) wisdom is to start with primitives (basic primitive structures) as simple as possible, and then introduce various operations for constructing more complex structures. An alternate approach is to start with complex (more complicated) primitives, which directly capture some crucial linguistic properties and then introduce some general operations for composing these complex structures. These two approaches provide different domains of locality, i.e., domains over which various types of (...) linguistic dependencies can be specified. The latter approach, characterized as complicate locally, simplify globally (CLSG), pushes non‐local dependencies to become local, i.e., they arise in the basic primitive structures to start with.The CLSG approach has led to some new insights into syntactic description, semantic composition, language generation, statistical processing, and psycholinguistic phenomena, all these with possible relevance to the cognitive architecture of language. In this paper, we will describe these results in an introductory manner making use of the framework of lexicalized tree‐adjoining grammar (LTAG), a key example of the CLSG approach, thereby describing the interplay between formal analysis on the one hand and linguistic and processing issues on the other hand. (shrink)