The study of different types of grammar and the devices that are necessary to produce them is part of the branch of mathematical linguistics or computational theory called:

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The study of grammar and the devices used to produce it falls under the purview of automata theory, which deals with the abstract computation models that underlie language structures. Other options like Linguistic Theory, Phonetics, and Phonology may be tangentially related to the study of language and its structures but do not specifically deal with the mathematical modeling of grammatical rules and their execution using abstract computational models. Syntax, while more similar to automata theory than other options, specifically concerns itself with sentence formation and word order, whereas automata theory deals with the broader set of rules that govern language as a whole.

The Study of Different Types of Grammar and the Devices