The formalism based on Noam Chomsky

The formalism of context-free grammars was developed in the mid-1950s by Noam Chomsky, and also their classification as a special type of formal grammar (which he called phrase-structure grammars).
A context-free grammar provides a simple and mathematically precise mechanism for describing the methods by which phrases in some natural language are built from smaller blocks, capturing the "block structure" of sentences in a natural way. Its simplicity makes the formalism amenable to rigorous mathematical study.

In Chomsky's generative grammar framework, the syntax of natural language was described by a context-free rules combined with transformation rules. In later work (e.g. Chomsky 1981), the idea of formulating a grammar consisting of explicit rewrite rules was abandoned. In other generative frameworks, e.g. Generalized Phrase Structure Grammar (Gazdar et al. 1985), context-free grammars were taken to be the mechanism for the entire syntax, eliminating transformations.

—A formal grammar (sometimes simply called a grammar) is a set of formation rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form. 

Formal language theory, the discipline which studies formal grammars and languages, is a branch of applied mathematics . Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas.
 

A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting must start. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a “recognizer"—a function in computing that determines whether a given string belongs to the language or is grammatically incorrect

Parsing is the process of recognizing an utterance (a string in natural languages) by breaking it down to a set of symbols and analyzing each one against the grammar of the language. Most languages have the meanings of their utterances structured according to their syntax—a practice known as compositional semantics.

The linguistic formalism derived from Chomsky can be characterized by a focus on innate, universal grammar (UG), and a disregard for the role of stimuli. According to this position, language use is only relevant in triggering the innate structures. With regard to the tradition, Chomsky’s position can be characterized as a continuation of essential principles of structuralist theory from Saussure (Givón 2001).

The formalist propositions regarding innateness and stimuli do fit extensively with the cognitive opposition
to behaviouristic psychology.

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