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What is a 'context-free grammar' in relation to natural languages? This Wikipedia article, gives a broad description, but it isn't clear exactly what features of a language would result in it not being considered context free, or being considered so.
What is the fundamental property of a grammar that would result in it being context-free or non-context-free? Why is this property of special interest to linguists?
Ideally an answer would be digestible by readers with no mathematical background.
Sometimes technical descriptions are made with technical vocabulary neologisms in order to capture complicated and nuanced concepts which might be ungainly to convey correctly with simpler non-technical terms. So it may be difficult to come up with a description of a technical concept using only non-technical terms.
A rule in a language is context-free if the rule (affecting some words together) does not use text surrounding those words in order to apply correctly.
For example, suppose you want to make a comparative out of an adjective. Just add '-er': hot -> hotter, tall -> taller. For longer words, you use 'more' first: 'more consistent', 'more independent'. And then for a few words, like 'good' you have a rule exception: 'better'. For all of these, it doesn't matter what the word is, you don't need to know anything else in the sentence to form that comparative: in 'he is good' and 'he is better', the comparative doesn't depend on anything else in that sentence outside of what happens to the adjective.
In contrast, one might say that conjugation of verbs is context-sensitive (the alternative to context-free). For example, in the present simple in English, it is 'I say' and 'he says': to get the right ending on the verb, you need to know the context (the pronoun that comes before). The pronoun itself is not part of the thing that changed, but its presence changed the following word.
There are no fundamental properties. Some/most (?) natural languages are mildly context-sensitive to allow for features such as cross-serial dependencies. Pure context-free grammars are too cumbersome to be used in linguistics, one needs to add a constraint system (in the form of a formal logic, typical an equational logic) which makes the whole system Turing-complete even if the backbone is a context-free grammar.
CFG (context free phrase structure grammar) is important in linguistics, even fundamental, because it is what allows us to describe natural language expressions with hierarchical tree structures, which have become the universal descriptive tool of grammarians. Many linguistic theories are variants of CFG; e.g., Reed-Kellogg diagramming, tagmemics, the base component of transformational grammar, GPSG (Generalized Phrase Structure Grammar), dependency grammar.
Among the grammatical constructions in human languages which seemingly cannot be properly described by a CFG are cross serial dependencies, mentioned by Atamin above, and various constructions with discontinuous constituents, such as RNR constructions (in McCawley's theory of them).
I am an enthusiast for CFG, personally, though the consensus of contemporary syntacticians is that CFG is not sufficient to describe human languages.
