When we have a decision problem, " does $f(x)=1$ hold?", we call the set of strings $x$ for which the answer is yes a "language".
Why this strange terminology?
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Sign up to join this communityWhen we have a decision problem, " does $f(x)=1$ hold?", we call the set of strings $x$ for which the answer is yes a "language".
Why this strange terminology?
I believe a lot of this came out of linguistics research. People wanted to write down a formal description (grammar) of, e.g., the English language. By extension, the set of strings defined by any grammar is a language and, since unrestricted grammars are Turing-powerful, any computable set of strings is the language of some grammar.
I think it is useful to note that most pioneering CS work, such as initial work by Turing and Church, had the intention to describe computation carried out by humans. (Recall that a computer was a human profession around that time!)
From this perspective, the set of strings that are output of a decision problem is the set of strings that a (precise and correct!) human 'computer' may 'utter' (i.e. output, most likely the computer would write his/her answers down).
Calling this set of strings that refers to structured, but human, communication a 'language' doesn't sound very far-fetched to me, so this could be a reason for the term.
My knowledge of history isn't good enough to check whether this is merely my interpretation or the common interpretation at the time, so take this with a grain of salt. Still, I hope this provides a reason to name the concept a 'language'.
From Wikipedia, “a formal language is a set of strings of symbols together with a set of rules that are specific to it.” To precisely specify what we are trying to decide, ambiguous natural languages won’t do, so instead mathematicians represent possible inputs with precise symbols. Together, the set of strings that result in Yes outputs comprise, in a poetic sense, the language of that problem, e.g., how 2, 3, 5, 7, and so on are the language of prime numbers.
Again according to Wikipedia
The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift (1879), literally meaning “concept writing,” and which Frege described as a “formal language of pure thought.”