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I understand "escaping data" as making an exception when matching data; for example, if a program can't match data wrapped in some character/s (such as single and/or double quotes) without an exception, than we make an exception when "escaping" such character/s to be matched.

Is escaping a concept in CS?
Is it "part of how any computer would work" or just a technical implementation in human-developed programming languages?

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Escaping characters is part of an abstraction, most commonly pertaining to programming languages. Computer architectures, specifically at the sub-ISA level, don't know anything about it.

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Yes, escaping is used in various contexts in CS. It is one possible solution that is used to address a particular problem. It's not something that would necessarily be part of "how any computer would work", and it's not the only possible solution; there are other ways to allow inclusion of quotes inside a quoted string without using escaping.

For more, see https://en.wikipedia.org/wiki/Escape_character and https://en.wikipedia.org/wiki/Delimiter#Delimiter_collision.

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