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I have some confusion about the differences between finite & infinite. can someone tell me how the Toll machine or Park meter, soda vending machine can be used with infinite language? if not, please give me an example of other application.

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As Yuval Filmus said, the KMP substring matching is widely used in practice because of its modularity and efficiency.

A maybe more "practical" exemple is lexical analysis. (https://en.wikipedia.org/wiki/Lexical_analysis)

Depending on your application (let say or the compilation of a programming language) the parsing phase can be done with some various generalisation of automatons (state transition machines or pile-automatons for instance). The language they accept is infinite (think of the set of all programs that are syntactically correct in your favorite language), yet the recognition of a correct program should be done by an automata (or a generalisation of it) which is finitely presented.

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