The main influence of Automata theory in programming was in the Compilers Theory.
It is used in the Lexical Analysis part, and it is used in the construction of parsers and compilers.
The finite automata allowed to implement such stuffs in a generic way.
As you say, the field wouldn't exist as we know it, but maybe another way.
Maybe the parsers, compilers would be very ugly. And the code would be full of conditional constructs (if, then, else).
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