May I chime in? A gentle introduction to process algebras [^1] shows some examples of how you can represent programs as (possibly infinite) state machines. Essentially you define some operators to compose smaller programs (aka. processes) into larger ones, hence the name "process algebra". These definitions allow you to "see" your process as a generalization of a state machine known as a labeled transition system [^2]. And then you can exploit this representation to verify properties about the process. Lots of fun!
[^2]: These rules are called an operational semantics. There are other ways of giving semantics to a formal system: for instance, denotational semantics exploit recursive mathematical definitions. Each way has its pros and cons.
State machines (that contain both labeled and unlabeled transition systems) may indeed serve as operational semantics of some formalisms, but they can be denotational semantics of others, or even studied completely separately from any programming formalism.
Not only process algebras, but even the lambda calculus is naturally expressed as a state machine, where the states represent the expression, and transitions represent possible reductions (lambda calculus is nondeterministic).
[^1]: https://pdfs.semanticscholar.org/12d9/eae1638729aeb237b5be44...
[^2]: These rules are called an operational semantics. There are other ways of giving semantics to a formal system: for instance, denotational semantics exploit recursive mathematical definitions. Each way has its pros and cons.