The processor chip itself is designed as a DFA (as is very common in digital logic design though EEs prefer the term finite state machine or FSM). These slides do a good job of explaining this perspective on digital design. [1]

It becomes much more powerful when given the ability to read and write from arbitrary storage, they same way theoretical DFAs become turing machines when given the same power.

I am glossing over the complications that on-chip caches and memory hierarchy introduce, but I feel the core idea of a processor chip as a DFA is still worthwhile.

[1]http://www-inst.eecs.berkeley.edu/~cs150/fa05/CLD_Supplement...