Physical computational devices have operational constraints that necessitate nonzero entropy production (EP). In particular, almost all real-world computers are "periodic" in that they iteratively apply the same physical process, and are "local" in that not all physical variables that are statistically coupled are also physically coupled. Here we investigate the nonzero EP that necessarily results from these constraints in deterministic finite automata (DFA), a foundational system of computer science theory. First we derive expressions for the minimal EP due to implementing a DFA with a periodic, local physical process. We then relate these costs to the computational characteristics of the DFA. Specifically, we show that these costs divide "regular languages" into two classes: those with an invertible local update map, which have zero minimal cost, and those with a non-invertible local update map, which have high cost. We also demonstrate the thermodynamic advantages of implementing a DFA with a physical process that is agnostic about which inputs it will receive.