How can I prove that every regular language $R$ has linear time complexity, i.e. every regular language satisfies $$R \in \text{TIME}(n)$$
2026-04-01 10:21:36.1775038896
Why every regular language is in $\text{TIME}(n)$?
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If $R$ is a regular language, then $R$ is recognized by some deterministic finite state machine. A finite state machine processes a string $x$ by reading each character and transitioning to the appropriate state—it takes exactly $|x|$ steps.
Every deterministic finite state machine $M$ is equivalent to a Turing machine that just reads its input in order and has the same finite control (states and transition rule) as $M$. Hence every regular language is recognized by a linear-time Turing machine.