Binary Coded Unary Regular Languages.

L ⊆ { 0 , 1 } ∗ is a binary coded unary regular language , if there exists a unary regular language L ′ ⊆ { a } ∗ such that a x is in L ′ if and only if the binary representation of x is in L . If a unary language L ′ is accepted by an optimal deterministic finite automaton (dfa) A ′ with n states, then its binary coded version L is regular and can be accepted by a dfa A using at most n states, but at least 1 + ⌈ log n ⌉ states. There are witness languages matching these upper and lower bounds exactly, for each n . More precisely, if A ′ uses σ ≥ 0 states in the initial segment and μ · 2 ℓ states in the loop, where μ is odd and ℓ ≥ 0 , then the optimal A for L consists of a preamble with at most σ but at least max { 1 , 1 + ⌈ log σ ⌉ - ℓ } states, except for σ = 0 with no preamble, and a kernel with at most μ · 2 ℓ but at least μ + ℓ states. Also these lower bounds are matched exactly by witness languages, for each σ , μ , ℓ . The conversion in the opposite way is not always granted: there are binary regular languages the unary versions of which are not even context free. The corresponding conversion of a unary nondeterministic finite automaton (nfa) to a binary nfa uses O ( n 2 ) states and introduces a binary version of Chrobak normal form.