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For a formal language L , the problem of language enumeration asks to compute the length-lexicographically smallest word in L larger than a given input w ∈ Σ ∗ (henceforth called the L -successor of w). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if L is recognized by a DFA with n states, then 2 Θ (n log n) states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute L -successors. As a byproduct, we obtain that if L is recognized by a DFA with n states, then 2 Θ (n log n) states are sufficient for a DFA to recognize the subset S (L) of L composed of its lexicographically smallest words. We give a matching lower bound that holds even if S (L) is represented as an NFA. It has been known that L -successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in N L. If the regular language L is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of L under D L O G T I M E -uniform A C 0 reductions. In particular, this implies that fixed star-free languages can be enumerated in A C 0 , arbitrary fixed regular languages can be enumerated in N C 1 and that there exist regular languages for which the problem is N C 1 -complete. [ABSTRACT FROM AUTHOR]

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number > 2 5 is the sum of at most three natural numbers whose base- 2 representation has an equal number of 0 's and 1 's. [ABSTRACT FROM AUTHOR]

We consider several language-theoretic aspects of various notions of unique decipherability (or unique factorization) in formal languages. Given a language L at some position within the Chomsky hierarchy, we investigate the language of words UD (L) in L ⁎ that have unique factorization over L. We also consider similar notions for weaker forms of unique decipherability, such as numerically decipherable words ND (L) , multiset decipherable words MSD (L) and set decipherable words SD (L). Although these notions of unique factorization have been considered before, it appears that the languages of words having these properties have not been positioned in the Chomsky hierarchy up until now. We show that UD (L) , ND (L) , MSD (L) and SD (L) need not be context-free if L is context-free. In fact ND (L) and MSD (L) need not be context-free even if L is finite, although UD (L) and SD (L) are regular in this case. We show that if L is context-sensitive, then so are UD (L) , ND (L) , MSD (L) and SD (L). We also prove that the membership problem (resp., emptiness problem) for these classes is PSPACE-complete (resp., undecidable). We finally determine upper and lower bounds on the length of the shortest word of L ⁎ not having the various forms of unique decipherability into elements of L. [ABSTRACT FROM AUTHOR]