Most Complex Non-returning Regular Languages
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Date
2017-07-03
Authors
Brzozowski, Janusz
Davies, Sylvie
Journal Title
Journal ISSN
Volume Title
Publisher
Springer
Abstract
A regular language L is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each n⩾4 there exists a ternary witness of state complexity n that meets the bound for reversal and that at least three letters are needed to meet this bound. Moreover, the restrictions of this witness to binary alphabets meet the bounds for product, star, and boolean operations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has (n−1)ⁿ elements and requres at least (n2) generators. We find the maximal state complexities of atoms of non-returning languages. Finally, we show that there exists a most complex non-returning language that meets the bounds for all these complexity measures.
Description
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-60252-3_7
Keywords
Atom, Boolean operation, Concatenation, Different alphabets, Most complex, Non-returning, Reversal, Regular, Star, State complexity, Syntactic semigroup, Transition semigroup, Unrestricted complexity