Most Complex Deterministic Union-Free Regular Languages
A regular language L is union-free if it can be represented by a regular expression without the union operation. A union-free language is deterministic if it can be accepted by a deterministic one-cycle-free-path finite automaton; this is an automaton which has one final state and exactly one cycle-free path from any state to the final state. Jirásková and Masopust proved that the state complexities of the basic operations reversal, star, product, and boolean operations in deterministic union-free languages are exactly the same as those in the class of all regular languages. To prove that the bounds are met they used five types of automata, involving eight types of transformations of the set of states of the automata. We show that for each \(n \geqslant 3\) there exists one ternary witness of state complexity n that meets the bound for reversal and product. Moreover, the restrictions of this witness to binary alphabets meet the bounds for star and boolean operations. We also show that the tight upper bounds on the state complexity of binary operations that take arguments over different alphabets are the same as those for arbitrary regular languages. Furthermore, we prove that the maximal syntactic semigroup of a union-free language has \(n^n\) elements, as in the case of regular languages, and that the maximal state complexities of atoms of union-free languages are the same as those for regular languages. Finally, we prove that there exists a most complex union-free language that meets the bounds for all these complexity measures. Altogether this proves that the complexity measures above cannot distinguish union-free languages from regular languages.
KeywordsAtom Boolean operation Concatenation Different alphabets Most complex One-cycle-free-path Regular Reversal Star State complexity Syntactic semigroup Transition semigroup Union-free
- 2.Bell, J., Brzozowski, J., Moreira, N., Reis, R.: Symmetric groups and quotient complexity of boolean operations. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 1–12. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43951-7_1CrossRefMATHGoogle Scholar
- 3.Brzozowski, J.A.: Regular expression techniques for sequential circuits. Ph.D. thesis, Princeton University, Princeton, NJ (1962). http://maveric.uwaterloo.ca/~brzozo/publication.html
- 6.Brzozowski, J.A., Davies, S., Liu, B.Y.V.: Most complex regular ideal languages. Discrete Math. Theor. Comput. Sci. 18(3) (2016). Paper #15Google Scholar
- 7.Brzozowski, J.A., Sinnamon, C.: Unrestricted state complexity of binary operations on regular and ideal languages (2016). http://arxiv.org/abs/1609.04439. Accessed 2017
- 23.Nagy, B.: On union complexity of regular languages. In: CINTI 2010, pp. 177–182. IEEE (2010)Google Scholar