Complexity of SuffixFree Regular Languages
Abstract
We study various complexity properties of suffixfree regular languages. The quotient complexity of a regular language $L$ is the number of left quotients of $L$; this is the same as the state complexity of $L$. A regular language $L'$ is a dialect of a regular language $L$ if it differs only slightly from $L$. The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands. A sequence $(L_k,L_{k+1},\dots)$ of regular languages in some class ${\mathcal C}$, where $n$ is the quotient complexity of $L_n$, is called a stream. A stream is most complex in class ${\mathcal C}$ if its languages $L_n$ meet the complexity upper bounds for all basic measures. It is known that there exist such most complex streams in the class of regular languages, in the class of prefixfree languages, and also in the classes of right, left, and twosided ideals. In contrast to this, we prove that there does not exist a most complex stream in the class of suffixfree regular languages. However, we do exhibit one ternary suffixfree stream that meets the bound for product and whose restrictions to binary alphabets meet the bounds for star and boolean operations. We also exhibit a quinary stream that meets the bounds for boolean operations, reversal, size of syntactic semigroup, and atom complexities. Moreover, we solve an open problem about the bound for the product of two languages of quotient complexities $m$ and $n$ in the binary case by showing that it can be met for infinitely many $m$ and $n$.
 Publication:

arXiv eprints
 Pub Date:
 April 2015
 arXiv:
 arXiv:1504.05159
 Bibcode:
 2015arXiv150405159B
 Keywords:

 Computer Science  Formal Languages and Automata Theory
 EPrint:
 27 pages, 7 figures, 2 tables