Abstract
We study descriptive complexity properties of the class of regular bifix-free languages, which is the intersection of prefix-free and suffix-free regular languages. We show that there exist universal bifix-free languages that meet all the bounds for the state complexity of basic operations (Boolean operations, product, star, and reversal). This is in contrast with suffix-free languages, where it is known that there does not exist such languages. Then we present a stream of bifix-free languages that is most complex in terms of all basic operations, syntactic complexity, and the number of atoms and their complexities, which requires a superexponential alphabet.We also complete the previous results by characterizing the state complexity of product, star, and reversal, and establishing tight upper bounds for atom complexities of bifix-free languages. Moreover, we consider the problem of the minimal size of an alphabet required to meet the bounds and the problem of attainable values of state complexities (magic numbers).
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