In search of most complex regular languages

Abstract

Regular languages that are most complex under common complexity measures are studied. In particular, certain ternary languages U n (a,b,c), n ≥ 3, over the alphabet {a,b,c} are examined. It is proved that the state complexity bounds that hold for arbitrary regular languages are also met by the languages U n (a,b,c) for union, intersection, difference, symmetric difference, product (concatenation) and star. Maximal bounds are also met by U n (a,b,c) for the number of atoms, the quotient complexity of atoms, the size of the syntactic semigroup, reversal, and 22 combined operations, 5 of which require slightly modified versions. The language U n (a,b,c,d) is an extension of U n (a,b,c), obtained by adding an identity input to the minimal DFA of U n (a,b,c). The witness U n (a,b,c,d) and its modified versions work for 14 more combined operations. Thus U n (a,b,c) and U n (a,b,c,d) appear to be universal witnesses for alphabets of size 3 and 4, respectively. © 2012 Springer-Verlag.