Aspects of Enumeration and Generation with a String

In general, the representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we show how a (unique) string representation for (complete) initially-connected deterministic automata (ICDFA’s) with n states over an alphabet of k symbols can be used for counting, exact enumeration, sampling and optimal coding, not only the set of ICDFA’s but, to some extent, the set of regular languages. An exact generation algorithm can be used to partition the set of ICDFA’s in order to parallelize the counting of minimal automata (and thus of regular languages). We present also a uniform random generator for ICDFA’s that uses a table of pre-calculated values. Based on the same table it is also possible to obtain an optimal coding for ICDFA’s. Keyword: regular languages, initially-connected deterministic finite automata, enumeration, random generation