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.
The quotient complexity of a regular language L is the number of left quotients of L, which is the same as the state complexity of L. Suppose that L and L ′ are binary regular languages with quotient complexities m and n, and that the transition semigroups of the minimal deterministic automata accepting L and L ′ are the symmetric groups Sm and Sn of degrees m and n, respectively. Denote by • any binary boolean operation that is not a constant and not a function of one argument only. For m, n ≥ 2 with (m, n) ∈ {(2, 2), (3, 4), (4, 3), (4, 4)} we prove that the quotient complexity of L • L ′ is mn if and only either (a) m = n or (b) m = n and the bases (ordered pairs of generators) of Sm and Sn are not conjugate. For (m, n) ∈ {(2, 2), (3, 4), (4, 3), (4, 4)} we give examples to show that this need not hold. In proving these results we generalize the notion of uniform minimality to direct products of automata. We also establish a non-trivial connection between complexity of boolean operations and group theory.
MotivationThe left quotient, or simply quotient, of a regular language L over an alphabet Σ by a word w ∈ Σ * is the regular language w −1 L = {x ∈ Σ * : wx ∈ L}. It is well known that a language is regular if and only if it has a finite number of ⋆
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.