We solve the following problem proposed by H. Straubing. Given a two letter alphabet A, what is the maximal number of states f(n) of the minimal automaton of a subset of An, the set of all words of length n. We give an explicit formula to compute f(n) and we show that 1 = lim n→∞ nf(n)/2n ≤ limn→∞ nf(n)/2 n = 2. The purpose of this note is to solve the following question, raised by H. Straubing...

We define compact automata and show that every language has a unique minimal compact automaton. We also define recognition of languages by compact left semitopological monoids and construct the analogue of the syntactic monoid in this context. For rational languages this reduces to the usual theory of finite automata and finite monoids.

We show that if a semisimple synchronizing automaton with $n$ states has a minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the $2$-packing number for families of $r$-subsets of $[1,n]$.

We inv~stigate-the relations be'·"'h.",pn the tree representation of a prefix code C, its patterns, the minimal automaton ~ * whi.ch recognizes C and the homomorphic images of the underlying algebra of ::G'. These relations provide us with an appealing formulation which allows us to state (and eventually solve) some problems concerning the structure of prefix codes and therefore the structure o...

Innnite leaf alphabets provide a new approach to remove the connection between the sizes of the leaf alphabet and the set of states of a tree automaton. Unique minimal tree automata can be constructed for classes of tree automata; however, such a class cannot be nite.

Let us consider a regular expression r of length m and text string T n over an alphabet Σ. Then, the RE minimal substring search problem is to find all substrings matching r. Yamamoto proposed O(mn) time O(m) space algorithm using Thompson automaton. In this paper, we improve Yamamoto's by introducing parallelism. The runs in worst case O(mn/p) best case, where p denotes number processors. Besi...

The following exercise is typical in introductory texts on deterministic finite automata (DFAs): “produce an automaton that recognizes the set of binary strings that, when interpreted as binary numbers, are divisible by k.” For example, exercise 1.30 in [2] asks the student to prove that the language {x |x is a binary number that is a multiple of k} is regular for each k ≥ 1; explicitly present...