Counting Subwords and Regular Languages

Abstract

Let x and y be words. We consider the languages whose words z are those for which the numbers of occurrences of x and y, as subwords of z, are the same (resp., the number of x's is less than the number of y's, resp., is less than or equal). We give a necessary and sufficient condition on x and y for these languages to be regular, and we show how to check this condition efficiently.

Note on the Infiniteness and Equivalence Problems for Word-MIX Languages

“…In the recent work [8] by Colbourn et. al., the counting feature of MIX is generalised from the counting letter occurrences to the counting of word occurrences.…”

“…The decidability of the infiniteness/equivalence turn to be non-trivial: L(0, 1, 00, 11) and L(0, 1, 01, 10) are finite but L(00, 11, 000, 111) is infinite over A = {0, 1} (example from [8]), and L(ab, ba, a) is infinite but L(ab, ba, a, b) is finite over A = {a, b} (these two examples appear again in Section 4.1), for example. In addition, while L(w 1 , w 2 ) is always deterministic context-free (DCFL), it can also be regular (L(ab, ba) ⊆ {a, b} * is regular, for example) [8]. This kind of generalisation (from letter occurrences to word occurrences) is also considered in the context of the Parikh map [9].…”

“…[8] provided a necessary and sufficient condition for w 1 and w 2 for these languages to be regular, and gave a polynomial time algorithm for testing that condition. The finiteness of L(w 1 , w 2 ) is also considered in [8] and they proved that, for any non-empty words w 1 , w 2 ∈ A * , L(w 1 , w 2 ) is finite if and only if the alphabet A consists of a single letter (i .e., A = {a}) and w 1 = w 2 . For more general case, allowing more than two parameter words L(w 1 , .…”

“…The (n-dimensional) de Bruijn graph [11] can track all information of subword occurrences (of length at most n), hence it is a very useful tool for counting subword occurrences and related problems. The de Bruijn graph also played a key role in the proof of Theorem 8 in [8].…”