On codes, ω-codes and ω-generators

Set Decipherable Languages and Generators

“…Theorem 11 (see [9]). Let C be a rational code such that C * is the greatest generator (with respect to inclusion).…”

“…Our starting point is a result proved in [9] stating that if L is a code such that L * is the greatest generator of L ω , then there exists an ω-code C such that C ω = L ω if and only if L itself is an ω-code. We try to extend this result by defining a class of languages that is as simple as possible after the codes, and we prove the similar result for this class.…”

“…Our technique. Our approach is different from [9]. Instead of manipulating directly the language L, we look at the minimal relations satisfied by L. These relations capture the structure of the double L-factorizations of words or infinite words, and they are easy to compute by using a variation of domino graph [6].…”

“…languages allowing to perform the decoding of infinite words after some bounded delay, without waiting the whole message). Julia et al [9] solved the problem for an interesting case: L ω has the greatest generator that is generated by a code. Notice that the condition of having the greatest generator is only a technical assumption; we can remove it by considering all maximal generators instead of the greatest one.…”

One-relation languages and code generators

“…Of course ω-codes are codes, but the converse does not hold. We investigate the open problem to characterize languages L such that L ω = G ω for some code or ω-code G. See [4,5,8] for partial answers and various approaches. This question is still open even if the language L is a finite language.…”