On effective construction of the greatest solution of language inequality XA⊆BXXA⊆BX

In this paper, we consider effective constructions of the greatest solution of the language inequality XA⊆BXXA⊆BX. It has been proved by Kunc in 2005 that the greatest solution of XA⊆BXXA⊆BX is regular provided that B is regular, no matter what A is. However this proof is based on Kruskal's tree theorem, and does not provide any effective way to construct the greatest solution.We focus on this gap in this paper. We give an effective construction of the greatest solution for the following two cases:(i)A,BA,B are regular and there exists k⩾1k⩾1 such that pref(B)Ak∩B⩽kpref(B)=∅pref(B)Ak∩B⩽kpref(B)=∅, where pref(B)pref(B) is the set of prefixes of words in B,(ii)A,BA,B are regular and B is a code with finite decoding delay.Our construction takes the point of view of games. As shown by Kunc in his regularity proof, the construction of the greatest solution can be reduced to the construction of the winning region of a two-player game. Our contribution is to show that the winning regions of the two-player game for the two cases can be constructed effectively.The main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy. While the construction for the second case is based on the observation that the two-player game can be reduced to a two-player reachability game played on the transition graph of a one-counter machine.