Observations on the disjointness problem for rational subsets of free partially commutative monoids

Let I be a partially commutative alphabet of size three. Let M denote the free partially commutative monoid generated by I. The disjointness problem for rational subsets of M is: \n \nfor two given rational (described by regular expressions) subsets, X, Y of M decide if XnY=0. \n \nIn this paper we show that the problem is decidable for every commutativity region over the alphabet I. It is known (see (3)) that the problem is undecidable in the case of the four letters alphabet. Hence we give a sharp bound on the number of letters for which the problem is decidable. A similar situation occurs for the unique decipherability problem with partially commutative alphabets. It was shown in (4) that this problem is decidable for alphabets of size three and that it is undecidable for alphabets of size four. We show that the unique decipherability problem with partially commutative alphabet I is a special case of the disjointness problem of rational subsets of the monoid generated by I. This and our algorithm for the disjointness problem give alternative and much simpler proof of the decidability of the unique decipherability problem with partially commutative alphabets of size three. Let I={a,b,c}. It was proved in (5) using multicounter machines that if a commutes with c and b, and b does not commute with c then the disjointness problem is decidable. We give here a simpler proof for this case and prove the decidability for all other possible commutativity relations for three letters alphabet.