Two techniques in the area of the star problem in trace monoids

This paper deals with decision problems related to the star problem in trace monoids which means to determine whether the iteration of a recognizable trace language is recognizable. Due to a theorem by Richomme (in: I. Privara et al. (Eds.), MFCS'94 Proc., Lecture Notes in Computer Science, vol. 841, Springer, Berlin, 1994, pp. 577-586), we know that the star problem is decidable in trace monoids which do not contain a submonoid of the form {a,c}* x {b,d}*. [cf. Theory Comput. Systems 34(3) (2001) 193-227]. Here, we consider a more general problem: Is it decidable whether for some recognizable trace language R and some recognizable or finite trace language P the intersection R boolean AND P* is recognizable? If P is recognizable, then we show that this problem is decidable iff the underlying trace monoid does not contain a submonoid of the form {a,c}* x b*. In the case of finite languages P, this problem is decidable in {a,c} * x b* but undecidable in {a,c}* x {b,d}*. (C) 2003 Elsevier B.V. All rights reserved.