On Exchange-Robust and Subst-Robust Primitive Partial Words

A partial word is a word that may have some unknown places known as “holes” and can be replaced by the symbols from the underlying alphabet. A partial word u is said to be primitive if there does not exist a word v such that u is contained in a nontrivial integer power of v. We study the preservation of primitivity in partial words by the effect of some point mutation operations. In this paper, we investigate the effect of exchanging two adjacent symbols and of substituting a symbol by another symbol from the alphabet. We characterize the classes of primitive partial words with one hole which are not exchange robust and not substitute robust. We prove that the language of exchange robust primitive partial words with one hole is not right 1-dense and also prove that the language of primitive partial words with one hole which is substitute robust is closed under conjugacy relation. We show that the language of non-exchangerobust primitive partial words is not context-free over a binary alphabet.