Complexity of Finding Alphabet Indexing

For two finite disjoint sets $P$ and $Q$ of strings over an alphabet $sum$, an alphabet indexing $psi$ for $P,Q$ by an indexing alphabet $Gamma$ with $mid Gamma mid < mid sum mid is a mapping $psi : sum \to Gamma$ satisfying $\tilde{psi}(P) cap \tilde{psi}(Q) = emptyset$, where $\tilde{psi} : sum^* \to Gamma^*$ is the homomorphism derived from $psi$. We defined this notion through experiments of knowledge acquisition from amino acid sequences of proteins by learning algorithms. This paper analyzes the complexity of finding an alphabet indexing. We first show that the problem is NP-complete. Then we give a local search algorithm for this problem and show a result on PLS-completeness