Separating Words Problem over Groups

Abstract

The separating words problem asks - given two words $w, x \in \{0,1\}^n$, the size of the smallest automaton (in terms of number of states, expressed as a function of n) which accepts one of them and rejects the other. The best lower bound known for the problem is $\varOmega (\log n)$, whereas the best upper bound known is $O(n^{1/3} \log ^7 n)$, due to (Chase 2021). Motivated by the applications to this problem, we study separating in the context of groups - a finite group G is said to separate w and x, if there is a substitution function from $\phi : \varSigma \rightarrow G$ such that the expressions $\phi (w)$ and $\phi (x)$ yield different elements in the group G. We show the following results:

• By a result of Robson [6], there is a permuting automaton of size $O(\sqrt{n})$ states which separate any two words w and x of length n. Hence, there is a group of size $2^{O(\sqrt{n}\log n)}$ which separate w and x. Using basic properties of one dimensional representations of the groups, we improve this to $O(\sqrt{n}2^{\sqrt{n}})$.

• A class of groups $\mathcal {G}$ is said to be universal if for any two words $w, x \in \varSigma ^*$, there exists a group $G \in \mathcal {G}$ for which a separating substitution map exists such that the yields of the words under the map are distinct. We show that the class of permutation groups, solvable groups, nilpotent groups and, in particular, p-groups, are universal.

• Class of Abelian groups and Dihedral groups are not universal. En route to this result, we derive sufficiency conditions for a class of groups to be non-universal.

• We can also translate separation using groups to separation using automaton. Any two words $w,x \in \varSigma ^n$ which are separated by a group G can be separated using an automaton of size |G|. We show better bounds for permutation groups. We also study the natural computational problem in the context and show it to be $\textsf {NP}$-complete.