m-Nearly k-Universal Words -- Investigating Simon Congruence
This work addresses a foundational problem in formal language theory, offering incremental progress on Simon congruence.
The paper tackles the open problem of determining the index of Simon congruence by investigating m-nearly k-universal words, providing a full characterization and index for m=1, and partial results for other cases.
Determining the index of the Simon congruence is a long outstanding open problem. Two words $u$ and $v$ are called Simon congruent if they have the same set of scattered factors, which are parts of the word in the correct order but not necessarily consecutive, e.g., $\mathtt{oath}$ is a scattered factor of $\mathtt{logarithm}$. Following the idea of scattered factor $k$-universality, we investigate $m$-nearly $k$-universality, i.e., words where $m$ scattered factors of length $k$ are absent, w.r.t. Simon congruence. We present a full characterisation as well as the index of the congruence for $m=1$. For $m\neq 1$, we show some results if in addition $w$ is $(k-1)$-universal as well as some further insights for different $m$.