Complexity of Regex Crosswords

Abstract

In a regular expression crossword puzzle, one is given two non-empty lists $\langle {} \langle {} R_1,\ldots , R_m \rangle {}$ and $\langle {} C_1, \ldots , C_n \rangle {} \rangle {}$ over some alphabet, and the challenge is to fill in an $m\times n$ grid of characters such that the string formed by the $i^\text {th}$ row is in $L(R_i)$ and the string in the $j^\text {th}$ column is in $L(C_j)$. We consider a restriction of this puzzle where all the $R_i$ are equal to one another and similarly the $C_j$. We consider a 2-player version of this puzzle, showing it to be -complete. Using a reduction from , we also give a new, simple proof of the known result that the existence problem of a solution for the restricted (1-player) puzzle is -complete.