[Paper Review] The complexity of some regex crossword problems

Stephen Fenner|arXiv (Cornell University)|Nov 20, 2014

semigroups and automata theory8 references3 citations

TL;DR

This paper investigates the computational complexity of regex crossword puzzles under structural restrictions, proving that even when all row and column regular expressions are identical, the existence of a solution remains NP-complete or undecidable depending on input encoding. It establishes key results: NP-completeness for fixed-column regex puzzles with unary dimensions, NP-hardness for symmetric puzzles, and undecidability for existence over arbitrary sizes using a Turing machine simulation via two-dimensional tableaux.

ABSTRACT

In a typical regular expression (regex) crossword puzzle, you are given two nonempty lists $R_1,\ldots,R_m$ and $C_1,\ldots,C_n$ of regular expressions over some alphabet, and your goal is to fill in an $m imes n$ grid with letters from that alphabet so that the string formed by the $i$th row is in $L(R_i)$, and the string formed by the $j$th column is in $L(C_j)$, for all $1\le i\le m$ and $1\le j\le n$. Such a grid is a solution to the puzzle. It is known that determining whether a solution exists is NP-complete. We consider a number of restrictions and variants to this problem where all the $R_i$ are equal to some regular expression $R$, and all the $C_j$ are equal to some regular expression $C$. We call the solution to such a puzzle an $(R,C)$-crossword. Our main results are the following: 1. There exists a fixed regular expression $C$ over the alphabet $\{0,1\}$ such that the following problem is NP-complete: "Given a regular expression $R$ over $\{0,1\}$ and positive integers $m$ and $n$ given in unary, does an $m imes n$ $(R,C)$-crossword exist?" This improves the result mentioned above. 2. The following problem is NP-hard: "Given a regular expression $E$ over $\{0,1\}$ and positive integers $m$ and $n$ given in unary, does an $m imes n$ $(E,E)$-crossword exist?" 3. There exists a fixed regular expression $C$ over $\{0,1\}$ such that the following problem is undecidable (equivalent to the Halting Problem): "Given a regular expression $R$ over $\{0,1\}$, does an $(R,C)$-crossword exist (of any size)?" 4. The following problem is undecidable (equivalent to the Halting Problem): "Given a regular expression $E$ over $\{0,1\}$, does an $(E,E)$-crossword exist (of any size)?"

Motivation & Objective

• To analyze the computational complexity of regex crossword puzzles under structural constraints where all row and column regular expressions are identical.
• To determine whether the existence of a solution remains NP-hard or becomes undecidable under such restrictions.
• To investigate the impact of input encoding (unary vs. binary) on the complexity of the problem.
• To explore the decidability of existence for crosswords of arbitrary size, particularly in relation to the Halting Problem.
• To extend results to symmetric cases where row and column expressions are the same, and to consider implications for two-player games and unbounded grid variants.

Proposed method

• Uses a direct encoding of a one-tape Turing machine’s computation as a two-dimensional tableau, where rows represent configurations and columns enforce transition legality.
• Employs a fixed regular expression $ C $ over $ \{0,1\} $ to simulate the transition function and tape head movement in the crossword grid.
• Applies the Cook-Levin theorem-style construction to reduce SAT to the $(R,C)$-crossword problem, proving NP-completeness under unary encoding.
• Demonstrates that when dimensions are given in unary, the problem remains NP-complete even with identical row and column expressions.
• Shows that if dimensions are given in binary, the problem becomes NEXP-complete, and with mixed unary/binary, it becomes PSPACE-complete.
• Uses a reduction from the Halting Problem to prove undecidability for existence of solutions over arbitrary-sized grids, even with fixed $ C $.

Experimental results

Research questions

• RQ1 Is the existence of an $(R,C)$-crossword NP-complete when all row expressions are identical and all column expressions are fixed to a single $ C $, with dimensions given in unary?
• RQ2 Is the problem of determining whether an $(E,E)$-crossword exists NP-hard, even when all expressions are the same?
• RQ3 For which fixed regular expressions $ C $ is the problem of existence of an $(R,C)$-crossword undecidable?
• RQ4 What is the complexity of the $(R,C)$-crossword existence problem when dimensions are given in binary rather than unary?
• RQ5 Can the existence of a solution for an $(E,E)$-crossword be undecidable, even when $ E $ is fixed?

Key findings

• There exists a fixed regular expression $ C $ over $ \{0,1\} $ such that the $(R,C)$-crossword existence problem with dimensions in unary is NP-complete.
• The problem of determining whether an $(E,E)$-crossword exists is NP-hard, even when all expressions are identical.
• For a fixed $ C $, the problem of whether an $(R,C)$-crossword exists of any size is undecidable and equivalent to the Halting Problem.
• The problem of whether an $(E,E)$-crossword exists of any size is also undecidable and equivalent to the Halting Problem.
• When dimensions are given in binary, the $(R,C)$-crossword existence problem becomes NEXP-complete under polynomial-time reductions.
• With one dimension in unary and the other in binary, the problem becomes PSPACE-complete.