Word equations in linear space

Word equations (in free semigroup) are an important problem on the intersection of formal languages and algebra. Given two sequences consisting of letters and variables we are to decide whether there is a substitution for the variables that turns this formal equation into true equality of strings. The exact computational complexity of this problem remains unknown, with the best known lower and upper bounds being NP and PSPACE, respectively. Recently, a novel technique of recompression was applied to this problem, simplifying the known proofs and lowering the space complexity to (nondeterministic) O(n log n), where n is the length of the input equation. In this paper we show that word equations are in nondeterministic linear space, thus the language of satisfiable word equations is context-sensitive. We use the known recompression-based algorithm and additionally employ Huffman coding for letters. The proof, however, uses analysis of how the fragments of the equation depend on each other as well as a new strategy for nondeterministic choices of the algorithm, which uses several new ideas to limit the space occupied by the letters.