Word equations in non-deterministic linear space

Satisfiability of word equations problem is: Given two sequences consisting of letters and variables decide whether there is a substitution for the variables that turns this equation into true equality. The exact computational complexity of this problem remains unknown, with the best lower and upper bounds being, respectively, NP and PSPACE. Recently, the novel technique of recompression was applied to this problem, simplifying the known proofs and lowering the space complexity to (non-deterministic) O(nlog⁡n). In this paper we show that satisfiability of word equations is in non-deterministic 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 non-deterministic choices of the algorithm.