Title: Word equations in linear space,
Abstract: Word equations 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 equation into true equality of strings. The computational complexity of
this problem remains unknown, with the best lower and upper bounds being NP and
PSPACE. 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). 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.