Abstract
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(nlogn). 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.
Highlights
Solving word equations was an intriguing problem since the dawn of computer science, motivated first by its ties to Hilbert’s 10th problem
In this paper we show that satisfiability of word equations can be tested in nondeterministic linear space in terms of the number of bits of the input, showing that the language of satisfiable word equations is context-sensitive
BlockComp is sound and complete; to be more precise, for any solution S of an equation U = V for the nondeterministic choices 95:4 Word Equations in Nondeterministic Linear Space corresponding to S the returned equation U = V has a solution S such that S (U ) is the Γ compression of S(U ) and S (X) is obtained from S(X) by removing the a-prefix and b-suffix, where a is the first letter of S(X) and b the last, and performing the Γ compression
Summary
Solving word equations was an intriguing problem since the dawn of computer science, motivated first by its ties to Hilbert’s 10th problem. 95:4 Word Equations in Nondeterministic Linear Space corresponding to S the returned equation U = V has a solution S such that S (U ) is the Γ compression of S(U ) and S (X) is obtained from S(X) by removing the a-prefix and b-suffix, where a is the first letter of S(X) and b the last, and performing the Γ compression. When Γ and Γr are disjoint, the PairComp(Γ , Γr) is sound and complete; to be more precise, for any solution S of an equation U = V for the nondeterministic choices corresponding to S the returned equation U = V has a solution S such that S (U ) is the (Γ , Γr) compression of S(U ) and S (X) is obtained from S(X) by removing the first letter of S(X), if it is in Γr, and the last, if it is in Γ , and performing the (Γ , Γr) compression. LinWordEq is sound, complete and terminates (for appropriate nondeterministic choices) for satisfiable equations
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