[Paper Review] Word equations in linear space

Artur Jeż|arXiv (Cornell University)|Feb 2, 2017

semigroups and automata theory18 references3 citations

TL;DR

This paper establishes that word equation satisfiability is decidable in nondeterministic linear space, placing the problem within the class of context-sensitive languages. By enhancing the recompression technique with Huffman coding and a novel dependency-based encoding strategy, the authors reduce space complexity from O(n log n) to O(n), proving that both satisfiable and unsatisfiable word equations are context-sensitive via the Immerman–Szelepcsényi theorem.

ABSTRACT

Satisfiability of word equations is an important problem in 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 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 (nondeterministic) O(n log n). In this paper we show that satisfiability of word equations is in nondeterministic linear space, thus the language of satisfiable word equations is context-sensitive, and by the famous Immerman-Szelepcsenyi theorem: the language of unsatisfiable word equations is also 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.

Motivation & Objective

• To close the gap between the known NP-hard lower bound and PSPACE upper bound for word equation satisfiability.
• To reduce the space complexity of word equation solving from O(n log n) to O(n) in nondeterministic space.
• To show that the language of satisfiable word equations is context-sensitive, leveraging the Immerman–Szelepcsényi theorem.
• To develop a space-efficient algorithm using recompression and dependency-aware encoding of letters.

Proposed method

• Extends the recompression-based algorithm of Jeż (2015) with Huffman coding for efficient letter representation.
• Introduces a new dependency-based encoding of letters using factors of the equation that they depend on.
• Employs a refined compression strategy that minimizes bit encoding size while limiting local changes and new letter introductions.
• Uses nondeterministic choices guided by a solution’s structure to ensure correctness and space efficiency.
• Applies block compression and pair compression in linear space, as established in Lemma 5.
• Maintains a compact representation of the alphabet and symbol mappings throughout phases to preserve linear space.

Experimental results

Research questions

• RQ1 Can word equation satisfiability be solved in nondeterministic linear space?
• RQ2 Is the language of satisfiable word equations context-sensitive?
• RQ3 Can the space complexity of recompression-based word equation solving be reduced to O(n)?
• RQ4 What role does dependency analysis among equation fragments play in space-efficient computation?
• RQ5 How can prefix coding, such as Huffman coding, be integrated into recompression to minimize space usage?

Key findings

• The satisfiability of word equations is decidable in nondeterministic linear space, i.e., in NSPACE(n).
• The language of satisfiable word equations is context-sensitive, and by the Immerman–Szelepcsényi theorem, so is the language of unsatisfiable word equations.
• The algorithm achieves O(n) space complexity by using Huffman coding and a dependency-based encoding of letters.
• The proof uses a novel strategy for nondeterministic choices that limits the number of new letters introduced and ensures local changes in the equation.
• The space bound holds for any prefix code, including Huffman coding, and the total space is bounded by a constant multiple of the input size.
• The algorithm terminates in O(log N) phases, where N is the length of a solution, and each phase reduces the solution length by a constant factor.