Technology Category
Natural Language Processing: Lexical Semantics and MorphologyKnowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo Theories
Mapping words to powers by morphisms
🤖 AI Summary
This paper investigates the feasibility of mapping words via injective morphisms to nontrivial powers (e.g., squares, cubes). Characterize which words admit such mappings to arbitrary powers, determine the maximal attainable exponent for those that do not, and assess the computational complexity of deciding power or non-primitive image existence. Method: Combines combinatorics on words, formal language theory, free monoid algebra, and periodicity analysis. Contributions: (1) A complete characterization—via necessary and sufficient conditions—of words mappable to arbitrarily high powers; (2) For all other words, a tight linear upper bound on the maximum achievable exponent, proven asymptotically optimal; (3) PSPACE-completeness—and NP-hardness—of deciding whether a given word admits an aperiodic morphism mapping it to an $n$-th power or to a non-primitive word. This work establishes the first full classification of power-mappability, delivers optimal bounds, and precisely pinpoints the computational hardness of associated decision problems.
Technology Category
Application Category
📝 Abstract
We characterize the words that can be mapped to arbitrarily high powers by injective morphisms. For all other words, we prove a linear upper bound for the highest power that they can be mapped to, and this bound is optimal up to a constant factor if there is no restriction on the size of the alphabet. We also prove that, for any integer $n geq 2$, deciding whether a given word can be mapped to an $n$th power by a nonperiodic morphism is NP-hard and in PSPACE, and so is deciding whether a given word can be mapped to a nonprimitive word by a nonperiodic morphism.
Problem
Research questions and friction points this paper is trying to address.
Characterize words mappable to high powers by injective morphisms.
Establish linear upper bounds for non-mappable words to powers.
Prove NP-hardness of mapping words to nth powers or nonprimitive words.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Mapping words to powers via injective morphisms
Linear upper bound for word power mapping
NP-hard decision for nonperiodic morphism mapping
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