Abstract
We investigate the complexity of the solvability problem for restricted classes of word equations with and without regular constraints. For general word equations, the solvability problem remains \({{\mathrm{\mathsf {NP}}}}\)-hard, even if the variables on both sides are ordered, and for word equations with regular constraints, the solvability problems remains \({{\mathrm{\mathsf {NP}}}}\)-hard for variable disjoint (i. e., the two sides share no variables) equations with two variables, only one of which is repeated. On the other hand, word equations with only one repeated variable (but an arbitrary number of variables) and at least one non-repeated variable on each side, can be solved in polynomial-time.
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Notes
- 1.
The use of the term regular in this context has historical reasons: the matching problem has been first investigated in terms of so-called pattern languages, i. e., the set of all words that match a given pattern \(\alpha \in (\varSigma \cup X)^*\), which are regular languages if \(\alpha \) is regular.
- 2.
We will also use minor modifications later on of this reduction in order to conclude corollaries of Theorem 3.
- 3.
In order to prove \({{\mathrm{\mathsf {NP}}}}\)-hardness, a simpler production would suffice, but we need a linear reduction in order to obtain the ETH lower bound.
- 4.
Note that this is just the reduction used by Schaefer [15] in order to reduce 3-\(\textsc {Sat}\) to 1-in-3 3-\(\textsc {Sat}\). We recall it here to observe that this reduction is linear.
- 5.
For the latter case, note that in the reduction of Theorem 14, we can add a non-repeated variable with regular constraint \(\emptyset \) to the left side.
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Acknowledgements
We are indebted to Artur Jeż for valuable discussions. Markus L. Schmid gratefully acknowledges partial support for this research from DFG, that in particular enabled his visit at the University of Kiel.
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Manea, F., Nowotka, D., Schmid, M.L. (2016). On the Solvability Problem for Restricted Classes of Word Equations. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_25
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