On the complexity of iterated shuffle

Abstract

It is demonstrated that the following problems are NP complete: 1. (1) Given words w and w 1, w 2, …, w n , is w in the shuffle of w 1, w 2, …, w n ? 2. (2) Given words w and v, is w in the iterated shuffle of v? From these results we show that the languages {$w¢w R: w σ {a, b} ∗} ∅, wσ{a,b} ∗ $w) ∅, {ab ncde nf: n ⩾ 0} ∅ , and { a n+1 b n c n f n : n ⩾ 0 ∅ are NP complete, where ∅ denotes the iterated shuffle. By representing these languages in various ways using the operations shuffle, iterated shuffle, union, concatenation, intersection, intersection with a regular set, non-erasing homomorphism and inverse homomorphism, results on the complexity of language classes generated using various subsets of these operations are obtained. Finally, it is shown that the iterated shuffle of a regular set can be recognized in deterministic polynomial time.