Let S be a finite set of words over a finite alphabet Σ. We say that S is complete if Fact(S^{*}) = Σ^{*}, where Fact(L) denotes the set of all factors of all words of L.
A word in Σ^{*} - Fact(S^{*}) is called uncompleteable. Restivo conjectured in 1981 that for any non-complete set S there is an uncompleteable word of length at most 2k^{2}, where k is the length of the longest word in S.

Restivo's conjecture is false, but the correct bound is still not known. Recently Gusev and Pribavkina have produced a set of counterexamples for which the bound is 5k^{2} - 17k + 13 for k ≥ 4.

A related open problem is the computational complexity of determining, given a finite set S of words over Σ, if Fact(S^{*}) = Σ^{*}. Here the complexity is measured in terms of the total size of S.

References:

V. V. Gusev and E. V. Pribavkina, On non-complete sets and Restivo's conjecture, in G. Mauri and A. Leporati, eds., *Developments in Language Theory*, 15th International conference, DLT 2011, Lect. Notes in Comp. Sci., Vol. 6795, Springer, 2011, pp. 239-250.

-- JeffreyShallit - 20 Jul 2011

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Topic revision: r2 - 2011-07-21 - JeffreyShallit

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