On incomplete and synchronizing finite sets

Carpi, Arturo; D'Alessandro, Flavio

doi:10.1016/j.tcs.2015.08.042

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…A recent article [11] describes a sophisticated computer-assisted search for sets X with long shortest uncompletable words. While these experiments do not formally disprove a quadratic upper bound in k, they seem to hint at an exponential behaviour in k. See also [5] for recent work and open problems related to Restivo's conjecture.…”

mentioning

confidence: 81%

On Nonnegative Integer Matrices and Short Killing Words

Kiefer¹,

Mascle²

2018

Preprint

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Let n be a natural number and M a set of n × n-matrices over the nonnegative integers such that the joint spectral radius of M is at most one. We show that if the zero matrix 0 is a product of matrices in M, then there are M 1 , . . . , M n 5 ∈ M with M 1 • • • M n 5 = 0. This result has applications in automata theory and the theory of codes. Specifically, if X ⊂ Σ * is a finite incomplete code, then there exists a word w ∈ Σ * of length polynomial in x∈X |x| such that w is not a factor of any word in X * . This proves a weak version of Restivo's conjecture.

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confidence: 81%

On Nonnegative Integer Matrices and Short Killing Words

Kiefer¹,

Mascle²

2018

Preprint

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“…Both computational complexity question and finding the tight upper bound on the length also appear as one of the Berstel and Perrin's research problems [4, Resarch problems] and on the Shallit's list [20]. The problem itself has been connected with a number of different problems, e.g., testing if all bi-infinite words can be generated by the given list of words [17], the famous Černý conjecture [7], and the matrix mortality problem [14] in a restricted setting. 1.3.…”

Section: We Can See That 111 /mentioning

confidence: 99%

The Frobenius and factor universality problems of the Kleene star of a finite set of words

Mika,

Szykuła

2019

Preprint

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We settle complexity questions of two problems about the free monoid L * generated by a finite set L of words over an alphabet Σ. The first one is the Frobenius monoid problem, which is whether for a given finite set of words L, the language L * is cofinite. The open question concerning its computational complexity was originally posed by Shallit and Xu in 2009. The second problem is whether L * is factor universal, which means that every word over Σ is a factor of some word from L * . It is related to the longstanding Restivo's open question from 1981 about the maximal length of the shortest words which are not factors of any word from L * . We show that both problems are PSPACE-complete, which holds even if the alphabet is binary. Additionally, we exhibit families of sets L that show exponential (in the sum of the lengths of words in L or in the length of the longest words in L) worst-case lower bounds on the lengths related to both problems: the length of the longest words not in L * when L * is cofinite, and the length of the shortest words that are not a factor of any word in L * when L * is not factor universal. The second family essentially settles in the negative the Restivo's conjecture and its weaker variations. As auxiliary tools, we introduce the concept of set rewriting systems. Finally, we note upper bounds on the computation time and the length for both problems, which are exponential only in the length of the longest words in L.

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“…This strong form of Restivo's conjecture was refuted in [9], with a lower bound of 5k 2 -O(k). See also [6] for more recent work and open problems related to Restivo's conjecture. The article [12] describes a sophisticated computer-assisted search for sets X with long shortest uncompletable words.…”

mentioning

confidence: 99%

On Nonnegative Integer Matrices and Short Killing Words

Kiefer¹,

Mascle²

2021

SIAM J. Discrete Math.

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Let n be a natural number, and let \scrM be a set of n\times n-matrices over the nonnegative integers such that the joint spectral radius of \scrM is at most one. We show that if the zero matrix 0 is a product of matrices in \scrM , then there are M 1 , . . . , M n 5 \in \scrM with M 1 \cdot \cdot \cdot M n 5 = 0. This result has applications in automata theory and the theory of codes. Specifically, if X \subset \Sigma \ast is a finite incomplete code, then there exists a word w \in \Sigma \ast of length polynomial in \sum x\in X | x| such that w is not a factor of any word in X \ast . This proves a weak version of Restivo's conjecture.

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On incomplete and synchronizing finite sets

Cited by 8 publications

References 17 publications

On Nonnegative Integer Matrices and Short Killing Words

On Nonnegative Integer Matrices and Short Killing Words

The Frobenius and factor universality problems of the Kleene star of a finite set of words

On Nonnegative Integer Matrices and Short Killing Words

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