On Nonnegative Integer Matrices and Short Killing Words

Let $n$ be a natural number and $\mathcal{M}$ a set of $n \times n$-matrices over the nonnegative integers such that the joint spectral radius of $\mathcal{M}$ is at most one. We show that if the zero matrix $0$ is a product of matrices in $\mathcal{M}$, then there are $M_1, \ldots, M_{n^5} \in \mathcal{M}$ with $M_1 \cdots M_{n^5} = 0$. This result has applications in automata theory and the theory of codes. Specifically, if $X \subset \Sigma^$ is a finite incomplete code, then there exists a word $w \in \Sigma^$ of length polynomial in $\sum_{x \in X} |x|$ such that $w$ is not a factor of any word in $X^*$. This proves a weak version of Restivo’s conjecture.

💡 Research Summary

The paper investigates finite sets of n×n non‑negative integer matrices 𝓜 under the assumption that their joint spectral radius ρ(𝓜) does not exceed one. The joint spectral radius measures the maximal asymptotic growth of products of matrices drawn from 𝓜; the condition ρ≤1 guarantees that the norm of any product of length k grows at most polynomially in k, whereas ρ>1 would cause exponential growth.

The central result (Theorem 1) states that if the zero matrix belongs to the semigroup generated by 𝓜, then there exists a product of at most ℓ≤(1/16)n⁵+(15/16)n⁴ matrices from 𝓜 that equals the zero matrix. Moreover, such a product can be found in time polynomial in the size of the description of 𝓜. The theorem is proved in two parts. First, the authors treat the case where 𝓜 is strongly connected: for every pair of indices (i,j) there is a word w with the (i,j) entry of M(w) positive. In this situation Lemma 10 shows that all matrices in the generated semigroup are binary (entries 0 or 1); thus the semigroup can be identified with the transition monoid of an unambiguous finite automaton (UFA). The problem of producing a zero matrix then becomes the problem of finding a killing word—a word that labels no path in the automaton. By extending Carpi’s earlier bound for minimum‑rank words in UFAs, the authors construct a killing word of length O(n⁵) in polynomial time.

The second part lifts the strong‑connectivity restriction. The authors decompose the matrix set into its strongly connected components, treat each component as above, and then concatenate suitable bridging words. Careful bookkeeping shows that the total length never exceeds the same O(n⁵) bound, and the construction remains polynomial‑time.

From an automata‑theoretic viewpoint, each matrix M(a) (a∈Σ) is interpreted as the adjacency matrix of a nondeterministic finite automaton (NFA) over alphabet Σ. When the NFA is unambiguous, its transition monoid consists of {0,1}‑matrices, and a zero matrix corresponds exactly to a killing word. The authors prove (Proposition 3) that under ρ≤1 one can decide in polynomial time whether a killing word exists, by checking whether the spectral radius of the average matrix A = (1/|Σ|)∑_{a∈Σ}M(a) is strictly smaller than one. This decision procedure does not construct the word, but Theorem 1 fills that gap by providing an explicit short killing word when it exists. The result also shows that the PSPACE‑complete killing‑word problem becomes tractable under the spectral‑radius condition.

The paper then applies these findings to the theory of finite codes. For a finite code X⊂Σ* the flower automaton A_X is a UFA whose states correspond to the words of X and whose transitions follow concatenations. A word is uncompletable in X (i.e., it does not appear as a factor of any word in X*) precisely when it is a killing word for A_X. Consequently, Theorem 1 yields a polynomial‑size bound on the length of the shortest uncompletable word: if X is incomplete, there exists such a word of length O(m⁵), where m = Σ_{x∈X}|x|. This gives a weak version of Restivo’s conjecture, which originally suggested a quadratic bound in the maximal word length k = max_{x∈X}|x|. By exploiting the special structure of flower automata, the authors improve the bound to (k+1)·k²·(m+2)(m+1), still polynomial but larger than k². The result does not contradict recent counter‑examples to Restivo’s original conjecture because those examples are not codes.

Finally, the authors show that not every minimum‑rank matrix in the generated monoid can be expressed by a short product. Theorem 6 constructs a family of binary matrix monoids where a particular rank‑1 matrix is the unique minimum‑rank element, yet any product representing it may require super‑polynomial length. This demonstrates that Theorem 1’s guarantee applies only to some minimum‑rank element, not to all.

Overall, the paper blends spectral matrix analysis, combinatorial automata theory, and coding theory to obtain new polynomial‑time algorithms and explicit length bounds for killing words and minimum‑rank words. It clarifies the crucial role of the joint spectral radius condition, provides the first O(n⁵) upper bound for shortest killing words in UFAs, and advances the understanding of Restivo’s conjecture for finite codes.