19.13 Does a $(\textsf{min,+})$-WA preserve REG by inverse image?

We are interested in functions realized by weighted automata over the semiring \(\mathbb N_{\mathsf{ min}}=\langle \mathbb N\cup\left\{ \infty \right\} , \mathsf{ min}, +, \infty,0\rangle\). An $\mathbb N\mathsf{ min}$-weighted automaton \(\mathcal A\) over alphabet \(\Sigma\) realizes a partial function \(\llbracket \mathcal A \rrbracket:\Sigma^*\rightarrow \mathbb N\). We want to decide given such a function if it preserves ``simple'' sets by inverse image. By simple we mean regular languages (\(\mathbb N\) is viewed a the set of words over a unary alphabet, hence the regular languages are the semilinear sets). Let us state the decision problem.

Input: \(\mathcal{A}\), weighted automaton over \(\mathbb{N}_{\mathsf{min}}\)

Question: Does it hold that for all \(S\subseteq \mathbb N\) semilinear, \(\llbracket \mathcal A \rrbracket^{-1}(S)\) is regular?

Remark. This is equivalent to solving the problem over the semiring \(\mathbb N_{\mathsf{ max}}=\langle \mathbb N\cup\left\{ -\infty \right\} , \mathsf{ max}, +, -\infty,0\rangle\).