Regular expressions and semi-linear sets

Question

In proving Parikh's Theorem, my Theory of Computer Science textbook defines a linear set as:

$u_0 + \langle u_1, \dots, u_m \rangle = \{u_0 + a_1u_1 + \dots + a_mu_m \mid a_1, \dots, a_m \in \mathbb{N}\}$ where $u_i$ are vectors of natural numbers.

and a semi-linear set as a union of finitely many linear sets. It goes on to say ''For every semilinear set $S \subset N^k$, it is not hard to construct a regular set $R \subset \Sigma^*$ such that $\psi(R) = S$'' (where $\psi$ is the Parikh map, taking strings over an alphabet $\Sigma$ to vectors where the first entry is the number of the first letter, the second entry is the number of the second letter, etc. So $\psi(\{a, ab, ba, aaa\})) = \{(1), (1, 1), (3)\}$.)

I was trying to think why regular languages would be semi-linear instead of just linear, and it seems like the + (or) operation in regular expressions is to blame. Is this correct: are languages described by regular expressions which use only concatenation and $^*$ linear?

Answer 1

Your claim is correct for a one letter alphabet $\Sigma = \{a\}$. We prove by induction that $\psi(r)$ is linear for every regular expression without addition. When $r = a$, $\psi(r) = 1$. When $r = r_1 r_2$, by the induction hypothesis $\psi(r_1) = u_0 + \sum_i \mathbb{N} u_i$ and $\psi(r_2) = v_0 + \sum_j \mathbb{N} v_j$, and so $\psi(r) = u_0 + v_0 + \sum_i \mathbb{N} u_i + \sum_j \mathbb{N} v_j$. When $r = r_1^*$, $\psi(r)$ is a subset of $\mathbb{N}$ closed under addition. Let $g = \operatorname{gcd}(\psi(r))$. Then $\psi(r)/g$ is a numerical semigroup, and so finitely generated, say $\psi(r)/g = \sum_i \mathbb{N} u_i$. So $\psi(r) = \sum_i \mathbb{N} gu_i$.

This argument doesn't quite work when $|\Sigma| \geq 2$, although there are some notions of GCD for multidimensional vectors, see for example this interesting presentation by Vadim Ponomarenko. Perhaps the argument above can be pushed through, or perhaps a different argument, which uses the fact that $\psi(r_1)$ is linear, exists. It's also possible that the claim is wrong in this case.

Answer 2

The claim fails for two symbols. The Parikh image $M\subseteq \mathbb{N}\times\mathbb{N}$ of $(abb^*)^*$ is not linear:

It is easy to see that for every finitely generated submonoid $N$ of $M$, there is a $k\ge 0$ such that for every $(x,y)\in N$, we have $x\le ky$ and $y\le kx$. This, however, is not satisfied by $M$ itself, meaning $M$ is not finitely generated.

Now as a submonoid of $\mathbb{N}\times\mathbb{N}$, $M$ is linear iff it is finitely generated.