AbstractFor a regular language L, let $${{\,\mathrm{Var}\,}}(L)$$
Var
(
L
)
be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers $$k_1,k_2,\ldots ,k_n$$
k
1
,
k
2
,
…
,
k
n
and an n-ary regularity preserving operation f, let $$g_f^{{{\,\mathrm{Var}\,}}}(k_1,k_2,\ldots ,k_n)$$
g
f
Var
(
k
1
,
k
2
,
…
,
k
n
)
be the set of all numbers k such that there are regular languages $$L_1,L_2,\ldots , L_n$$
L
1
,
L
2
,
…
,
L
n
such that $${{\,\mathrm{Var}\,}}(L_i)=k_i$$
Var
(
L
i
)
=
k
i
for $$1\le i\le n$$
1
≤
i
≤
n
and $${{\,\mathrm{Var}\,}}(f(L_1,L_2,\ldots , L_n))=k$$
Var
(
f
(
L
1
,
L
2
,
…
,
L
n
)
)
=
k
. We completely determine the sets $$g_f^{{{\,\mathrm{Var}\,}}}$$
g
f
Var
for the operations reversal, Kleene-closures $$+$$
+
and $$*$$
∗
, and union; and we give partial results for product and intersection.
Finitely generated subgroups of free groups as formal languages and
their cogrowth
