You want me to count the arrows? I’ll C2 it!
Tom Bourne
School of Mathematics and Statistics
University of St Andrews
CIRCA Lunch Seminar
6th April 2017
Generalised star-height
Consider all regular expressions for a regular language.
Minimum nesting-depth of stars is star-height.
(a ∪ b)∗
(a∗ ∪ b ∗ )∗
Lemma
The class of regular languages is closed under complementation.
∅c
A∗ has star-height 1 and generalised star-height 0.
Definitions
A – finite alphabet.
L ⊆ A∗ – language.
M – monoid.
ϕ : A∗ → M – function.
Image – im(L) = {m ∈ M | m = w ϕ for some w ∈ L}.
ϕ is a homomorphism if (vw )ϕ = (v ϕ)(w ϕ) ∀v , w ∈ A∗ .
Recognition by a monoid
Let X ⊆ M.
Preimage – X ϕ−1 = {w ∈ A∗ | w ϕ ∈ X }.
L ⊆ A∗ is recognised by M if ∃ a homomorphism ϕ : A∗ → M such
that L = (Lϕ)ϕ−1 .
Theorem
L is regular ⇔ it is recognised by a finite monoid.
Groups of order less than 12
Theorem (Pin, Straubing and Thérien (1992))
Every regular language recognised by a finite group of order less
than 12 is of generalised star-height at most one.
Most of these groups are:
I
abelian; or,
I
nilpotent of class 2.
Exceptions: Dih3 and Dih5 .
Groups of order less than 12
Both Dih3 and Dih5 can be decomposed as A o Z2 , where A is an
abelian group.
Theorem (Pin, Straubing and Thérien (1989))
Every regular language recognised by a finite monoid of the form
A o Z2 is of generalised star-height at most one.
Proof relies on ‘cyclic’ automata and counting arrows.
Cyclic automata – definition
Let p, r ∈ Z+ .
Define A = (S, A, s0 , δ, T ) by
I
S = (Zp )r ;
I
A is a finite alphabet;
I
s0 = (0, 0, . . . , 0);
I
T = ∅; and,
for each a ∈ A ∃ an r -tuple ta ∈ S s.t. δ(s, a) = s + ta ∀s ∈ S .
A is said to be cyclic.
Cyclic automata – example
A = {a, b, c, d}. p = r = 2 ⇒ S = (Z2 )2 and s0 = (0, 0).
ta = td = (0, 0), tb = (0, 1), tc = (1, 0).
b
a, d
(0, 0)
(0, 1)
a, d
b
c
c
c
c
b
a, d
(1, 0)
(1, 1)
b
a, d
Z2 automaton
Since p = 2 is prime, can assume r = 1; i.e. S = {0, 1}.
Want to find all words such that the number of times the arrow
(0, a) is traversed in the automaton below is equal to k (mod n):
a, B
C
0
1
a, B
C
Z2 automaton
Can simplify to
a, b
0
1
a, b
without altering the count of (0, a).
Consider the set X = {w ∈ {a, b}∗ | δ(0, w ) = 0}.
Every word in X can be uniquely factorised as a product of words
of length two; i.e.
X = (aa ∪ ab ∪ ba ∪ bb)∗ .
Groups of order less than 16
Question
Is every regular language recognised by a finite group of order less
than 16 of generalised star-height at most one?
Most groups already covered except for:
I
A4 = (Z2 × Z2 ) o Z3 ; and,
I
Dic3 = Z3 o Z4 .
Question
Is every language recognised by a finite monoid of the form A o Zr
of generalised star-height at most one?
Z3 automaton
Since p = 3 is prime, can assume r = 1; i.e. S = {0, 1, 2}.
Want to find all words such that the number of times the arrow
(0, a) is traversed in the automaton below is equal to k (mod n).
a, B
D
0
C
a, B
1
C
C
2
D
a, B
D
Z3 automaton
Can simplify to
a, b
0
c
a, b
1
c
c
2
a, b
without altering the count of (0, a).
Consider the set X = {w ∈ {a, b, c}∗ | δ(0, w ) = 0}.
Z3 automaton
a, b
0
1
c
c
a, b
E = a((a ∪ b)c)∗ ((a ∪ b)2 ∪ c),
c
2
a, b
F = b((a ∪ b)c)∗ ((a ∪ b)2 ∪ c)
G = c(c(a ∪ b))∗ (a ∪ b ∪ c 2 )
X = (E ∪ F ∪ G )∗
We cannot uniquely factorise the words in X into products of
words of fixed length.