# automaton.has_twins_property¶

Whether the automaton has the twins property.

• Sibling states: Two states $p$, $q$ are siblings if there exist two labels $x$ and $y$ such that $p$ and $q$ can be reached from an initial state by path labeled with $x$ and there is a cycle at $p$ and $q$ both labeled with $y$.
• Twins states: Two sibling states $p$ and $q$ are twins iff for any label $y$: $w[P(p, y, p)] = w(P[q, y, q])$
• Has twins property: An automaton has the twins property if any two sibling states of this automaton are twins.

Preconditions:

• The automaton is not cycle ambiguous

## Examples¶

In [1]:
import vcsn
q = vcsn.context('lal_char(ab), q')
def std(e):
return q.expression(e, 'binary').standard()


Consider the following $\mathbb{Q}$ automaton:

In [2]:
a = std('(ab)* + (ab)*')
a

Out[2]:

State $1$ and $3$ are siblings: they can be reached from $0$ with label "a" and there are two cycles in them with the same label "ba". Since the weights of these cycles equals $1$ (in $\mathbb{Q}$), they are twins. This automaton has two sibling states only and they are twins so it has twins property.

In [3]:
a.has_twins_property()

Out[3]:
True

Conversely, the following automaton does not have the twins property because state $1$ and state $4$ are siblings but not twins: the weights of cycles differ ($1$ != $2$).

In [4]:
a = std('(<2>ab)* + (ab)*')
a

Out[4]:
In [5]:
a.has_twins_property()

Out[5]:
False

When the automaton has no sibling states, it has the twins property.

In [6]:
a = std("(aa)*+(ab)*")
a

Out[6]:
In [7]:
a.has_twins_property()

Out[7]:
True

In the tropical semiring ($\mathbb{Z}_{\text{min}}$), an automaton is determinizable iff the automaton has the twins property.

In [8]:
%%automaton a
context = "lal_char(abcd), zmin"
$-> 0 0 -> 1 <1>a 0 -> 2 <2>a 1 -> 1 <3>b 1 -> 3 <5>c 2 -> 2 <3>b 2 -> 3 <6>d 3 ->$


This automaton has the twins property (the two sibling states $1$ and $2$ are twins), so it is determinizable (in $\mathbb{Z}_{\text{min}}$).

In [9]:
a.determinize()

Out[9]:

The twins property can also be check in $\mathbb{Z}$:

In [10]:
%%automaton a
context = "letterset<char_letters(abcd)>, z"
$-> 0 0 -> 1 a 0 -> 2 <2>a 1 -> 1 <3>b 1 -> 3 <5>c 2 -> 2 <3>b 2 -> 3 <6>d 3 ->$

In [11]:
a.has_twins_property()

Out[11]:
True

Or with tuples of weightsets:

In [12]:
%%automaton a
context = "lal_char(abc), lat<z,zmin>"
$-> 0 0 -> 1 <(1, 3)>a 0 -> 2 <(1, 5)>a 1 -> 3 <(4, 8)>b 3 ->$
2 -> 4 <(6, 4)>b
4 -> \$
3 -> 1 <(9, 3)>a
4 -> 2 <(6, 7)>a

In [13]:
a.has_twins_property()

Out[13]:
True