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).standard()









    



:0: FutureWarning: IPython widgets are experimental and may change in the future.

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



In [2]:

    
a = std('(ab)*')+std('(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