Infinite word

AUTHORS:

  • Sebastien Labbe

  • Franco Saliola

EXAMPLES:

Creation of an infinite word

Periodic infinite words:

sage: v = Word([0, 4, 8, 8, 3])
sage: vv = v^Infinity
sage: vv
word: 0488304883048830488304883048830488304883...

>>> from sage.all import *
>>> v = Word([Integer(0), Integer(4), Integer(8), Integer(8), Integer(3)])
>>> vv = v**Infinity
>>> vv
word: 0488304883048830488304883048830488304883...

v = Word([0, 4, 8, 8, 3])
vv = v^Infinity
vv

Infinite words from a function \(f:\mathbb{N}\rightarrow A\) over an alphabet \(A\):

sage: Word(lambda n: n%3)
word: 0120120120120120120120120120120120120120...

>>> from sage.all import *
>>> Word(lambda n: n%Integer(3))
word: 0120120120120120120120120120120120120120...

Word(lambda n: n%3)
sage: def t(n):
....:     return add(Integer(n).digits(base=2)) % 2
sage: Word(t, alphabet = [0, 1])
word: 0110100110010110100101100110100110010110...

>>> from sage.all import *
>>> def t(n):
...     return add(Integer(n).digits(base=Integer(2))) % Integer(2)
>>> Word(t, alphabet = [Integer(0), Integer(1)])
word: 0110100110010110100101100110100110010110...

def t(n):
    return add(Integer(n).digits(base=2)) % 2
Word(t, alphabet = [0, 1])
>>> from sage.all import *
>>> def t(n):
...     return add(Integer(n).digits(base=Integer(2))) % Integer(2)
>>> Word(t, alphabet = [Integer(0), Integer(1)])
word: 0110100110010110100101100110100110010110...

def t(n):
    return add(Integer(n).digits(base=2)) % 2
Word(t, alphabet = [0, 1])

or as a one-liner:

sage: Word(lambda n : add(Integer(n).digits(base=2)) % 2, alphabet = [0, 1])
word: 0110100110010110100101100110100110010110...

>>> from sage.all import *
>>> Word(lambda n : add(Integer(n).digits(base=Integer(2))) % Integer(2), alphabet = [Integer(0), Integer(1)])
word: 0110100110010110100101100110100110010110...

Word(lambda n : add(Integer(n).digits(base=2)) % 2, alphabet = [0, 1])

Infinite words from iterators:

sage: from itertools import count,repeat
sage: Word( repeat(4) )
word: 4444444444444444444444444444444444444444...
sage: Word( count() )
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...

>>> from sage.all import *
>>> from itertools import count,repeat
>>> Word( repeat(Integer(4)) )
word: 4444444444444444444444444444444444444444...
>>> Word( count() )
word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...

from itertools import count,repeat
Word( repeat(4) )
Word( count() )

Infinite words from morphism

For example, let \(A=\{a,b\}\) and \(\mu : A^* \rightarrow A^*\) be the morphism defined by \(a\mapsto ab, b\mapsto ba\):

sage: mu = WordMorphism('a->ab,b->ba'); mu
WordMorphism: a->ab, b->ba
sage: mu.fixed_point('a')
word: abbabaabbaababbabaababbaabbabaabbaababba...

>>> from sage.all import *
>>> mu = WordMorphism('a->ab,b->ba'); mu
WordMorphism: a->ab, b->ba
>>> mu.fixed_point('a')
word: abbabaabbaababbabaababbaabbabaabbaababba...

mu = WordMorphism('a->ab,b->ba'); mu
mu.fixed_point('a')

Infinite words in a specific combinatorial class:

sage: W = InfiniteWords("ab"); W
Infinite words over {'a', 'b'}
sage: f = lambda n : 'a' if n % 2 == 1 else 'b'
sage: W(f)
word: babababababababababababababababababababa...

>>> from sage.all import *
>>> W = InfiniteWords("ab"); W
Infinite words over {'a', 'b'}
>>> f = lambda n : 'a' if n % Integer(2) == Integer(1) else 'b'
>>> W(f)
word: babababababababababababababababababababa...

W = InfiniteWords("ab"); W
f = lambda n : 'a' if n % 2 == 1 else 'b'
W(f)
class sage.combinat.words.infinite_word.InfiniteWord_class[source]

Bases: Word_class

length()[source]

Return the length of self.

EXAMPLES:

sage: f = lambda n : n % 6
sage: w = Word(f); w
word: 0123450123450123450123450123450123450123...
sage: w.length()
+Infinity

>>> from sage.all import *
>>> f = lambda n : n % Integer(6)
>>> w = Word(f); w
word: 0123450123450123450123450123450123450123...
>>> w.length()
+Infinity

f = lambda n : n % 6
w = Word(f); w
w.length()