Abstract word (finite or infinite)

This module gathers functions that works for both finite and infinite words.

AUTHORS:

• Sebastien Labbe
• Franco Saliola

EXAMPLES:

sage: a = 0.618
sage: g = words.CodingOfRotationWord(alpha=a, beta=1-a, x=a)
sage: f = words.FibonacciWord()
sage: p = f.longest_common_prefix(g, length='finite')
sage: p
word: 0100101001001010010100100101001001010010...
sage: p.length()
231

class sage.combinat.words.abstract_word.Word_class

Bases: sage.structure.sage_object.SageObject

alphabet()

EXAMPLES:

sage: w = Word('abaccefa')
sage: w. alphabet()
doctest:1: DeprecationWarning: alphabet() is deprecated, use parent().alphabet() instead
Python objects
sage: y = Words('456')('64654564')
sage: y.alphabet()
Ordered Alphabet ['4', '5', '6']

apply_morphism(morphism)

Returns the word obtained by applying the morphism to self.

INPUT:

• morphism - Can be an instance of WordMorphism, or anything that can be used to construct one.

EXAMPLES:

sage: w = Word("ab")
sage: d = {'a':'ab', 'b':'ba'}
sage: w.apply_morphism(d)
word: abba
sage: w.apply_morphism(WordMorphism(d))
word: abba

sage: w = Word('ababa')
sage: d = dict(a='ab', b='ba')
sage: d
{'a': 'ab', 'b': 'ba'}
sage: w.apply_morphism(d)
word: abbaabbaab

For infinite words:

sage: t = words.ThueMorseWord([0,1]); t
word: 0110100110010110100101100110100110010110...
sage: t.apply_morphism({0:8,1:9})
word: 8998988998898998988989988998988998898998...

delta()

Returns the image of self under the delta morphism. This is the word composed of the length of consecutive runs of the same letter in a given word.

OUTPUT:

Word over integers

EXAMPLES:

For finite words:

sage: W = Words('0123456789')
sage: W('22112122').delta()
word: 22112
sage: W('555008').delta()
word: 321
sage: W().delta()
word: 
sage: Word('aabbabaa').delta()
word: 22112

For infinite words:

sage: t = words.ThueMorseWord()
sage: t.delta()
word: 1211222112112112221122211222112112112221...

finite_differences(mod=None)

Returns the word obtained by the diffences of consecutive letters of self.

INPUT:

• self - A word over the integers.

• mod - (default: None) It can be one of the following:

  • None or 0 : result is over the integers
  • integer : result is over the integers modulo mod.

EXAMPLES:

sage: w = Word([x^2 for x in range(10)])
sage: w.finite_differences()
word: 1,3,5,7,9,11,13,15,17
sage: w.finite_differences(mod=4)
word: 131313131
sage: w.finite_differences(mod=0)
word: 1,3,5,7,9,11,13,15,17

TESTS:

sage: w = Word([2,3,6])
sage: w.finite_differences()
word: 13
sage: w = Word([2,6])
sage: w.finite_differences()
word: 4
sage: w = Word([2])
sage: w.finite_differences()
word:
sage: w = Word()
sage: w.finite_differences()
word:

If the word is infinite, so is the result:

sage: w = Word(lambda n:n)
sage: u = w.finite_differences()
sage: u
word: 1111111111111111111111111111111111111111...
sage: type(u)
<class 'sage.combinat.words.word.InfiniteWord_iter_with_caching'>

is_empty()

Returns True if the length of self is zero, and False otherwise.

EXAMPLES:

sage: it = iter([])
sage: Word(it).is_empty()
True
sage: it = iter([1,2,3])
sage: Word(it).is_empty()
False
sage: from itertools import count
sage: Word(count()).is_empty()
False

iterated_right_palindromic_closure(f=None, algorithm='recursive')

Returns the iterated (-)palindromic closure of self.

INPUT:

• f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).
• algorithm - string (default: 'recursive') specifying which algorithm to be used when computing the iterated palindromic closure. It must be one of the two following values:
  • 'definition' - computed using the definition
  • 'recursive' - computation based on an efficient formula that recursively computes the iterated right palindromic closure without having to recompute the longest -palindromic suffix at each iteration [2].

OUTPUT:

word – the iterated (-)palindromic closure of self

EXAMPLES:

sage: w = Word('abc')
sage: w.iterated_right_palindromic_closure()
word: abacaba

sage: w = Word('aaa')
sage: w.iterated_right_palindromic_closure()
word: aaa

sage: w = Word('abbab')
sage: w.iterated_right_palindromic_closure()
word: ababaabababaababa

A right -palindromic closure:

sage: f = WordMorphism('a->b,b->a')
sage: w = Word('abbab')
sage: w.iterated_right_palindromic_closure(f=f)
word: abbaabbaababbaabbaabbaababbaabbaab

An infinite word:

sage: t = words.ThueMorseWord('ab')
sage: t.iterated_right_palindromic_closure()
word: ababaabababaababaabababaababaabababaabab...

There are two implementations computing the iterated right -palindromic closure, the latter being much more efficient:

sage: w = Word('abaab')
sage: u = w.iterated_right_palindromic_closure(algorithm='definition')
sage: v = w.iterated_right_palindromic_closure(algorithm='recursive')
sage: u
word: abaabaababaabaaba
sage: u == v
True
sage: w = words.RandomWord(8)
sage: u = w.iterated_right_palindromic_closure(algorithm='definition')
sage: v = w.iterated_right_palindromic_closure(algorithm='recursive')
sage: u == v
True

TESTS:

The empty word:

sage: w = Word()
sage: w.iterated_right_palindromic_closure()
word:

If the word is finite, so is the result:

sage: w = Word([0,1]*7)
sage: c = w.iterated_right_palindromic_closure()
sage: type(c)
<class 'sage.combinat.words.word.FiniteWord_iter_with_caching'>

REFERENCES:

• [1] A. de Luca, A. De Luca, Pseudopalindrome closure operators in free monoids, Theoret. Comput. Sci. 362 (2006) 282–300.
• [2] J. Justin, Episturmian morphisms and a Galois theorem on continued fractions, RAIRO Theoret. Informatics Appl. 39 (2005) 207-215.

length()

Returns the length of self.

TESTS:

sage: from sage.combinat.words.word import Word_class
sage: w = Word(iter('abba'*100), length="unknown")
sage: w.length() is None
True
sage: w = Word(iter('abba'), length="finite")
sage: w.length()
4
sage: w = Word(iter([0,1,1,0,1,0,0,1]*100), length="unknown")
sage: w.length() is None
True
sage: w = Word(iter([0,1,1,0,1,0,0,1]), length="finite")
sage: w.length()
8

lex_greater(other)

Returns True if self is lexicographically greater than other.

EXAMPLES:

sage: w = Word([1,2,3])
sage: u = Word([1,3,2])
sage: v = Word([3,2,1])
sage: w.lex_greater(u)
False
sage: v.lex_greater(w)
True
sage: a = Word("abba")
sage: b = Word("abbb")
sage: a.lex_greater(b)
False
sage: b.lex_greater(a)
True

For infinite words:

sage: t = words.ThueMorseWord()
sage: t[:10].lex_greater(t)
False
sage: t.lex_greater(t[:10])
True

lex_less(other)

Returns True if self is lexicographically less than other.

EXAMPLES:

sage: w = Word([1,2,3])
sage: u = Word([1,3,2])
sage: v = Word([3,2,1])
sage: w.lex_less(u)
True
sage: v.lex_less(w)
False
sage: a = Word("abba")
sage: b = Word("abbb")
sage: a.lex_less(b)
True
sage: b.lex_less(a)
False

For infinite words:

sage: t = words.ThueMorseWord()
sage: t.lex_less(t[:10])
False
sage: t[:10].lex_less(t)
True

longest_common_prefix(other, length='unknown')

Returns the longest common prefix of self and other.

INPUT:

• other - word
• length - str or +Infinity (optional, default: 'unknown') the length type of the resulting word if known. It may be one of the following:
  • 'unknown'
  • 'finite'
  • 'infinite' or Infinity

OUTPUT:

word

EXAMPLES:

sage: f = lambda n : add(Integer(n).digits(2)) % 2
sage: t = Word(f)
sage: u = t[:10]
sage: t.longest_common_prefix(u)
word: 0110100110

The longest common prefix of two equal infinite words:

sage: t1 = Word(f)
sage: t2 = Word(f)
sage: t1.longest_common_prefix(t2)
word: 0110100110010110100101100110100110010110...

Useful to study the approximation of an infinite word:

sage: a = 0.618
sage: g = words.CodingOfRotationWord(alpha=a, beta=1-a, x=a)
sage: f = words.FibonacciWord()
sage: p = f.longest_common_prefix(g, length='finite')
sage: p.length()
231

TESTS:

sage: w = Word('12345')
sage: y = Word('1236777')
sage: w.longest_common_prefix(y)
word: 123
sage: w.longest_common_prefix(w)
word: 12345
sage: y.longest_common_prefix(w)
word: 123
sage: y.longest_common_prefix(y)
word: 1236777
sage: Word().longest_common_prefix(w)
word: 
sage: w.longest_common_prefix(Word())
word: 
sage: w.longest_common_prefix(w[:3])
word: 123
sage: Word("11").longest_common_prefix(Word("1"))
word: 1
sage: Word("1").longest_common_prefix(Word("11"))
word: 1

With infinite words:

sage: t = words.ThueMorseWord('ab')
sage: u = t[:10]
sage: u.longest_common_prefix(t)
word: abbabaabba
sage: u.longest_common_prefix(u)
word: abbabaabba

longest_periodic_prefix(period=1)

Returns the longest prefix of self having the given period.

INPUT:

• period - positive integer (optional, default 1)

OUTPUT:

word

EXAMPLES:

sage: Word([]).longest_periodic_prefix()
word: 
sage: Word([1]).longest_periodic_prefix()
word: 1
sage: Word([1,2]).longest_periodic_prefix()
word: 1
sage: Word([1,1,2]).longest_periodic_prefix()
word: 11
sage: Word([1,2,1,2,1,3]).longest_periodic_prefix(2)
word: 12121
sage: type(_)
<class 'sage.combinat.words.word.FiniteWord_iter_with_caching'>
sage: Word(lambda n:0).longest_periodic_prefix()
word: 0000000000000000000000000000000000000000...

palindrome_prefixes_iterator(max_length=None)

Returns an iterator over the palindrome prefixes of self.

INPUT:

• max_length - non negative integer or None (optional, default: None) the maximum length of the prefixes

OUTPUT:

iterator

EXAMPLES:

sage: w = Word('abaaba')
sage: for pp in w.palindrome_prefixes_iterator(): print pp
word: 
word: a
word: aba
word: abaaba
sage: for pp in w.palindrome_prefixes_iterator(max_length=4): print pp
word: 
word: a
word: aba

You can iterate over the palindrome prefixes of an infinite word:

sage: f = words.FibonacciWord()
sage: for pp in f.palindrome_prefixes_iterator(max_length=20): print pp
word: 
word: 0
word: 010
word: 010010
word: 01001010010
word: 0100101001001010010

parent()

Returns the parent of self.

TESTS:

sage: Word(iter([1,2,3]), length="unknown").parent()
Words
sage: Word(range(12)).parent()
Words
sage: Word(range(4), alphabet=range(6)).parent()
Words over Ordered Alphabet [0, 1, 2, 3, 4, 5]
sage: Word(iter('abac'), alphabet='abc').parent()
Words over Ordered Alphabet ['a', 'b', 'c']

partial_sums(start, mod=None)

Returns the word defined by the partial sums of its prefixes.

INPUT:

• self - A word over the integers.

• start - integer, the first letter of the resulting word.

• mod - (default: None) It can be one of the following:

  • None or 0 : result is over the integers
  • integer : result is over the integers modulo mod.

EXAMPLES:

sage: w = Word(range(10))
sage: w.partial_sums(0)
word: 0,0,1,3,6,10,15,21,28,36,45
sage: w.partial_sums(1)
word: 1,1,2,4,7,11,16,22,29,37,46

sage: w = Word([1,2,3,1,2,3,2,2,2,2])
sage: w.partial_sums(0, mod=None)
word: 0,1,3,6,7,9,12,14,16,18,20
sage: w.partial_sums(0, mod=0)
word: 0,1,3,6,7,9,12,14,16,18,20
sage: w.partial_sums(0, mod=8)
word: 01367146024
sage: w.partial_sums(0, mod=4)
word: 01323102020
sage: w.partial_sums(0, mod=2)
word: 01101100000
sage: w.partial_sums(0, mod=1)
word: 00000000000

TESTS:

If the word is infinite, so is the result:

sage: w = Word(lambda n:1)
sage: u = w.partial_sums(0)
sage: type(u)
<class 'sage.combinat.words.word.InfiniteWord_iter_with_caching'>

prefixes_iterator(max_length=None)

Returns an iterator over the prefixes of self.

INPUT:

• max_length - non negative integer or None (optional, default: None) the maximum length of the prefixes

OUTPUT:

iterator

EXAMPLES:

sage: w = Word('abaaba')
sage: for p in w.prefixes_iterator(): print p
word: 
word: a
word: ab
word: aba
word: abaa
word: abaab
word: abaaba
sage: for p in w.prefixes_iterator(max_length=3): print p
word: 
word: a
word: ab
word: aba

You can iterate over the prefixes of an infinite word:

sage: f = words.FibonacciWord()
sage: for p in f.prefixes_iterator(max_length=8): print p
word: 
word: 0
word: 01
word: 010
word: 0100
word: 01001
word: 010010
word: 0100101
word: 01001010

TESTS:

sage: list(f.prefixes_iterator(max_length=0))
[word: ]

string_rep()

Returns the raw sequence of letters as a string.

EXAMPLES:

sage: Word('abbabaab').string_rep()
'abbabaab'
sage: Word([0, 1, 0, 0, 1]).string_rep()
'01001'
sage: Word([0,1,10,101]).string_rep()
'0,1,10,101'
sage: WordOptions(letter_separator='-')
sage: Word([0,1,10,101]).string_rep()
'0-1-10-101'
sage: WordOptions(letter_separator=',')

to_integer_word()

Returns a word over the integers whose letters are those output by self._to_integer_iterator()

EXAMPLES:

sage: from itertools import count
sage: w = Word(count()); w
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,...
sage: w.to_integer_word()
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,...
sage: w = Word(iter("abbacabba"), length="finite"); w
word: abbacabba
sage: w.to_integer_word()
word: 011020110
sage: w = Word(iter("abbacabba"), length="unknown"); w
word: abbacabba
sage: w.to_integer_word()
word: 011020110