Word morphisms/substitutions

This modules implements morphisms over finite and infinite words.

AUTHORS:

  • Sebastien Labbe (2007-06-01): initial version

  • Sebastien Labbe (2008-07-01): merged into sage-words

  • Sebastien Labbe (2008-12-17): merged into sage

  • Sebastien Labbe (2009-02-03): words next generation

  • Sebastien Labbe (2009-11-20): allowing the choice of the datatype of the image. Doc improvements.

  • Stepan Starosta (2012-11-09): growing letters

EXAMPLES:

Creation of a morphism from a dictionary or a string:

sage: n = WordMorphism({0:[0,2,2,1],1:[0,2],2:[2,2,1]})

sage: m = WordMorphism('x->xyxsxss,s->xyss,y->ys')

sage: n
WordMorphism: 0->0221, 1->02, 2->221
sage: m
WordMorphism: s->xyss, x->xyxsxss, y->ys

The codomain may be specified:

sage: WordMorphism({0:[0,2,2,1],1:[0,2],2:[2,2,1]}, codomain=Words([0,1,2,3,4]))
WordMorphism: 0->0221, 1->02, 2->221

Power of a morphism:

sage: n^2
WordMorphism: 0->022122122102, 1->0221221, 2->22122102

Image under a morphism:

sage: m('y')
word: ys
sage: m('xxxsy')
word: xyxsxssxyxsxssxyxsxssxyssys

Iterated image under a morphism:

sage: m('y', 3)
word: ysxyssxyxsxssysxyssxyss

See more examples in the documentation of the call method (m.__call__?).

Infinite fixed point of morphism:

sage: fix = m.fixed_point('x')
sage: fix
word: xyxsxssysxyxsxssxyssxyxsxssxyssxyssysxys...
sage: fix.length()
+Infinity

Incidence matrix:

sage: matrix(m)
[2 3 1]
[1 3 0]
[1 1 1]

Many other functionalities…:

sage: m.is_identity()
False
sage: m.is_endomorphism()
True
class sage.combinat.words.morphism.PeriodicPointIterator(m, cycle)

Bases: object

(Lazy) constructor of the periodic points of a word morphism.

This class is mainly used in WordMorphism.periodic_point and WordMorphism.periodic_points.

EXAMPLES:

sage: from sage.combinat.words.morphism import PeriodicPointIterator
sage: s = WordMorphism('a->bacca,b->cba,c->aab')
sage: p = PeriodicPointIterator(s, ['a','b','c'])
sage: p._cache[0]
lazy list ['a', 'a', 'b', ...]
sage: p._cache[1]
lazy list ['b', 'a', 'c', ...]
sage: p._cache[2]
lazy list ['c', 'b', 'a', ...]
get_iterator(i)

Internal method.

EXAMPLES:

sage: from sage.combinat.words.morphism import PeriodicPointIterator
sage: s = WordMorphism('a->bacca,b->cba,c->aab')
sage: p = PeriodicPointIterator(s, ['a','b','c'])
sage: p.get_iterator(0)
<generator object ...get_iterator at ...>
class sage.combinat.words.morphism.WordMorphism(data, domain=None, codomain=None)

Bases: sage.structure.sage_object.SageObject

WordMorphism class

INPUT:

  • data – dict or str or an instance of WordMorphism, the map giving the image of letters

  • domain – (optional:None) set of words over a given alphabet. If None, the domain alphabet is computed from data and is sorted.

  • codomain – (optional:None) set of words over a given alphabet. If None, the codomain alphabet is computed from data and is sorted.

Note

When the domain or the codomain are not explicitly given, it is expected that the letters are comparable because the alphabets of the domain and of the codomain are sorted.

EXAMPLES:

From a dictionary:

sage: n = WordMorphism({0:[0,2,2,1],1:[0,2],2:[2,2,1]})
sage: n
WordMorphism: 0->0221, 1->02, 2->221

From a string with '->' as separation:

sage: m = WordMorphism('x->xyxsxss,s->xyss,y->ys')
sage: m
WordMorphism: s->xyss, x->xyxsxss, y->ys
sage: m.domain()
Finite words over {'s', 'x', 'y'}
sage: m.codomain()
Finite words over {'s', 'x', 'y'}

Specifying the domain and codomain:

sage: W = FiniteWords([0,1,2])
sage: d = {0:[0,1], 1:[0,1,0], 2:[0]}
sage: m = WordMorphism(d, domain=W, codomain=W)
sage: m([0]).parent()
Finite words over {0, 1, 2}

When the alphabet is non-sortable, the domain and/or codomain must be explicitly given:

sage: W = FiniteWords(['a',6])
sage: d = {'a':['a',6,'a'],6:[6,6,6,'a']}
sage: WordMorphism(d, domain=W, codomain=W)
WordMorphism: 6->666a, a->a6a
abelian_rotation_subspace()

Returns the subspace on which the incidence matrix of self acts by roots of unity.

EXAMPLES:

sage: WordMorphism('0->1,1->0').abelian_rotation_subspace()
Vector space of degree 2 and dimension 2 over Rational Field
Basis matrix:
[1 0]
[0 1]
sage: WordMorphism('0->01,1->10').abelian_rotation_subspace()
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
sage: WordMorphism('0->01,1->1').abelian_rotation_subspace()
Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[0 1]
sage: WordMorphism('1->122,2->211').abelian_rotation_subspace()
Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[ 1 -1]
sage: WordMorphism('0->1,1->102,2->3,3->4,4->2').abelian_rotation_subspace()
Vector space of degree 5 and dimension 3 over Rational Field
Basis matrix:
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]

The domain needs to be equal to the codomain:

sage: WordMorphism('0->1,1->',codomain=Words('01')).abelian_rotation_subspace()
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]
codomain()

Returns the codomain of self.

EXAMPLES:

sage: WordMorphism('a->ab,b->a').codomain()
Finite words over {'a', 'b'}
sage: WordMorphism('6->ab,y->5,0->asd').codomain()
Finite words over {'5', 'a', 'b', 'd', 's'}
conjugate(pos)

Returns the morphism where the image of the letter by self is conjugated of parameter pos.

INPUT:

  • pos - integer

EXAMPLES:

sage: m = WordMorphism('a->abcde')
sage: m.conjugate(0) == m
True
sage: m.conjugate(1)
WordMorphism: a->bcdea
sage: m.conjugate(3)
WordMorphism: a->deabc
sage: WordMorphism('').conjugate(4)
WordMorphism:
sage: m = WordMorphism('a->abcde,b->xyz')
sage: m.conjugate(2)
WordMorphism: a->cdeab, b->zxy
domain()

Returns domain of self.

EXAMPLES:

sage: WordMorphism('a->ab,b->a').domain()
Finite words over {'a', 'b'}
sage: WordMorphism('b->ba,a->ab').domain()
Finite words over {'a', 'b'}
sage: WordMorphism('6->ab,y->5,0->asd').domain()
Finite words over {'0', '6', 'y'}
dual_map(k=1)

Return the dual map \(E_k^*\) of self (see [1]).

Note

It is actually implemented only for \(k=1\).

INPUT:

  • self - unimodular endomorphism defined on integers 1, 2, \ldots, d

  • k - integer (optional, default: 1)

OUTPUT:

an instance of E1Star - the dual map

EXAMPLES:

sage: sigma = WordMorphism({1:[2],2:[3],3:[1,2]})
sage: sigma.dual_map()
E_1^*(1->2, 2->3, 3->12)

sage: sigma.dual_map(k=2)
Traceback (most recent call last):
...
NotImplementedError: The dual map E_k^* is implemented only for k = 1 (not 2)

REFERENCES:

  • [1] Sano, Y., Arnoux, P. and Ito, S., Higher dimensional extensions of substitutions and their dual maps, Journal d’Analyse Mathematique 83 (2001), 183-206.

extend_by(other)

Returns self extended by other.

Let \(\varphi_1:A^*\rightarrow B^*\) and \(\varphi_2:C^*\rightarrow D^*\) be two morphisms. A morphism \(\mu:(A\cup C)^*\rightarrow (B\cup D)^*\) corresponds to \(\varphi_1\) extended by \(\varphi_2\) if \(\mu(a)=\varphi_1(a)\) if \(a\in A\) and \(\mu(a)=\varphi_2(a)\) otherwise.

INPUT:

  • other - a WordMorphism.

OUTPUT:

WordMorphism

EXAMPLES:

sage: m = WordMorphism('a->ab,b->ba')
sage: n = WordMorphism({'0':'1','1':'0','a':'5'})
sage: m.extend_by(n)
WordMorphism: 0->1, 1->0, a->ab, b->ba
sage: n.extend_by(m)
WordMorphism: 0->1, 1->0, a->5, b->ba
sage: m.extend_by(m)
WordMorphism: a->ab, b->ba
fixed_point(letter)

Returns the fixed point of self beginning by the given letter.

A fixed point of morphism \(\varphi\) is a word \(w\) such that \(\varphi(w) = w\).

INPUT:

  • self - an endomorphism, must be prolongable on letter

  • letter - in the domain of self, the first letter of the fixed point.

OUTPUT:

  • word - the fixed point of self beginning with letter.

EXAMPLES:

sage: W = FiniteWords('abc')

  1. Infinite fixed point:

    sage: WordMorphism('a->ab,b->ba').fixed_point(letter='a')
word: abbabaabbaababbabaababbaabbabaabbaababba...
sage: WordMorphism('a->ab,b->a').fixed_point(letter='a')
word: abaababaabaababaababaabaababaabaababaaba...
sage: WordMorphism('a->ab,b->b,c->ba', codomain=W).fixed_point(letter='a')
word: abbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb...

  2. Infinite fixed point of an erasing morphism:

    sage: WordMorphism('a->ab,b->,c->ba', codomain=W).fixed_point(letter='a')
word: ab

  3. Finite fixed point:

    sage: WordMorphism('a->ab,b->b,c->ba', codomain=W).fixed_point(letter='b')
word: b
sage: _.parent()
Finite words over {'a', 'b', 'c'}

sage: WordMorphism('a->ab,b->cc,c->', codomain=W).fixed_point(letter='a')
word: abcc
sage: _.parent()
Finite words over {'a', 'b', 'c'}

sage: m = WordMorphism('a->abc,b->,c->')
sage: fp = m.fixed_point('a'); fp
word: abc

sage: m = WordMorphism('a->ba,b->')
sage: m('ba')
word: ba
sage: m.fixed_point('a') #todo: not implemented
word: ba

  1. Fixed point of a power of a morphism:

    sage: m = WordMorphism('a->ba,b->ab')
sage: (m^2).fixed_point(letter='a')
word: abbabaabbaababbabaababbaabbabaabbaababba...
fixed_points()

Returns the list of all fixed points of self.

EXAMPLES:

sage: f = WordMorphism('a->ab,b->ba')
sage: for w in f.fixed_points(): print(w)
abbabaabbaababbabaababbaabbabaabbaababba...
baababbaabbabaababbabaabbaababbaabbabaab...

sage: f = WordMorphism('a->ab,b->c,c->a')
sage: for w in f.fixed_points(): print(w)
abcaababcabcaabcaababcaababcabcaababcabc...

sage: f = WordMorphism('a->ab,b->cab,c->bcc')
sage: for w in f.fixed_points(): print(w)
abcabbccabcabcabbccbccabcabbccabcabbccab...

This shows that ticket trac ticket #13668 has been resolved:

sage: d = {1:[1,2],2:[2,3],3:[4],4:[5],5:[6],6:[7],7:[8],8:[9],9:[10],10:[1]}
sage: s = WordMorphism(d)
sage: s7 = s^7
sage: s7.fixed_points()
[word: 12232342..., word: 2,3,4,5,6,7,8...]
sage: s7r = s7.reversal()
sage: s7r.periodic_point(2)
word: 2,1,1,10,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3,2,10,9,8,7,6,5,4,3,2,9,8,7,6,5,4,3,2,8,...

This shows that ticket trac ticket #13668 has been resolved:

sage: s = "1->321331332133133,2->133321331332133133,3->2133133133321331332133133"
sage: s = WordMorphism(s)
sage: (s^2).fixed_points()
[]
growing_letters()

Returns the list of growing letters.

See is_growing() for more information.

EXAMPLES:

sage: WordMorphism('0->01,1->10').growing_letters()
['0', '1']
sage: WordMorphism('0->01,1->1').growing_letters()
['0']
sage: WordMorphism('0->01,1->0,2->1',codomain=Words('012')).growing_letters()
['0', '1', '2']
has_conjugate_in_classP(f=None)

Returns True if self has a conjugate in class \(f\)-\(P\).

DEFINITION : Let \(A\) be an alphabet. We say that a primitive substitution \(S\) is in the class P if there exists a palindrome \(p\) and for each \(b\in A\) a palindrome \(q_b\) such that \(S(b)=pq_b\) for all \(b\in A\). [1]

Let \(f\) be an involution on \(A\). We say that a morphism \(\varphi\) is in class \(f\)-\(P\) if there exists an \(f\)-palindrome \(p\) and for each \(\alpha \in A\) there exists an \(f\)-palindrome \(q_\alpha\) such that \(\varphi(\alpha)=pq_\alpha\). [2]

INPUT:

  • f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

REFERENCES:

  • [1] Hof, A., O. Knill et B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995) 149-159.

  • [2] Labbe, Sebastien. Proprietes combinatoires des \(f\)-palindromes, Memoire de maitrise en Mathematiques, Montreal, UQAM, 2008, 109 pages.

EXAMPLES:

sage: fibo = WordMorphism('a->ab,b->a')
sage: fibo.has_conjugate_in_classP()
True
sage: (fibo^2).is_in_classP()
False
sage: (fibo^2).has_conjugate_in_classP()
True
has_left_conjugate()

Returns True if all the non empty images of self begins with the same letter.

EXAMPLES:

sage: m = WordMorphism('a->abcde,b->xyz')
sage: m.has_left_conjugate()
False
sage: WordMorphism('b->xyz').has_left_conjugate()
True
sage: WordMorphism('').has_left_conjugate()
True
sage: WordMorphism('a->,b->xyz').has_left_conjugate()
True
sage: WordMorphism('a->abbab,b->abb').has_left_conjugate()
True
sage: WordMorphism('a->abbab,b->abb,c->').has_left_conjugate()
True
has_right_conjugate()

Returns True if all the non empty images of self ends with the same letter.

EXAMPLES:

sage: m = WordMorphism('a->abcde,b->xyz')
sage: m.has_right_conjugate()
False
sage: WordMorphism('b->xyz').has_right_conjugate()
True
sage: WordMorphism('').has_right_conjugate()
True
sage: WordMorphism('a->,b->xyz').has_right_conjugate()
True
sage: WordMorphism('a->abbab,b->abb').has_right_conjugate()
True
sage: WordMorphism('a->abbab,b->abb,c->').has_right_conjugate()
True
image(letter)

Return the image of a letter.

INPUT:

  • letter – a letter in the domain alphabet

OUTPUT:

word

Note

The letter is assumed to be in the domain alphabet (no check done). Hence, this method is faster than the __call__ method suitable for words input.

EXAMPLES:

sage: m = WordMorphism('a->ab,b->ac,c->a')
sage: m.image('b')
word: ac

sage: s = WordMorphism({('a', 1):[('a', 1), ('a', 2)], ('a', 2):[('a', 1)]})
sage: s.image(('a',1))
word: ('a', 1),('a', 2)

sage: s = WordMorphism({'b':[1,2], 'a':(2,3,4), 'z':[9,8,7]})
sage: s.image('b')
word: 12
sage: s.image('a')
word: 234
sage: s.image('z')
word: 987
images()

Returns the list of all the images of the letters of the alphabet under self.

EXAMPLES:

sage: sorted(WordMorphism('a->ab,b->a').images())
[word: a, word: ab]
sage: sorted(WordMorphism('6->ab,y->5,0->asd').images())
[word: 5, word: ab, word: asd]
incidence_matrix()

Returns the incidence matrix of the morphism. The order of the rows and column are given by the order defined on the alphabet of the domain and the codomain.

The matrix returned is over the integers. If a different ring is desired, use either the change_ring function or the matrix function.

EXAMPLES:

sage: m = WordMorphism('a->abc,b->a,c->c')
sage: m.incidence_matrix()
[1 1 0]
[1 0 0]
[1 0 1]
sage: m = WordMorphism('a->abc,b->a,c->c,d->abbccccabca,e->abc')
sage: m.incidence_matrix()
[1 1 0 3 1]
[1 0 0 3 1]
[1 0 1 5 1]
is_empty()

Returns True if the cardinality of the domain is zero and False otherwise.

EXAMPLES:

sage: WordMorphism('').is_empty()
True
sage: WordMorphism('a->a').is_empty()
False
is_endomorphism()

Returns True if the codomain is a subset of the domain.

EXAMPLES:

sage: WordMorphism('a->ab,b->a').is_endomorphism()
True
sage: WordMorphism('6->ab,y->5,0->asd').is_endomorphism()
False
sage: WordMorphism('a->a,b->aa,c->aaa').is_endomorphism()
False
sage: Wabc = Words('abc')
sage: m = WordMorphism('a->a,b->aa,c->aaa',codomain = Wabc)
sage: m.is_endomorphism()
True

We check that trac ticket #8674 is fixed:

sage: P = WordPaths('abcd')
sage: m = WordMorphism('a->adab,b->ab,c->cbcd,d->cd', domain=P, codomain=P)
sage: m.is_endomorphism()
True
is_erasing()

Returns True if self is an erasing morphism, i.e. the image of a letter is the empty word.

EXAMPLES:

sage: WordMorphism('a->ab,b->a').is_erasing()
False
sage: WordMorphism('6->ab,y->5,0->asd').is_erasing()
False
sage: WordMorphism('6->ab,y->5,0->asd,7->').is_erasing()
True
sage: WordMorphism('').is_erasing()
False
is_growing(letter=None)

Return True if letter is a growing letter.

A letter \(a\) is growing for the morphism \(s\) if the length of the iterates of \(| s^n(a) |\) tend to infinity as \(n\) goes to infinity.

INPUT:

  • letter – None or a letter in the domain of self

Note

If letter is None, this returns True if self is everywhere growing, i.e., all letters are growing letters (see [CassNic10]), and that self must be an endomorphism.

EXAMPLES:

sage: WordMorphism('0->01,1->1').is_growing('0')
True
sage: WordMorphism('0->01,1->1').is_growing('1')
False
sage: WordMorphism('0->01,1->10').is_growing()
True
sage: WordMorphism('0->1,1->2,2->01').is_growing()
True
sage: WordMorphism('0->01,1->1').is_growing()
False

The domain needs to be equal to the codomain:

sage: WordMorphism('0->01,1->0,2->1',codomain=Words('012')).is_growing()
True

Test of erasing morphisms:

sage: WordMorphism('0->01,1->').is_growing('0')
False
sage: m = WordMorphism('a->bc,b->bcc,c->',codomain=Words('abc'))
sage: m.is_growing('a')
False
sage: m.is_growing('b')
False
sage: m.is_growing('c')
False

REFERENCES:

CassNic10

Cassaigne J., Nicolas F. Factor complexity. Combinatorics, automata and number theory, 163–247, Encyclopedia Math. Appl., 135, Cambridge Univ. Press, Cambridge, 2010.

is_identity()

Returns True if self is the identity morphism.

EXAMPLES:

sage: m = WordMorphism('a->a,b->b,c->c,d->e')
sage: m.is_identity()
False
sage: WordMorphism('a->a,b->b,c->c').is_identity()
True
sage: WordMorphism('a->a,b->b,c->cb').is_identity()
False
sage: m = WordMorphism('a->b,b->c,c->a')
sage: (m^2).is_identity()
False
sage: (m^3).is_identity()
True
sage: (m^4).is_identity()
False
sage: WordMorphism('').is_identity()
True
sage: WordMorphism({0:[0],1:[1]}).is_identity()
True

We check that trac ticket #8618 is fixed:

sage: t = WordMorphism({'a1':['a2'], 'a2':['a1']})
sage: (t*t).is_identity()
True
is_in_classP(f=None)

Returns True if self is in class \(P\) (or \(f\)-\(P\)).

DEFINITION : Let \(A\) be an alphabet. We say that a primitive substitution \(S\) is in the class P if there exists a palindrome \(p\) and for each \(b\in A\) a palindrome \(q_b\) such that \(S(b)=pq_b\) for all \(b\in A\). [1]

Let \(f\) be an involution on \(A\). “We say that a morphism \(\varphi\) is in class \(f\)-\(P\) if there exists an \(f\)-palindrome \(p\) and for each \(\alpha \in A\) there exists an \(f\)-palindrome \(q_\alpha\) such that \(\varphi(\alpha)=pq_\alpha\). [2]

INPUT:

  • f - involution (default: None) on the alphabet of self. It must be callable on letters as well as words (e.g. WordMorphism).

REFERENCES:

  • [1] Hof, A., O. Knill et B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995) 149-159.

  • [2] Labbe, Sebastien. Proprietes combinatoires des \(f\)-palindromes, Memoire de maitrise en Mathematiques, Montreal, UQAM, 2008, 109 pages.

EXAMPLES:

sage: WordMorphism('a->bbaba,b->bba').is_in_classP()
True
sage: tm = WordMorphism('a->ab,b->ba')
sage: tm.is_in_classP()
False
sage: f = WordMorphism('a->b,b->a')
sage: tm.is_in_classP(f=f)
True
sage: (tm^2).is_in_classP()
True
sage: (tm^2).is_in_classP(f=f)
False
sage: fibo = WordMorphism('a->ab,b->a')
sage: fibo.is_in_classP()
True
sage: fibo.is_in_classP(f=f)
False
sage: (fibo^2).is_in_classP()
False
sage: f = WordMorphism('a->b,b->a,c->c')
sage: WordMorphism('a->acbcc,b->acbab,c->acbba').is_in_classP(f)
True
is_involution()

Returns True if self is an involution, i.e. its square is the identity.

INPUT:

  • self - an endomorphism

EXAMPLES:

sage: WordMorphism('a->b,b->a').is_involution()
True
sage: WordMorphism('a->b,b->ba').is_involution()
False
sage: WordMorphism({0:[1],1:[0]}).is_involution()
True
is_primitive()

Returns True if self is primitive.

A morphism \(\varphi\) is primitive if there exists an positive integer \(k\) such that for all \(\alpha\in\Sigma\), \(\varphi^k(\alpha)\) contains all the letters of \(\Sigma\).

INPUT:

  • self - an endomorphism

ALGORITHM:

Exercices 8.7.8, p.281 in [1]: (c) Let \(y(M)\) be the least integer \(e\) such that \(M^e\) has all positive entries. Prove that, for all primitive matrices \(M\), we have \(y(M) \leq (d-1)^2 + 1\). (d) Prove that the bound \(y(M)\leq (d-1)^2+1\) is best possible.

EXAMPLES:

sage: tm = WordMorphism('a->ab,b->ba')
sage: tm.is_primitive()
True
sage: fibo = WordMorphism('a->ab,b->a')
sage: fibo.is_primitive()
True
sage: m = WordMorphism('a->bb,b->aa')
sage: m.is_primitive()
False
sage: f = WordMorphism({0:[1],1:[0]})
sage: f.is_primitive()
False

sage: s = WordMorphism('a->b,b->c,c->ab')
sage: s.is_primitive()
True
sage: s = WordMorphism('a->b,b->c,c->d,d->e,e->f,f->g,g->h,h->ab')
sage: s.is_primitive()
True

REFERENCES:

  • [1] Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, 2003.

is_prolongable(letter)

Returns True if self is prolongable on letter.

A morphism \(\varphi\) is prolongable on a letter \(a\) if \(a\) is a prefix of \(\varphi(a)\).

INPUT:

  • self - its codomain must be an instance of Words

  • letter - a letter in the domain alphabet

OUTPUT:

Boolean

EXAMPLES:

sage: WordMorphism('a->ab,b->a').is_prolongable(letter='a')
True
sage: WordMorphism('a->ab,b->a').is_prolongable(letter='b')
False
sage: WordMorphism('a->ba,b->ab').is_prolongable(letter='b')
False
sage: (WordMorphism('a->ba,b->ab')^2).is_prolongable(letter='b')
True
sage: WordMorphism('a->ba,b->').is_prolongable(letter='b')
False
sage: WordMorphism('a->bb,b->aac').is_prolongable(letter='a')
False

We check that trac ticket #8595 is fixed:

sage: s = WordMorphism({('a', 1) : [('a', 1), ('a', 2)], ('a', 2) : [('a', 1)]})
sage: s.is_prolongable(('a',1))
True
is_uniform(k=None)

Returns True if self is a \(k\)-uniform morphism.

Let \(k\) be a positive integer. A morphism \(\phi\) is called \(k\)-uniform if for every letter \(\alpha\), we have \(|\phi(\alpha)| = k\). In other words, all images have length \(k\). A morphism is called uniform if it is \(k\)-uniform for some positive integer \(k\).

INPUT:

  • k - a positive integer or None. If set to a positive integer, then the function return True if self is \(k\)-uniform. If set to None, then the function return True if self is uniform.

EXAMPLES:

sage: phi = WordMorphism('a->ab,b->a')
sage: phi.is_uniform()
False
sage: phi.is_uniform(k=1)
False
sage: tau = WordMorphism('a->ab,b->ba')
sage: tau.is_uniform()
True
sage: tau.is_uniform(k=1)
False
sage: tau.is_uniform(k=2)
True
language(n, u=None)

Return the words of length n in the language generated by this substitution.

Given a non-erasing substitution \(s\) and a word \(u\) the DOL-language generated by \(s\) and \(u\) is the union of the factors of \(s^n(u)\) where \(n\) is a non-negative integer.

INPUT:

  • n – non-negative integer - length of the words in the language

  • u – a word or None (optional, default None) - if set to None some letter of the alphabet is used

OUTPUT: a Python set

EXAMPLES:

The fibonacci morphism:

sage: s = WordMorphism({0: [0,1], 1:[0]})
sage: sorted(s.language(3))
[word: 001, word: 010, word: 100, word: 101]
sage: len(s.language(1000))
1001
sage: all(len(s.language(n)) == n+1 for n in range(100))
True

A growing but non-primitive example. The DOL-languages generated by 0 and 2 are different:

sage: s = WordMorphism({0: [0,1], 1:[0], 2:[2,0,2]})

sage: u = s.fixed_point(0)
sage: A0 = u[:200].factor_set(5)
sage: B0 = s.language(5, [0])
sage: set(A0) == B0
True

sage: v = s.fixed_point(2)
sage: A2 = v[:200].factor_set(5)
sage: B2 = s.language(5, [2])
sage: set(A2) == B2
True

sage: len(A0), len(A2)
(6, 20)

The Chacon transformation (non-primitive):

sage: s = WordMorphism({0: [0,0,1,0], 1:[1]})
sage: sorted(s.language(10))
[word: 0001000101,
 word: 0001010010,
 ...
 word: 1010010001,
 word: 1010010100]
latex_layout(layout=None)

Get or set the actual latex layout (oneliner vs array).

INPUT:

  • layout - string (default: None), can take one of the following values:

    • None - Returns the actual latex layout. By default, the layout is 'array'

    • 'oneliner' - Set the layout to 'oneliner'

    • 'array' - Set the layout to 'array'

EXAMPLES:

sage: s = WordMorphism('a->ab,b->ba')
sage: s.latex_layout()
'array'
sage: s.latex_layout('oneliner')
sage: s.latex_layout()
'oneliner'
list_of_conjugates()

Returns the list of all the conjugate morphisms of self.

DEFINITION:

Recall from Lothaire [1] (Section 2.3.4) that \(\varphi\) is right conjugate of \(\varphi'\), noted \(\varphi\triangleleft\varphi'\), if there exists \(u \in \Sigma^*\) such that

\[\varphi(\alpha)u = u\varphi'(\alpha),\]

for all \(\alpha \in \Sigma\), or equivalently that \(\varphi(x)u = u\varphi'(x)\), for all words \(x \in \Sigma^*\). Clearly, this relation is not symmetric so that we say that two morphisms \(\varphi\) and \(\varphi'\) are conjugate, noted \(\varphi\bowtie\varphi'\), if \(\varphi\triangleleft\varphi'\) or \(\varphi'\triangleleft\varphi\). It is easy to see that conjugacy of morphisms is an equivalence relation.

REFERENCES:

  • [1] M. Lothaire, Algebraic Combinatorics on words, Cambridge University Press, 2002.

EXAMPLES:

sage: m = WordMorphism('a->abbab,b->abb')
sage: m.list_of_conjugates()
[WordMorphism: a->babba, b->bab,
WordMorphism: a->abbab, b->abb,
WordMorphism: a->bbaba, b->bba,
WordMorphism: a->babab, b->bab,
WordMorphism: a->ababb, b->abb,
WordMorphism: a->babba, b->bba,
WordMorphism: a->abbab, b->bab]
sage: m = WordMorphism('a->aaa,b->aa')
sage: m.list_of_conjugates()
[WordMorphism: a->aaa, b->aa]
sage: WordMorphism('').list_of_conjugates()
[WordMorphism: ]
sage: m = WordMorphism('a->aba,b->aba')
sage: m.list_of_conjugates()
[WordMorphism: a->baa, b->baa,
WordMorphism: a->aab, b->aab,
WordMorphism: a->aba, b->aba]
sage: m = WordMorphism('a->abb,b->abbab,c->')
sage: m.list_of_conjugates()
[WordMorphism: a->bab, b->babba, c->,
WordMorphism: a->abb, b->abbab, c->,
WordMorphism: a->bba, b->bbaba, c->,
WordMorphism: a->bab, b->babab, c->,
WordMorphism: a->abb, b->ababb, c->,
WordMorphism: a->bba, b->babba, c->,
WordMorphism: a->bab, b->abbab, c->]
partition_of_domain_alphabet()

Returns a partition of the domain alphabet.

Let \(\varphi:\Sigma^*\rightarrow\Sigma^*\) be an involution. There exists a triple of sets \((A, B, C)\) such that

  • \(A \cup B \cup C =\Sigma\);

  • \(A\), \(B\) and \(C\) are mutually disjoint and

  • \(\varphi(A)= B\), \(\varphi(B)= A\), \(\varphi(C)= C\).

These sets are not unique.

INPUT:

  • self - An involution.

OUTPUT:

A tuple of three sets

EXAMPLES:

sage: m = WordMorphism('a->b,b->a')
sage: m.partition_of_domain_alphabet() #random ordering
({'a'}, {'b'}, {})
sage: m = WordMorphism('a->b,b->a,c->c')
sage: m.partition_of_domain_alphabet() #random ordering
({'a'}, {'b'}, {'c'})
sage: m = WordMorphism('a->a,b->b,c->c')
sage: m.partition_of_domain_alphabet() #random ordering
({}, {}, {'a', 'c', 'b'})
sage: m = WordMorphism('A->T,T->A,C->G,G->C')
sage: m.partition_of_domain_alphabet() #random ordering
({'A', 'C'}, {'T', 'G'}, {})
sage: I = WordMorphism({0:oo,oo:0,1:-1,-1:1,2:-2,-2:2,3:-3,-3:3})
sage: I.partition_of_domain_alphabet() #random ordering
({0, -1, -3, -2}, {1, 2, 3, +Infinity}, {})
periodic_point(letter)

Return the periodic point of self that starts with letter.

EXAMPLES:

sage: f = WordMorphism('a->bab,b->ab')
sage: f.periodic_point('a')
word: abbababbababbabababbababbabababbababbaba...
sage: f.fixed_point('a')
Traceback (most recent call last):
...
TypeError: self must be prolongable on a
periodic_points()

Return the periodic points of f as a list of tuples where each tuple is a periodic orbit of f.

EXAMPLES:

sage: f = WordMorphism('a->aba,b->baa')
sage: for p in f.periodic_points():
....:     print("{} , {}".format(len(p), p[0]))
1 , ababaaababaaabaabaababaaababaaabaabaabab...
1 , baaabaabaababaaabaababaaabaababaaababaaa...

sage: f = WordMorphism('a->bab,b->aa')
sage: for p in f.periodic_points():
....:     print("{} , {}".format(len(p), p[0]))
2 , aababaaaababaababbabaababaababbabaababaa...
sage: f.fixed_points()
[]

This shows that ticket trac ticket #13668 has been resolved:

sage: d = {1:[1,2],2:[2,3],3:[4],4:[5],5:[6],6:[7],7:[8],8:[9],9:[10],10:[1]}
sage: s = WordMorphism(d)
sage: s7 = s^7
sage: s7r = s7.reversal()
sage: for p in s7r.periodic_points(): p
[word: 1,10,9,8,7,6,5,4,3,2,10,9,8,7,6,5,4,3,2,...,
 word: 8765432765432654325432432322176543265432...,
 word: 5,4,3,2,4,3,2,3,2,2,1,4,3,2,3,2,2,1,3,2,...,
 word: 2,1,1,10,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,...,
 word: 9876543287654327654326543254324323221876...,
 word: 6543254324323221543243232214323221322121...,
 word: 3,2,2,1,2,1,1,10,9,8,7,6,5,4,3,2,2,1,1,1...,
 word: 10,9,8,7,6,5,4,3,2,9,8,7,6,5,4,3,2,8,7,6...,
 word: 7654326543254324323221654325432432322154...,
 word: 4,3,2,3,2,2,1,3,2,2,1,2,1,1,10,9,8,7,6,5...]
pisot_eigenvector_left()

Returns the left eigenvector of the incidence matrix associated to the largest eigenvalue (in absolute value).

Unicity of the result is guaranteed when the multiplicity of the largest eigenvalue is one, for example when self is a Pisot irreductible substitution.

A substitution is Pisot irreducible if the characteristic polynomial of its incidence matrix is irreducible over \(\QQ\) and has all roots, except one, of modulus strictly smaller than 1.

INPUT:

  • self - a Pisot irreducible substitution.

EXAMPLES:

sage: m = WordMorphism('a->aaaabbc,b->aaabbc,c->aabc')
sage: matrix(m)
[4 3 2]
[2 2 1]
[1 1 1]
sage: m.pisot_eigenvector_left()
(1, 0.8392867552141611?, 0.5436890126920763?)
pisot_eigenvector_right()

Returns the right eigenvector of the incidence matrix associated to the largest eigenvalue (in absolute value).

Unicity of the result is guaranteed when the multiplicity of the largest eigenvalue is one, for example when self is a Pisot irreductible substitution.

A substitution is Pisot irreducible if the characteristic polynomial of its incidence matrix is irreducible over \(\QQ\) and has all roots, except one, of modulus strictly smaller than 1.

INPUT:

  • self - a Pisot irreducible substitution.

EXAMPLES:

sage: m = WordMorphism('a->aaaabbc,b->aaabbc,c->aabc')
sage: matrix(m)
[4 3 2]
[2 2 1]
[1 1 1]
sage: m.pisot_eigenvector_right()
(1, 0.5436890126920763?, 0.2955977425220848?)
rauzy_fractal_plot(n=None, exchange=False, eig=None, translate=None, prec=53, colormap='hsv', opacity=None, plot_origin=None, plot_basis=False, point_size=None)

Returns a plot of the Rauzy fractal associated with a substitution.

The substitution does not have to be irreducible. The usual definition of a Rauzy fractal requires that its dominant eigenvalue is a Pisot number but the present method doesn’t require this, allowing to plot some interesting pictures in the non-Pisot case (see the examples below).

For more details about the definition of the fractal and the projection which is used, see Section 3.1 of [1].

Plots with less than 100,000 points take a few seconds, and several millions of points can be plotted in reasonable time.

Other ways to draw Rauzy fractals (and more generally projections of paths) can be found in sage.combinat.words.paths.FiniteWordPath_all.plot_projection() or in sage.combinat.e_one_star().

OUTPUT:

A Graphics object.

INPUT:

  • n - integer (default: None) The number of points used to plot the fractal. Default values: 1000 for a 1D fractal, 50000 for a 2D fractal, 10000 for a 3D fractal.

  • exchange - boolean (default: False). Plot the Rauzy fractal with domain exchange.

  • eig - a real element of QQbar of degree >= 2 (default: None). The eigenvalue used to plot the fractal. It must be an eigenvalue of self.incidence_matrix(). The one used by default the maximal eigenvalue of self.incidence_matrix() (usually a Pisot number), but for substitutions with more than 3 letters other interesting choices are sometimes possible.

  • translate - a list of vectors of RR^size_alphabet, or a dictionary from the alphabet to lists of vectors (default: None). Plot translated copies of the fractal. This option allows to plot tilings easily. The projection used for these vectors is the same as the projection used for the canonical basis to plot the fractal. If the input is a list, all the pieces will be translated and plotted. If the input is a dictionary, each piece will be translated and plotted accordingly to the vectors associated with each letter in the dictionary. Note: by default, the Rauzy fractal placed at the origin is not plotted with the translate option; the vector (0,0,...,0) has to be added manually.

  • prec - integer (default: 53). The number of bits used in the floating point representations of the points of the fractal.

  • colormap - color map or dictionary (default: 'hsv'). It can be one of the following:

    • string - a coloring map. For available coloring map names type: sorted(colormaps)

    • dict - a dictionary of the alphabet mapped to colors.

  • opacity - a dictionary from the alphabet to the real interval [0,1] (default: None). If none is specified, all letters are plotted with opacity 1.

  • plot_origin - a couple (k,c) (default: None). If specified, mark the origin by a point of size k and color c.

  • plot_basis - boolean (default: False). Plot the projection of the canonical basis with the fractal.

  • point_size - float (default: None). The size of the points used to plot the fractal.

EXAMPLES:

  1. The Rauzy fractal of the Tribonacci substitution:

    sage: s = WordMorphism('1->12,2->13,3->1')
sage: s.rauzy_fractal_plot()     # long time
Graphics object consisting of 3 graphics primitives

  2. The “Hokkaido” fractal. We tweak the plot using the plotting options to get a nice reusable picture, in which we mark the origin by a black dot:

    sage: s = WordMorphism('a->ab,b->c,c->d,d->e,e->a')
sage: G = s.rauzy_fractal_plot(n=100000, point_size=3, plot_origin=(50,"black"))  # not tested
sage: G.show(figsize=10, axes=false) # not tested

  3. Another “Hokkaido” fractal and its domain exchange:

    sage: s = WordMorphism({1:[2], 2:[4,3], 3:[4], 4:[5,3], 5:[6], 6:[1]})
sage: s.rauzy_fractal_plot()                  # not tested takes > 1 second
sage: s.rauzy_fractal_plot(exchange=True)     # not tested takes > 1 second

  4. A three-dimensional Rauzy fractal:

    sage: s = WordMorphism('1->12,2->13,3->14,4->1')
sage: s.rauzy_fractal_plot()     # not tested takes > 1 second

  5. A one-dimensional Rauzy fractal (very scattered):

    sage: s = WordMorphism('1->2122,2->1')
sage: s.rauzy_fractal_plot().show(figsize=20)     # not tested takes > 1 second

  6. A high resolution plot of a complicated fractal:

    sage: s = WordMorphism('1->23,2->123,3->1122233')
sage: G = s.rauzy_fractal_plot(n=300000)  # not tested takes > 1 second
sage: G.show(axes=false, figsize=20)      # not tested takes > 1 second

  7. A nice colorful animation of a domain exchange:

    sage: s = WordMorphism('1->21,2->3,3->4,4->25,5->6,6->7,7->1')
sage: L = [s.rauzy_fractal_plot(), s.rauzy_fractal_plot(exchange=True)]     # not tested takes > 1 second
sage: animate(L, axes=false).show(delay=100)     # not tested takes > 1 second

  8. Plotting with only one color:

    sage: s = WordMorphism('1->12,2->31,3->1')
sage: s.rauzy_fractal_plot(colormap={'1':'black', '2':'black', '3':'black'})     # not tested takes > 1 second

  9. Different fractals can be obtained by choosing another (non-Pisot) eigenvalue:

    sage: s = WordMorphism('1->12,2->3,3->45,4->5,5->6,6->7,7->8,8->1')
sage: E = s.incidence_matrix().eigenvalues()
sage: x = [x for x in E if -0.8 < x < -0.7][0]
sage: s.rauzy_fractal_plot()          # not tested takes > 1 second
sage: s.rauzy_fractal_plot(eig=x)     # not tested takes > 1 second

  10. A Pisot reducible substitution with seemingly overlapping tiles:

    sage: s = WordMorphism({1:[1,2], 2:[2,3], 3:[4], 4:[5], 5:[6], 6:[7], 7:[8], 8:[9], 9:[10], 10:[1]})
sage: s.rauzy_fractal_plot()     # not tested takes > 1 second

  11. A non-Pisot reducible substitution with a strange Rauzy fractal:

    sage: s = WordMorphism({1:[3,2], 2:[3,3], 3:[4], 4:[1]})
sage: s.rauzy_fractal_plot()     # not tested takes > 1 second

  12. A substitution with overlapping tiles. We use the options colormap and opacity to study how the tiles overlap:

    sage: s = WordMorphism('1->213,2->4,3->5,4->1,5->21')
sage: s.rauzy_fractal_plot()                                   # not tested takes > 1 second
sage: s.rauzy_fractal_plot(colormap={'1':'red', '4':'purple'})     # not tested takes > 1 second
sage: s.rauzy_fractal_plot(opacity={'1':0.1,'2':1,'3':0.1,'4':0.1,'5':0.1}, n=150000)     # not tested takes > 1 second

  13. Funny experiments by playing with the precision of the float numbers used to plot the fractal:

    sage: s = WordMorphism('1->12,2->13,3->1')
sage: s.rauzy_fractal_plot(prec=6)      # not tested
sage: s.rauzy_fractal_plot(prec=9)      # not tested
sage: s.rauzy_fractal_plot(prec=15)     # not tested
sage: s.rauzy_fractal_plot(prec=19)     # not tested
sage: s.rauzy_fractal_plot(prec=25)     # not tested

  14. Using the translate option to plot periodic tilings:

    sage: s = WordMorphism('1->12,2->13,3->1')
sage: s.rauzy_fractal_plot(n=10000, translate=[(0,0,0),(-1,0,1),(0,-1,1),(1,-1,0),(1,0,-1),(0,1,-1),(-1,1,0)])     # not tested takes > 1 second

    sage: t = WordMorphism("a->aC,b->d,C->de,d->a,e->ab")   # substitution found by Julien Bernat
sage: V = [vector((0,0,1,0,-1)), vector((0,0,1,-1,0))]
sage: S = set(map(tuple, [i*V[0] + j*V[1] for i in [-1,0,1] for j in [-1,0,1]]))
sage: t.rauzy_fractal_plot(n=10000, translate=S, exchange=true)     # not tested takes > 1 second

  15. Using the translate option to plot arbitrary tilings with the fractal pieces. This can be used for example to plot the self-replicating tiling of the Rauzy fractal:

    sage: s = WordMorphism({1:[1,2], 2:[3], 3:[4,3], 4:[5], 5:[6], 6:[1]})
sage: s.rauzy_fractal_plot()     # not tested takes > 1 second
sage: D = {1:[(0,0,0,0,0,0), (0,1,0,0,0,0)], 3:[(0,0,0,0,0,0), (0,1,0,0,0,0)], 6:[(0,1,0,0,0,0)]}
sage: s.rauzy_fractal_plot(n=30000, translate=D)     # not tested takes > 1 second

  16. Plot the projection of the canonical basis with the fractal:

    sage: s = WordMorphism({1:[2,1], 2:[3], 3:[6,4], 4:[5,1], 5:[6], 6:[7], 7:[8], 8:[9], 9:[1]})
sage: s.rauzy_fractal_plot(plot_basis=True)     # not tested takes > 1 second

REFERENCES:

AUTHOR:

Timo Jolivet (2012-06-16)

rauzy_fractal_points(n=None, exchange=False, eig=None, translate=None, prec=53)

Returns a dictionary of list of points associated with the pieces of the Rauzy fractal of self.

INPUT:

See the method rauzy_fractal_plot() for a description of the options and more examples.

OUTPUT:

dictionary of list of points

EXAMPLES:

The Rauzy fractal of the Tribonacci substitution and the number of points in the piece of the fractal associated with '1', '2' and '3' are respectively:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: D = s.rauzy_fractal_points(n=100)
sage: len(D['1'])
54
sage: len(D['2'])
30
sage: len(D['3'])
16

AUTHOR:

Timo Jolivet (2012-06-16)

rauzy_fractal_projection(eig=None, prec=53)

Returns a dictionary giving the projection of the canonical basis.

See the method rauzy_fractal_plot() for more details about the projection.

INPUT:

  • eig - a real element of QQbar of degree >= 2 (default: None). The eigenvalue used for the projection. It must be an eigenvalue of self.incidence_matrix(). The one used by default is the maximal eigenvalue of self.incidence_matrix() (usually a Pisot number), but for substitutions with more than 3 letters other interesting choices are sometimes possible.

  • prec - integer (default: 53). The number of bits used in the floating point representations of the coordinates.

OUTPUT:

dictionary, letter -> vector, giving the projection

EXAMPLES:

The projection for the Rauzy fractal of the Tribonacci substitution is:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: s.rauzy_fractal_projection()
{'1': (1.00000000000000, 0.000000000000000),
 '2': (-1.41964337760708, -0.606290729207199),
 '3': (-0.771844506346038, 1.11514250803994)}

AUTHOR:

Timo Jolivet (2012-06-16)

restrict_domain(alphabet)

Returns a restriction of self to the given alphabet.

INPUT:

  • alphabet - an iterable

OUTPUT:

WordMorphism

EXAMPLES:

sage: m = WordMorphism('a->b,b->a')
sage: m.restrict_domain('a')
WordMorphism: a->b
sage: m.restrict_domain('')
WordMorphism:
sage: m.restrict_domain('A')
WordMorphism:
sage: m.restrict_domain('Aa')
WordMorphism: a->b

The input alphabet must be iterable:

sage: m.restrict_domain(66)
Traceback (most recent call last):
...
TypeError: 'sage.rings.integer.Integer' object is not iterable
reversal()

Returns the reversal of self.

EXAMPLES:

sage: WordMorphism('6->ab,y->5,0->asd').reversal()
WordMorphism: 0->dsa, 6->ba, y->5
sage: WordMorphism('a->ab,b->a').reversal()
WordMorphism: a->ba, b->a
sage.combinat.words.morphism.get_cycles(f, domain=None)

Return the cycle of the function f on the finite set domain. It is assumed that f is an endomorphism.

INPUT:

  • f - function.

  • domain - set (default: None) - the domain of f. If none, then tries to use f.domain().

EXAMPLES:

sage: from sage.combinat.words.morphism import get_cycles
sage: get_cycles(lambda i: (i+1)%3, domain=[0,1,2])
[(0, 1, 2)]
sage: get_cycles(lambda i: [0,0,0][i], domain=[0,1,2])
[(0,)]
sage: get_cycles(lambda i: [1,1,1][i], domain=[0,1,2])
[(1,)]