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# -*- coding: utf-8 -*-

r"""

Word classes

AUTHORS:

- Arnaud Bergeron

- Amy Glen

- Sébastien Labbé

- Franco Saliola

"""

from __future__ import absolute_import

#*****************************************************************************

# Amy Glen <amy.glen@gmail.com>,

# Sébastien Labbé <slabqc@gmail.com>,

# Franco Saliola <saliola@gmail.com>

# This program is free software: you can redistribute it and/or modify

# it under the terms of the GNU General Public License as published by

# the Free Software Foundation, either version 2 of the License, or

# (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from sage.combinat.words.word_char import WordDatatype_char

from sage.combinat.words.abstract_word import Word_class

from sage.combinat.words.finite_word import FiniteWord_class

from sage.combinat.words.infinite_word import InfiniteWord_class

from .word_datatypes import (WordDatatype_str,

WordDatatype_list,

WordDatatype_tuple)

from .word_infinite_datatypes import (

WordDatatype_iter_with_caching,

WordDatatype_iter,

WordDatatype_callable_with_caching,

WordDatatype_callable)

from sage.monoids.free_monoid_element import FreeMonoidElement

# TODO. Word needs to be replaced by Word. Consider renameing

# Word_class to Word and imbedding Word as its __call__ method.

def Word(data=None, alphabet=None, length=None, datatype=None, caching=True, RSK_data=None):

r"""

Construct a word.

INPUT:

- ``data`` -- (default: ``None``) list, string, tuple, iterator, free

monoid element, ``None`` (shorthand for ``[]``), or a callable defined

on ``[0,1,...,length]``.

- ``alphabet`` -- any argument accepted by Words

- ``length`` -- (default: ``None``) This is dependent on the type of data.

It is ignored for words defined by lists, strings, tuples,

etc., because they have a naturally defined length.

For callables, this defines the domain of definition,

which is assumed to be ``[0, 1, 2, ..., length-1]``.

For iterators: Infinity if you know the iterator will not

terminate (default); ``"unknown"`` if you do not know whether the

iterator terminates; ``"finite"`` if you know that the iterator

terminates, but do not know the length.

- ``datatype`` -- (default: ``None``) ``None``, ``"list"``, ``"str"``,

``"tuple"``, ``"iter"``, ``"callable"``. If ``None``, then the function

tries to guess this from the data.

- ``caching`` -- (default: ``True``) ``True`` or ``False``. Whether to

keep a cache of the letters computed by an iterator or callable.

- ``RSK_data`` -- (Optional. Default: ``None``) A semistandard and a

standard Young tableau to run the inverse RSK bijection on.

.. NOTE::

Be careful when defining words using callables and iterators. It

appears that islice does not pickle correctly causing various errors

when reloading. Also, most iterators do not support copying and

should not support pickling by extension.

EXAMPLES:

Empty word::

sage: Word()

word:

Word with string::

sage: Word("abbabaab")

word: abbabaab

Word with string constructed from other types::

sage: Word([0,1,1,0,1,0,0,1], datatype="str")

word: 01101001

sage: Word((0,1,1,0,1,0,0,1), datatype="str")

word: 01101001

Word with list::

sage: Word([0,1,1,0,1,0,0,1])

word: 01101001

Word with list constructed from other types::

sage: Word("01101001", datatype="list")

word: 01101001

sage: Word((0,1,1,0,1,0,0,1), datatype="list")

word: 01101001

Word with tuple::

sage: Word((0,1,1,0,1,0,0,1))

word: 01101001

Word with tuple constructed from other types::

sage: Word([0,1,1,0,1,0,0,1], datatype="tuple")

word: 01101001

sage: Word("01101001", datatype="str")

word: 01101001

Word with iterator::

sage: from itertools import count

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,...

sage: Word(iter("abbabaab")) # iterators default to infinite words

word: abbabaab

sage: Word(iter("abbabaab"), length="unknown")

word: abbabaab

sage: Word(iter("abbabaab"), length="finite")

word: abbabaab

Word with function (a 'callable')::

sage: f = lambda n : add(Integer(n).digits(2)) % 2

sage: Word(f)

word: 0110100110010110100101100110100110010110...

sage: Word(f, length=8)

word: 01101001

Word over a string with a parent::

sage: w = Word("abbabaab", alphabet="abc"); w

word: abbabaab

sage: w.parent()

Finite words over {'a', 'b', 'c'}

Word from a free monoid element::

sage: M.<x,y,z> = FreeMonoid(3)

sage: Word(x^3*y*x*z^2*x)

word: xxxyxzzx

The default parent is the combinatorial class of all words::

sage: w = Word("abbabaab"); w

word: abbabaab

sage: w.parent()

Finite words over Set of Python objects of class 'object'

We can also input a semistandard tableau and a standard tableau to

obtain a word from the inverse RSK algorithm using the

``RSK_data`` option::

sage: p = Tableau([[1,2,2],[3]]); q = Tableau([[1,2,4],[3]])

sage: Word(RSK_data=[p, q])

word: 1322

TESTS::

sage: Word(5)

Traceback (most recent call last):

...

ValueError: Cannot guess a datatype from data (=5); please specify one

sage: W = Words()

sage: w = W('abc')

sage: w is W(w)

True

sage: w is Word(w, alphabet='abc')

False

"""

if isinstance(data, FreeMonoidElement):

return data.to_word(alphabet)

if RSK_data is not None:

#if a list of a semistandard and a standard tableau or a pair of lists

from sage.combinat.tableau import Tableau

if isinstance(RSK_data, (tuple, list)) and len(RSK_data) == 2 and \

all((isinstance(x, Tableau) for x in RSK_data)):

from sage.combinat.rsk import RSK_inverse

return RSK_inverse(*RSK_data, output='word')

elif isinstance(RSK_data, (tuple, list)) and len(RSK_data) == 2 and \

all((isinstance(x, (list, tuple)) for x in RSK_data)):

from sage.combinat.rsk import RSK_inverse

P,Q = map(Tableau, RSK_data)

return RSK_inverse(P, Q, 'word')

raise ValueError("Invalid input. Must be a pair of tableaux")

# Create the parent object

from .words import Words

parent = Words(alphabet)

return parent(data=data, length=length, datatype=datatype, caching=caching)

#######################################################################

# #

# Concrete word classes #

# #

#######################################################################

##### Finite Words #####

class FiniteWord_char(WordDatatype_char, FiniteWord_class):

r"""

Finite word represented by an array ``unsigned char *`` (i.e. integers

between 0 and 255).

For any word ``w``, type ``w.<TAB>`` to see the functions that can be applied

to ``w``.

EXAMPLES::

sage: W = Words(range(20))

sage: w = W(list(range(1, 10)) * 2)

sage: type(w)

sage: w

word: 123456789123456789

sage: w.is_palindrome()

False

sage: (w*w[::-1]).is_palindrome()

True

sage: (w[:-1:]*w[::-1]).is_palindrome()

True

sage: w.is_lyndon()

False

sage: W(list(range(10)) + [10, 10]).is_lyndon()

True

sage: w.is_square_free()

False

sage: w[:-1].is_square_free()

True

sage: u = W([randint(0,10) for i in range(10)])

sage: (u*u).is_square()

True

sage: (u*u*u).is_cube()

True

sage: len(w.factor_set())

127

sage: w.rauzy_graph(5)

Looped digraph on 9 vertices

sage: u = W([1,2,3])

sage: u.first_pos_in(w)

sage: u.first_pos_in(w[1:])

TESTS::

sage: W = Words([0,1,2])

sage: w = W([0,1,1,0])

sage: w == loads(dumps(w))

True

"""

pass

class FiniteWord_list(WordDatatype_list, FiniteWord_class):

r"""

Finite word represented by a Python list.

For any word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: w = Word(range(10))

sage: w.iterated_right_palindromic_closure()

word: 0102010301020104010201030102010501020103...

TESTS::

sage: w = Word([0,1,1,0])

sage: w == loads(dumps(w))

True

"""

pass

class FiniteWord_str(WordDatatype_str, FiniteWord_class):

r"""

Finite word represented by a Python str.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: w = Word('abcdef')

sage: w.is_square()

False

TESTS::

sage: w = Word('abba')

sage: w == loads(dumps(w))

True

"""

pass

class FiniteWord_tuple(WordDatatype_tuple, FiniteWord_class):

r"""

Finite word represented by a Python tuple.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: w = Word(())

sage: w.is_empty()

True

TESTS::

sage: w = Word((0,1,1,0))

sage: w == loads(dumps(w))

True

"""

pass

class FiniteWord_iter_with_caching(WordDatatype_iter_with_caching, FiniteWord_class):

r"""

Finite word represented by an iterator (with caching).

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: w = Word(iter('abcdef'))

sage: w.conjugate(2)

word: cdefab

TESTS::

sage: w = Word(iter(range(10)))

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

"""

pass

class FiniteWord_iter(WordDatatype_iter, FiniteWord_class):

r"""

Finite word represented by an iterator.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: w = Word(iter(range(10)), caching=False)

sage: w

word: 0123456789

sage: w.finite_differences()

word: 111111111

TESTS::

sage: w = Word(iter(range(10)), caching=False)

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

"""

pass

class FiniteWord_callable_with_caching(WordDatatype_callable_with_caching, FiniteWord_class):

r"""

Finite word represented by a callable (with caching).

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: f = lambda n : n % 3

sage: w = Word(f, length=32)

sage: w

word: 01201201201201201201201201201201

sage: w.border()

word: 01201201201201201201201201201

TESTS::

sage: w = Word(lambda n:n, length=10)

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

Pickle also works for concatenation of words::

sage: w = Word(range(10)) * Word('abcdef')

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

Pickle also works for power of words::

sage: w = Word(range(10)) ^ 2

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

"""

pass

class FiniteWord_callable(WordDatatype_callable, FiniteWord_class):

r"""

Finite word represented by a callable.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

EXAMPLES::

sage: f = lambda n : 3 if n > 8 else 6

sage: w = Word(f, length=30, caching=False)

sage: w

word: 666666666333333333333333333333

sage: w.is_symmetric()

True

TESTS::

sage: w = Word(lambda n:n, length=10, caching=False)

sage: type(w)

sage: z = loads(dumps(w))

sage: w == z

True

sage: type(z)

"""

pass

##### Infinite Words #####

class InfiniteWord_iter_with_caching(WordDatatype_iter_with_caching, InfiniteWord_class):

r"""

Infinite word represented by an iterable (with caching).

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Infinite words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: from itertools import cycle

sage: w = Word(cycle([9,8,4]))

sage: w

word: 9849849849849849849849849849849849849849...

sage: prefix = w[:23]

sage: prefix

word: 98498498498498498498498

sage: prefix.minimal_period()

TESTS::

sage: from itertools import count

sage: w = Word(count())

sage: type(w)

Pickle is not supported for infinite word defined by an iterator::

sage: dumps(w)

Traceback (most recent call last):

...

TypeError: can't pickle generator objects

"""

pass

class InfiniteWord_iter(WordDatatype_iter, InfiniteWord_class):

r"""

Infinite word represented by an iterable.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Infinite words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: from itertools import chain, cycle

sage: w = Word(chain('letsgo', cycle('forever')), caching=False)

sage: w

word: letsgoforeverforeverforeverforeverforeve...

sage: prefix = w[:100]

sage: prefix

word: letsgoforeverforeverforeverforeverforeve...

sage: prefix.is_lyndon()

False

TESTS::

sage: from itertools import count

sage: w = Word(count(), caching=False)

sage: type(w)

Pickle is not supported for infinite word defined by an iterator::

sage: dumps(w)

Traceback (most recent call last):

...

TypeError: can't pickle generator objects

"""

pass

class InfiniteWord_callable_with_caching(WordDatatype_callable_with_caching, InfiniteWord_class):

r"""

Infinite word represented by a callable (with caching).

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Infinite words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: w = Word(lambda n:n)

sage: factor = w[4:13]

sage: factor

word: 4,5,6,7,8,9,10,11,12

TESTS::

sage: w = Word(lambda n:n)

sage: type(w)

sage: z = loads(dumps(w))

sage: z

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: type(z)

"""

pass

class InfiniteWord_callable(WordDatatype_callable, InfiniteWord_class):

r"""

Infinite word represented by a callable.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Infinite words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: w = Word(lambda n:n, caching=False)

sage: 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.iterated_right_palindromic_closure()

word: 0102010301020104010201030102010501020103...

TESTS::

sage: w = Word(lambda n:n, caching=False)

sage: type(w)

sage: z = loads(dumps(w))

sage: z

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: type(z)

"""

pass

##### Words of unknown length #####

class Word_iter_with_caching(WordDatatype_iter_with_caching, Word_class):

r"""

Word of unknown length (finite or infinite) represented by an

iterable (with caching).

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: w = Word(iter([1,2,3]*1000), length='unknown')

sage: w

word: 1231231231231231231231231231231231231231...

sage: w.finite_differences(mod=2)

word: 1101101101101101101101101101101101101101...

TESTS::

sage: w = Word(iter('abcd'*100), length='unknown')

sage: type(w)

sage: w

word: abcdabcdabcdabcdabcdabcdabcdabcdabcdabcd...

Pickle is not supported for word of unknown length defined by an iterator::

sage: dumps(w)

Traceback (most recent call last):

...

TypeError: can't pickle generator objects

"""

pass

class Word_iter(WordDatatype_iter, Word_class):

r"""

Word of unknown length (finite or infinite) represented by an

iterable.

For such word `w`, type ``w.`` and hit TAB key to see the list of

functions defined on `w`.

Words behave like a Python list : they can be sliced using

square braquets to define for example a prefix or a factor.

EXAMPLES::

sage: w = Word(iter([1,1,4,9]*1000), length='unknown', caching=False)

sage: w

word: 1149114911491149114911491149114911491149...

sage: w.delta()

word: 2112112112112112112112112112112112112112...

TESTS::

sage: w = Word(iter('abcd'*100), length='unknown', caching=False)

sage: type(w)

sage: w

word: abcdabcdabcdabcdabcdabcdabcdabcdabcdabcd...

Pickle is not supported for word of unknown length defined by an iterator::

sage: dumps(w)

Traceback (most recent call last):

...

TypeError: can't pickle generator objects

"""

pass

Coverage for local/lib/python2.7/site-packages/sage/combinat/words/word.py : 19%

53 statements 10 run 43 missing 0 excluded