Recursively enumerated set

Recursively enumerated set

A set \(S\) is called recursively enumerable if there is an algorithm that enumerates the members of \(S\). We consider here the recursively enumerated sets that are described by some seeds and a successor function successors. The successor function may have some structure (symmetric, graded, forest) or not. The elements of a set having a symmetric, graded or forest structure can be enumerated uniquely without keeping all of them in memory. Many kinds of iterators are provided in this module: depth first search, breadth first search or elements of given depth.

See Wikipedia article Recursively_enumerable_set.

See documentation of RecursivelyEnumeratedSet() below for the description of the inputs.

AUTHORS:

  • Sebastien Labbe, April 2014, at Sage Days 57, Cernay-la-ville

EXAMPLES:

Forest structure

The set of words over the alphabet \(\{a,b\}\) can be generated from the empty word by appending letter \(a\) or \(b\) as a successor function. This set has a forest structure:

sage: seeds = ['']
sage: succ = lambda w: [w+'a', w+'b']
sage: C = RecursivelyEnumeratedSet(seeds, succ, structure='forest')
sage: C
An enumerated set with a forest structure

Depth first search iterator:

sage: it = C.depth_first_search_iterator()
sage: [next(it) for _ in range(6)]
['', 'a', 'aa', 'aaa', 'aaaa', 'aaaaa']

Breadth first search iterator:

sage: it = C.breadth_first_search_iterator()
sage: [next(it) for _ in range(6)]
['', 'a', 'b', 'aa', 'ab', 'ba']

Symmetric structure

The origin (0, 0) as seed and the upper, lower, left and right lattice point as successor function. This function is symmetric:

sage: succ = lambda a: [(a[0]-1,a[1]), (a[0],a[1]-1), (a[0]+1,a[1]), (a[0],a[1]+1)]
sage: seeds = [(0,0)]
sage: C = RecursivelyEnumeratedSet(seeds, succ, structure='symmetric', enumeration='depth')
sage: C
A recursively enumerated set with a symmetric structure (depth first search)

In this case, depth first search is the default enumeration for iteration:

sage: it_depth = iter(C)
sage: [next(it_depth) for _ in range(10)]
[(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9)]

Breadth first search:

sage: it_breadth = C.breadth_first_search_iterator()
sage: [next(it_breadth) for _ in range(10)]
[(0, 0), (0, 1), (0, -1), (1, 0), (-1, 0), (-1, 1), (-2, 0), (0, 2), (2, 0), (-1, -1)]

Levels (elements of given depth):

sage: sorted(C.graded_component(0))
[(0, 0)]
sage: sorted(C.graded_component(1))
[(-1, 0), (0, -1), (0, 1), (1, 0)]
sage: sorted(C.graded_component(2))
[(-2, 0), (-1, -1), (-1, 1), (0, -2), (0, 2), (1, -1), (1, 1), (2, 0)]

Graded structure

Identity permutation as seed and permutohedron_succ as successor function:

sage: succ = attrcall("permutohedron_succ")
sage: seed = [Permutation([1..5])]
sage: R = RecursivelyEnumeratedSet(seed, succ, structure='graded')
sage: R
A recursively enumerated set with a graded structure (breadth first search)

Depth first search iterator:

sage: it_depth = R.depth_first_search_iterator()
sage: [next(it_depth) for _ in range(5)]
[[1, 2, 3, 4, 5],
 [1, 2, 3, 5, 4],
 [1, 2, 5, 3, 4],
 [1, 2, 5, 4, 3],
 [1, 5, 2, 4, 3]]

Breadth first search iterator:

sage: it_breadth = R.breadth_first_search_iterator()
sage: [next(it_breadth) for _ in range(5)]
[[1, 2, 3, 4, 5],
 [1, 3, 2, 4, 5],
 [1, 2, 4, 3, 5],
 [2, 1, 3, 4, 5],
 [1, 2, 3, 5, 4]]

Elements of given depth iterator:

sage: list(R.elements_of_depth_iterator(9))
[[5, 4, 2, 3, 1], [4, 5, 3, 2, 1], [5, 3, 4, 2, 1], [5, 4, 3, 1, 2]]
sage: list(R.elements_of_depth_iterator(10))
[[5, 4, 3, 2, 1]]

Graded components (set of elements of the same depth):

sage: sorted(R.graded_component(0))
[[1, 2, 3, 4, 5]]
sage: sorted(R.graded_component(1))
[[1, 2, 3, 5, 4], [1, 2, 4, 3, 5], [1, 3, 2, 4, 5], [2, 1, 3, 4, 5]]
sage: sorted(R.graded_component(9))
[[4, 5, 3, 2, 1], [5, 3, 4, 2, 1], [5, 4, 2, 3, 1], [5, 4, 3, 1, 2]]
sage: sorted(R.graded_component(10))
[[5, 4, 3, 2, 1]]

No hypothesis on the structure

By “no hypothesis” is meant neither a forest, neither symmetric neither graded, it may have other structure like not containing oriented cycle but this does not help for enumeration.

In this example, the seed is 0 and the successor function is either +2 or +3. This is the set of non negative linear combinations of 2 and 3:

sage: succ = lambda a:[a+2,a+3]
sage: C = RecursivelyEnumeratedSet([0], succ)
sage: C
A recursively enumerated set (breadth first search)

Breadth first search:

sage: it = C.breadth_first_search_iterator()
sage: [next(it) for _ in range(10)]
[0, 2, 3, 4, 5, 6, 8, 9, 7, 10]

Depth first search:

sage: it = C.depth_first_search_iterator()
sage: [next(it) for _ in range(10)]
[0, 3, 6, 9, 12, 15, 18, 21, 24, 27]
sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet(seeds, successors, structure=None, enumeration=None, max_depth=None, post_process=None, facade=None, category=None)

Return a recursively enumerated set.

A set \(S\) is called recursively enumerable if there is an algorithm that enumerates the members of \(S\). We consider here the recursively enumerated set that are described by some seeds and a successor function successors.

Let \(U\) be a set and successors \(:U \to 2^U\) be a successor function associating to each element of \(U\) a subset of \(U\). Let seeds be a subset of \(U\). Let \(S\subseteq U\) be the set of elements of \(U\) that can be reached from a seed by applying recursively the successors function. This class provides different kinds of iterators (breadth first, depth first, elements of given depth, etc.) for the elements of \(S\).

See Wikipedia article Recursively_enumerable_set.

INPUT:

  • seeds – list (or iterable) of hashable objects
  • successors – function (or callable) returning a list (or iterable) of hashable objects
  • structure – string (optional, default: None), structure of the set, possible values are:
    • None – nothing is known about the structure of the set.
    • 'forest' – if the successors function generates a forest, that is, each element can be reached uniquely from a seed.
    • 'graded' – if the successors function is graded, that is, all paths from a seed to a given element have equal length.
    • 'symmetric' – if the relation is symmetric, that is, y in successors(x) if and only if x in successors(y)
  • enumeration'depth', 'breadth', 'naive' or None (optional, default: None). The default enumeration for the __iter__ function.
  • max_depth – integer (optional, default: float("inf")), limit the search to a certain depth, currently works only for breadth first search
  • post_process – (optional, default: None), for forest only
  • facade – (optional, default: None)
  • category – (optional, default: None)

EXAMPLES:

A recursive set with no other information:

sage: f = lambda a: [a+3, a+5]
sage: C = RecursivelyEnumeratedSet([0], f)
sage: C
A recursively enumerated set (breadth first search)
sage: it = iter(C)
sage: [next(it) for _ in range(10)]
[0, 3, 5, 8, 10, 6, 9, 11, 13, 15]

A recursive set with a forest structure:

sage: f = lambda a: [2*a,2*a+1]
sage: C = RecursivelyEnumeratedSet([1], f, structure='forest')
sage: C
An enumerated set with a forest structure
sage: it = C.depth_first_search_iterator()
sage: [next(it) for _ in range(7)]
[1, 2, 4, 8, 16, 32, 64]
sage: it = C.breadth_first_search_iterator()
sage: [next(it) for _ in range(7)]
[1, 2, 3, 4, 5, 6, 7]

A recursive set given by a symmetric relation:

sage: f = lambda a: [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([10, 15], f, structure='symmetric')
sage: C
A recursively enumerated set with a symmetric structure (breadth first search)
sage: it = iter(C)
sage: [next(it) for _ in range(7)]
[10, 15, 16, 9, 11, 14, 8]

A recursive set given by a graded relation:

sage: f = lambda a: [a+1, a+I]
sage: C = RecursivelyEnumeratedSet([0], f, structure='graded')
sage: C
A recursively enumerated set with a graded structure (breadth first search)
sage: it = iter(C)
sage: [next(it) for _ in range(7)]
[0, 1, I, I + 1, 2, 2*I, I + 2]

Warning

If you do not set the good structure, you might obtain bad results, like elements generated twice:

sage: f = lambda a: [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([0], f, structure='graded')
sage: it = iter(C)
sage: [next(it) for _ in range(7)]
[0, 1, -1, 0, 2, -2, 1]

TESTS:

The succesors method is an attribute:

sage: R = RecursivelyEnumeratedSet([1], lambda x: [x+1, x-1])
sage: R.successors(4)
[5, 3]
sage: C = RecursivelyEnumeratedSet((1, 2, 3), factor)
sage: C.successors
<function factor at ...>
sage: C._seeds
(1, 2, 3)
class sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_generic

Bases: sage.structure.parent.Parent

A generic recursively enumerated set.

For more information, see RecursivelyEnumeratedSet().

EXAMPLES:

sage: f = lambda a:[a+1]

Different structure for the sets:

sage: RecursivelyEnumeratedSet([0], f, structure=None)
A recursively enumerated set (breadth first search)
sage: RecursivelyEnumeratedSet([0], f, structure='graded')
A recursively enumerated set with a graded structure (breadth first search)
sage: RecursivelyEnumeratedSet([0], f, structure='symmetric')
A recursively enumerated set with a symmetric structure (breadth first search)
sage: RecursivelyEnumeratedSet([0], f, structure='forest')
An enumerated set with a forest structure

Different default enumeration algorithms:

sage: RecursivelyEnumeratedSet([0], f, enumeration='breadth')
A recursively enumerated set (breadth first search)
sage: RecursivelyEnumeratedSet([0], f, enumeration='naive')
A recursively enumerated set (naive search)
sage: RecursivelyEnumeratedSet([0], f, enumeration='depth')
A recursively enumerated set (depth first search)
breadth_first_search_iterator(max_depth=None)

Iterate on the elements of self (breadth first).

This code remembers every elements generated.

INPUT:

  • max_depth – (Default: None) specifies the maximal depth to which elements are computed; if None, the value of self._max_depth is used

EXAMPLES:

sage: f = lambda a: [a+3, a+5]
sage: C = RecursivelyEnumeratedSet([0], f)
sage: it = C.breadth_first_search_iterator()
sage: [next(it) for _ in range(10)]
[0, 3, 5, 8, 10, 6, 9, 11, 13, 15]
depth_first_search_iterator()

Iterate on the elements of self (depth first).

This code remembers every elements generated.

See Wikipedia article Depth-first_search.

EXAMPLES:

sage: f = lambda a: [a+3, a+5]
sage: C = RecursivelyEnumeratedSet([0], f)
sage: it = C.depth_first_search_iterator()
sage: [next(it) for _ in range(10)]
[0, 5, 10, 15, 20, 25, 30, 35, 40, 45]
elements_of_depth_iterator(depth)

Iterate over the elements of self of given depth.

An element of depth \(n\) can be obtained applying \(n\) times the successor function to a seed.

INPUT:

  • depth – integer

OUTPUT:

An iterator.

EXAMPLES:

sage: f = lambda a: [a-1, a+1]
sage: S = RecursivelyEnumeratedSet([5, 10], f, structure='symmetric')
sage: it = S.elements_of_depth_iterator(2)
sage: sorted(it)
[3, 7, 8, 12]
graded_component(depth)

Return the graded component of given depth.

This method caches each lower graded component.

A graded component is a set of elements of the same depth where the depth of an element is its minimal distance to a root.

INPUT:

  • depth – integer

OUTPUT:

A set.

EXAMPLES:

sage: f = lambda a: [a+3, a+5]
sage: C = RecursivelyEnumeratedSet([0], f)
sage: C.graded_component(0)
Traceback (most recent call last):
...
NotImplementedError: graded_component_iterator method currently implemented only for graded or symmetric structure

When the structure is symmetric:

sage: f = lambda a: [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([10, 15], f, structure='symmetric')
sage: for i in range(5): sorted(C.graded_component(i))
[10, 15]
[9, 11, 14, 16]
[8, 12, 13, 17]
[7, 18]
[6, 19]

When the structure is graded:

sage: f = lambda a: [a+1, a+I]
sage: C = RecursivelyEnumeratedSet([0], f, structure='graded')
sage: for i in range(5): sorted(C.graded_component(i))
[0]
[I, 1]
[2*I, I + 1, 2]
[3*I, 2*I + 1, I + 2, 3]
[4*I, 3*I + 1, 2*I + 2, I + 3, 4]
graded_component_iterator()

Iterate over the graded components of self.

A graded component is a set of elements of the same depth.

It is currently implemented only for herited classes.

OUTPUT:

An iterator of sets.

EXAMPLES:

sage: f = lambda a: [a+3, a+5]
sage: C = RecursivelyEnumeratedSet([0], f)
sage: it = C.graded_component_iterator()    # todo: not implemented
naive_search_iterator()

Iterate on the elements of self (in no particular order).

This code remembers every elements generated.

TESTS:

We compute all the permutations of 3:

sage: seeds = [Permutation([1,2,3])]
sage: succ = attrcall("permutohedron_succ")
sage: R = RecursivelyEnumeratedSet(seeds, succ)
sage: list(R.naive_search_iterator())
[[1, 2, 3], [2, 1, 3], [1, 3, 2], [2, 3, 1], [3, 2, 1], [3, 1, 2]]
seeds()

Return an iterable over the seeds of self.

EXAMPLES:

sage: R = RecursivelyEnumeratedSet([1], lambda x: [x+1, x-1])
sage: R.seeds()
[1]
successors
class sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_graded

Bases: sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_generic

Generic tool for constructing ideals of a graded relation.

INPUT:

  • seeds – list (or iterable) of hashable objects
  • successors – function (or callable) returning a list (or iterable)
  • enumeration'depth', 'breadth' or None (default: None)
  • max_depth – integer (default: float("inf"))

EXAMPLES:

sage: f = lambda a: [(a[0]+1,a[1]), (a[0],a[1]+1)]
sage: C = RecursivelyEnumeratedSet([(0,0)], f, structure='graded', max_depth=3)
sage: C
A recursively enumerated set with a graded structure (breadth first search)
sage: sorted(C)
[(0, 0), (0, 1), (0, 2), (0, 3), (1, 0),
 (1, 1), (1, 2), (2, 0), (2, 1), (3, 0)]
breadth_first_search_iterator(max_depth=None)

Iterate on the elements of self (breadth first).

This iterator make use of the graded structure by remembering only the elements of the current depth.

INPUT:

  • max_depth – (Default: None) Specifies the maximal depth to which elements are computed. If None, the value of self._max_depth is used.

EXAMPLES:

sage: f = lambda a: [(a[0]+1,a[1]), (a[0],a[1]+1)]
sage: C = RecursivelyEnumeratedSet([(0,0)], f, structure='graded')
sage: it = C.breadth_first_search_iterator(max_depth=3)
sage: list(it)
[(0, 0), (0, 1), (1, 0), (2, 0), (1, 1),
 (0, 2), (3, 0), (1, 2), (0, 3), (2, 1)]
graded_component_iterator()

Iterate over the graded components of self.

A graded component is a set of elements of the same depth.

The algorithm remembers only the current graded component generated since the structure is graded.

OUTPUT:

An iterator of sets.

EXAMPLES:

sage: f = lambda a: [(a[0]+1,a[1]), (a[0],a[1]+1)]
sage: C = RecursivelyEnumeratedSet([(0,0)], f, structure='graded', max_depth=3)
sage: it = C.graded_component_iterator()
sage: for _ in range(4): sorted(next(it))
[(0, 0)]
[(0, 1), (1, 0)]
[(0, 2), (1, 1), (2, 0)]
[(0, 3), (1, 2), (2, 1), (3, 0)]
class sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_symmetric

Bases: sage.sets.recursively_enumerated_set.RecursivelyEnumeratedSet_generic

Generic tool for constructing ideals of a symmetric relation.

INPUT:

  • seeds – list (or iterable) of hashable objects
  • successors – function (or callable) returning a list (or iterable)
  • enumeration'depth', 'breadth' or None (default: None)
  • max_depth – integer (default: float("inf"))

EXAMPLES:

sage: f = lambda a: [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([0], f, structure='symmetric')
sage: C
A recursively enumerated set with a symmetric structure (breadth first search)
sage: it = iter(C)
sage: [next(it) for _ in range(7)]
[0, 1, -1, 2, -2, 3, -3]

TESTS:

Do not use lambda functions for saving purposes:

sage: f = lambda a: [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([0], f, structure='symmetric')
sage: loads(dumps(C))
Traceback (most recent call last):
...
PicklingError: Can't pickle <type 'function'>: attribute lookup __builtin__.function failed

This works in the command line but apparently not as a doctest:

sage: def f(a): return [a-1,a+1]
sage: C = RecursivelyEnumeratedSet([0], f, structure='symmetric')
sage: loads(dumps(C))
Traceback (most recent call last):
...
PicklingError: Can't pickle <type 'function'>: attribute lookup __builtin__.function failed
breadth_first_search_iterator(max_depth=None)

Iterate on the elements of self (breadth first).

INPUT:

  • max_depth – (Default: None) specifies the maximal depth to which elements are computed; if None, the value of self._max_depth is used

Note

It should be slower than the other one since it must generates the whole graded component before yielding the first element of each graded component. It is used for test only.

EXAMPLES:

sage: f = lambda a: [(a[0]+1,a[1]), (a[0],a[1]+1)]
sage: C = RecursivelyEnumeratedSet([(0,0)], f, structure='graded')
sage: it = C._breadth_first_search_iterator_from_graded_component_iterator(max_depth=3)
sage: list(it)
[(0, 0), (0, 1), (1, 0), (2, 0), (1, 1), (0, 2)]

This iterator is used by default for symmetric structure:

sage: f = lambda a: [a-1,a+1]
sage: S = RecursivelyEnumeratedSet([10], f, structure='symmetric')
sage: it = iter(S)
sage: [next(it) for _ in range(7)]
[10, 9, 11, 8, 12, 13, 7]
graded_component_iterator()

Iterate over the graded components of self.

A graded component is a set of elements of the same depth.

The enumeration remembers only the last two graded components generated since the structure is symmetric.

OUTPUT:

An iterator of sets.

EXAMPLES:

sage: f = lambda a: [a-1, a+1]
sage: S = RecursivelyEnumeratedSet([10], f, structure='symmetric')
sage: it = S.graded_component_iterator()
sage: [sorted(next(it)) for _ in range(5)]
[[10], [9, 11], [8, 12], [7, 13], [6, 14]]

Starting with two generators:

sage: f = lambda a: [a-1, a+1]
sage: S = RecursivelyEnumeratedSet([5, 10], f, structure='symmetric')
sage: it = S.graded_component_iterator()
sage: [sorted(next(it)) for _ in range(5)]
[[5, 10], [4, 6, 9, 11], [3, 7, 8, 12], [2, 13], [1, 14]]

Gaussian integers:

sage: f = lambda a: [a+1, a+I]
sage: S = RecursivelyEnumeratedSet([0], f, structure='symmetric')
sage: it = S.graded_component_iterator()
sage: [sorted(next(it)) for _ in range(7)]
[[0],
 [I, 1],
 [2*I, I + 1, 2],
 [3*I, 2*I + 1, I + 2, 3],
 [4*I, 3*I + 1, 2*I + 2, I + 3, 4],
 [5*I, 4*I + 1, 3*I + 2, 2*I + 3, I + 4, 5],
 [6*I, 5*I + 1, 4*I + 2, 3*I + 3, 2*I + 4, I + 5, 6]]

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