def PebblingFormula(digraph):
"""Pebbling formula
Build a pebbling formula from the directed graph. If the graph has
an `ordered_vertices` attribute, then it is used to enumerate the
vertices (and the corresponding variables).
Arguments:
- `digraph`: directed acyclic graph.
"""
if not is_dag(digraph):
raise ValueError(
"Pebbling formula is defined only for directed acyclic graphs")
peb = CNF()
peb.mode_unchecked()
if hasattr(digraph, 'name'):
peb.header = "Pebbling formula of: "+digraph.name+"\n\n"+peb.header
else:
peb.header = "Pebbling formula\n\n"+peb.header
# add variables in the appropriate order
vertices = enumerate_vertices(digraph)
position = dict((v, i) for (i, v) in enumerate(vertices))
for v in vertices:
peb.add_variable(
v, description="There is a pebble on vertex ${}$".format(v))
# add the clauses
for v in vertices:
# If predecessors are pebbled the vertex must be pebbled
pred = sorted(digraph.predecessors(v), key=lambda x: position[x])
peb.add_clause([(False, p) for p in pred]+[(True, v)])
if digraph.out_degree(v) == 0: # the sink
peb.add_clause([(False, v)])
return peb
def PebblingFormula(digraph):
"""Pebbling formula
Build a pebbling formula from the directed graph. If the graph has
an `ordered_vertices` attribute, then it is used to enumerate the
vertices (and the corresponding variables).
Arguments:
- `digraph`: directed acyclic graph.
"""
if not is_dag(digraph):
raise ValueError("Pebbling formula is defined only for directed acyclic graphs")
peb=CNF()
peb.mode_unchecked()
if hasattr(digraph,'name'):
peb.header="Pebbling formula of: "+digraph.name+"\n\n"+peb.header
else:
peb.header="Pebbling formula\n\n"+peb.header
# add variables in the appropriate order
vertices=enumerate_vertices(digraph)
position=dict((v,i) for (i,v) in enumerate(vertices))
for v in vertices:
peb.add_variable(v,description="There is a pebble on vertex ${}$".format(v))
# add the clauses
for v in vertices:
# If predecessors are pebbled the vertex must be pebbled
pred=sorted(digraph.predecessors(v),key=lambda x:position[x])
peb.add_clause([(False,p) for p in pred]+[(True,v)])
if digraph.out_degree(v)==0: #the sink
peb.add_clause([(False,v)])
return peb
def SparseStoneFormula(D, B):
"""Sparse Stone formulas
This is a variant of the :py:func:`StoneFormula`. See that for
a description of the formula. This variant is such that each
vertex has only a small selection of which stone can go to that
vertex. In particular which stones are allowed on each vertex is
specified by a bipartite graph :math:`B` on which the left
vertices represent the vertices of DAG :math:`D` and the right
vertices are the stones.
If a vertex of :math:`D` correspond to the left vertex :math:`v`
in :math:`B`, then its neighbors describe which stones are allowed
for it.
The vertices in :math:`D` do not need to have the same name as the
one on the left side of :math:`B`. It is only important that the
number of vertices in :math:`D` is the same as the vertices in the
left side of :math:`B`.
In that case the element at position :math:`i` in the ordered
sequence ``enumerate_vertices(D)`` corresponds to the element of
rank :math:`i` in the sequence of left side vertices of
:math:`B` according to the output of ``Left, Right =
bipartite_sets(B)``.
Standard :py:func:`StoneFormula` is essentially equivalent to
a sparse stone formula where :math:`B` is the complete graph.
Parameters
----------
D : a directed acyclic graph
it should be a directed acyclic graph.
B : bipartite graph
Raises
------
ValueError
if :math:`D` is not a directed acyclic graph
ValueError
if :math:`B` is not a bipartite graph
ValueError
when size differs between :math:`D` and the left side of
:math:`B`
See Also
--------
StoneFormula
"""
if not is_dag(D):
raise ValueError(
"Stone formulas are defined only for directed acyclic graphs.")
if not has_bipartition(B):
raise ValueError(
"Vertices to stones mapping must be specified with a bipartite graph"
)
Left, Right = bipartite_sets(B)
nstones = len(Right)
if len(Left) != D.order():
raise ValueError(
"Formula requires the bipartite left side to match #vertices of the DAG."
)
cnf = CNF()
if hasattr(D, 'name'):
cnf.header = "Sparse Stone formula of: " + D.name + "\nwith " + str(
nstones) + " stones\n" + cnf.header
else:
cnf.header = "Sparse Stone formula with " + str(
nstones) + " stones\n" + cnf.header
# add variables in the appropriate order
vertices = enumerate_vertices(D)
stones = list(range(1, nstones + 1))
mapping = cnf.unary_mapping(vertices, stones, sparsity_pattern=B)
stone_formula_helper(cnf, D, mapping)
return cnf
def StoneFormula(D, nstones):
"""Stone formulas
The stone formulas have been introduced in [2]_ and generalized in
[1]_. They are one of the classic examples that separate regular
resolutions from general resolution [1]_.
A \"Stones formula\" from a directed acyclic graph :math:`D`
claims that each vertex of the graph is associated with one on
:math:`s` stones (not necessarily in an injective way).
In particular for each vertex :math:`v` in :math:`V(D)` and each
stone :math:`j` we have a variable :math:`P_{v,j}` that claims
that stone :math:`j` is associated to vertex :math:`v`.
Each stone can be either red or blue, and not both.
The propositional variable :math:`R_j` if true when the stone
:math:`j` is red and false otherwise.
The clauses of the formula encode the following constraints.
If a stone is on a source vertex (i.e. a vertex with no incoming
edges), then it must be red. If all stones on the predecessors of
a vertex are red, then the stone of the vertex itself must be red.
The formula furthermore enforces that the stones on the sinks
(i.e. vertices with no outgoing edges) are blue.
.. note:: The exact formula structure depends by the graph and on
its topological order, which is determined by the
``enumerate_vertices(D)``.
Parameters
----------
D : a directed acyclic graph
it should be a directed acyclic graph.
nstones : int
the number of stones.
Raises
------
ValueError
if :math:`D` is not a directed acyclic graph
ValueError
if the number of stones is negative
References
----------
.. [1] M. Alekhnovich, J. Johannsen, T. Pitassi and A. Urquhart
An Exponential Separation between Regular and General Resolution.
Theory of Computing (2007)
.. [2] R. Raz and P. McKenzie
Separation of the monotone NC hierarchy.
Combinatorica (1999)
"""
if not is_dag(D):
raise ValueError(
"Stone formulas are defined only for directed acyclic graphs.")
if nstones < 0:
raise ValueError("There must be at least one stone.")
cnf = CNF()
if hasattr(D, 'name'):
cnf.header = "Stone formula of: " + D.name + "\nwith " + str(
nstones) + " stones\n" + cnf.header
else:
cnf.header = "Stone formula with " + str(
nstones) + " stones\n" + cnf.header
# Add variables in the appropriate order
vertices = enumerate_vertices(D)
position = dict((v, i) for (i, v) in enumerate(vertices))
stones = list(range(1, nstones + 1))
# Caching variable names
color_vn = {}
stone_vn = {}
# Stones->Vertices variables
for v in vertices:
for j in stones:
stone_vn[(v, j)] = "P_{{{0},{1}}}".format(v, j)
cnf.add_variable(stone_vn[(v, j)],
description="Stone ${1}$ on vertex ${0}$".format(
v, j))
# Color variables
for j in stones:
color_vn[j] = "R_{{{0}}}".format(j)
cnf.add_variable(color_vn[j],
description="Stone ${}$ is red".format(j))
# Each vertex has some stone
for v in vertices:
cnf.add_clause_unsafe([(True, stone_vn[(v, j)]) for j in stones])
# If predecessors have red stones, the sink must have a red stone
for v in vertices:
for j in stones:
pred = sorted(D.predecessors(v), key=lambda x: position[x])
for stones_tuple in product([s for s in stones if s != j],
repeat=len(pred)):
cnf.add_clause_unsafe([(False, stone_vn[(p, s)])
for (p, s) in zip(pred, stones_tuple)] +
[(False, stone_vn[(v, j)])] +
[(False, color_vn[s])
for s in _uniqify_list(stones_tuple)] +
[(True, color_vn[j])])
if D.out_degree(v) == 0: #the sink
for j in stones:
cnf.add_clause_unsafe([(False, stone_vn[(v, j)]),
(False, color_vn[j])])
return cnf
def SparseStoneFormula(D,B):
"""Sparse Stone formulas
This is a variant of the :py:func:`StoneFormula`. See that for
a description of the formula. This variant is such that each
vertex has only a small selection of which stone can go to that
vertex. In particular which stones are allowed on each vertex is
specified by a bipartite graph :math:`B` on which the left
vertices represent the vertices of DAG :math:`D` and the right
vertices are the stones.
If a vertex of :math:`D` correspond to the left vertex :math:`v`
in :math:`B`, then its neighbors describe which stones are allowed
for it.
The vertices in :math:`D` do not need to have the same name as the
one on the left side of :math:`B`. It is only important that the
number of vertices in :math:`D` is the same as the vertices in the
left side of :math:`B`.
In that case the element at position :math:`i` in the ordered
sequence ``enumerate_vertices(D)`` corresponds to the element of
rank :math:`i` in the sequence of left side vertices of
:math:`B` according to the output of ``Left, Right =
bipartite_sets(B)``.
Standard :py:func:`StoneFormula` is essentially equivalent to
a sparse stone formula where :math:`B` is the complete graph.
Parameters
----------
D : a directed acyclic graph
it should be a directed acyclic graph.
B : bipartite graph
Raises
------
ValueError
if :math:`D` is not a directed acyclic graph
ValueError
if :math:`B` is not a bipartite graph
ValueError
when size differs between :math:`D` and the left side of
:math:`B`
See Also
--------
StoneFormula
"""
if not is_dag(D):
raise ValueError("Stone formulas are defined only for directed acyclic graphs.")
if not has_bipartition(B):
raise ValueError("Vertices to stones mapping must be specified with a bipartite graph")
Left, Right = bipartite_sets(B)
nstones = len(Right)
if len(Left) != D.order():
raise ValueError("Formula requires the bipartite left side to match #vertices of the DAG.")
cnf = CNF()
if hasattr(D, 'name'):
cnf.header = "Sparse Stone formula of: " + D.name + "\nwith " + str(nstones) + " stones\n" + cnf.header
else:
cnf.header = "Sparse Stone formula with " + str(nstones) + " stones\n" + cnf.header
# add variables in the appropriate order
vertices=enumerate_vertices(D)
stones=range(1,nstones+1)
mapping = cnf.unary_mapping(vertices,stones,sparsity_pattern=B)
stone_formula_helper(cnf,D,mapping)
return cnf
def StoneFormula(D,nstones):
"""Stone formulas
The stone formulas have been introduced in [2]_ and generalized in
[1]_. They are one of the classic examples that separate regular
resolutions from general resolution [1]_.
A \"Stones formula\" from a directed acyclic graph :math:`D`
claims that each vertex of the graph is associated with one on
:math:`s` stones (not necessarily in an injective way).
In particular for each vertex :math:`v` in :math:`V(D)` and each
stone :math:`j` we have a variable :math:`P_{v,j}` that claims
that stone :math:`j` is associated to vertex :math:`v`.
Each stone can be either red or blue, and not both.
The propositional variable :math:`R_j` if true when the stone
:math:`j` is red and false otherwise.
The clauses of the formula encode the following constraints.
If a stone is on a source vertex (i.e. a vertex with no incoming
edges), then it must be red. If all stones on the predecessors of
a vertex are red, then the stone of the vertex itself must be red.
The formula furthermore enforces that the stones on the sinks
(i.e. vertices with no outgoing edges) are blue.
.. note:: The exact formula structure depends by the graph and on
its topological order, which is determined by the
``enumerate_vertices(D)``.
Parameters
----------
D : a directed acyclic graph
it should be a directed acyclic graph.
nstones : int
the number of stones.
Raises
------
ValueError
if :math:`D` is not a directed acyclic graph
ValueError
if the number of stones is negative
References
----------
.. [1] M. Alekhnovich, J. Johannsen, T. Pitassi and A. Urquhart
An Exponential Separation between Regular and General Resolution.
Theory of Computing (2007)
.. [2] R. Raz and P. McKenzie
Separation of the monotone NC hierarchy.
Combinatorica (1999)
"""
if not is_dag(D):
raise ValueError("Stone formulas are defined only for directed acyclic graphs.")
if nstones<0:
raise ValueError("There must be at least one stone.")
cnf = CNF()
if hasattr(D, 'name'):
cnf.header = "Stone formula of: " + D.name + "\nwith " + str(nstones) + " stones\n" + cnf.header
else:
cnf.header = "Stone formula with " + str(nstones) + " stones\n" + cnf.header
# Add variables in the appropriate order
vertices=enumerate_vertices(D)
position=dict((v,i) for (i,v) in enumerate(vertices))
stones=range(1,nstones+1)
# Caching variable names
color_vn = {}
stone_vn = {}
# Stones->Vertices variables
for v in vertices:
for j in stones:
stone_vn[(v,j)] = "P_{{{0},{1}}}".format(v,j)
cnf.add_variable(stone_vn[(v,j)],
description="Stone ${1}$ on vertex ${0}$".format(v,j))
# Color variables
for j in stones:
color_vn[j] = "R_{{{0}}}".format(j)
cnf.add_variable(color_vn[j],
description="Stone ${}$ is red".format(j))
# Each vertex has some stone
for v in vertices:
cnf.add_clause_unsafe([(True,stone_vn[(v,j)]) for j in stones])
# If predecessors have red stones, the sink must have a red stone
for v in vertices:
for j in stones:
pred=sorted(D.predecessors(v),key=lambda x:position[x])
for stones_tuple in product([s for s in stones if s!=j],repeat=len(pred)):
cnf.add_clause_unsafe([(False, stone_vn[(p,s)]) for (p,s) in zip(pred,stones_tuple)] +
[(False, stone_vn[(v,j)])] +
[(False, color_vn[s]) for s in _uniqify_list(stones_tuple)] +
[(True, color_vn[j])])
if D.out_degree(v)==0: #the sink
for j in stones:
cnf.add_clause_unsafe([ (False,stone_vn[(v,j)]), (False,color_vn[j])])
return cnf
