def BinaryPigeonholePrinciple(pigeons,holes):
"""Binary Pigeonhole Principle CNF formula
The pigeonhole principle claims that no M pigeons can sit in
N pigeonholes without collision if M>N. This formula encodes the
principle using binary strings to identify the holes.
Parameters
----------
pigeon : int
number of pigeons
holes : int
number of holes
"""
bphp=CNF()
bphp.header="Binary Pigeonhole Principle for {0} pigeons and {1} holes\n".format(pigeons,holes)\
+ bphp.header
bphpgen=bphp.binary_mapping(xrange(1,pigeons+1), xrange(1,holes+1), injective = True)
for v in bphpgen.variables():
bphp.add_variable(v)
for c in bphpgen.clauses():
bphp.add_clause_unsafe(c)
return bphp
def BinaryPigeonholePrinciple(pigeons, holes):
"""Binary Pigeonhole Principle CNF formula
The pigeonhole principle claims that no M pigeons can sit in
N pigeonholes without collision if M>N. This formula encodes the
principle using binary strings to identify the holes.
Parameters
----------
pigeon : int
number of pigeons
holes : int
number of holes
"""
bphp = CNF()
bphp.header="Binary Pigeonhole Principle for {0} pigeons and {1} holes\n".format(pigeons,holes)\
+ bphp.header
bphpgen = bphp.binary_mapping(xrange(1, pigeons + 1),
xrange(1, holes + 1),
injective=True)
for v in bphpgen.variables():
bphp.add_variable(v)
for c in bphpgen.clauses():
bphp.add_clause_unsafe(c)
return bphp
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()
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_unsafe([(False, p) for p in pred] + [(True, v)])
if digraph.out_degree(v) == 0: #the sink
peb.add_clause_unsafe([(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()
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_unsafe([(False,p) for p in pred]+[(True,v)])
if digraph.out_degree(v)==0: #the sink
peb.add_clause_unsafe([(False,v)])
return peb
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 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
def GraphPigeonholePrinciple(graph,functional=False,onto=False):
"""Graph Pigeonhole Principle CNF formula
The graph pigeonhole principle CNF formula, defined on a bipartite
graph G=(L,R,E), claims that there is a subset E' of the edges such that
every vertex on the left size L has at least one incident edge in E' and
every edge on the right side R has at most one incident edge in E'.
This is possible only if the graph has a matching of size |L|.
There are different variants of this formula, depending on the
values of `functional` and `onto` argument.
- PHP(G): each left vertex can be incident to multiple edges in E'
- FPHP(G): each left vertex must be incident to exaclty one edge in E'
- onto-PHP: all right vertices must be incident to some vertex
- matching: E' must be a perfect matching between L and R
Arguments:
- `graph` : bipartite graph
- `functional`: add clauses to enforce at most one edge per left vertex
- `onto`: add clauses to enforce that any right vertex has one incident edge
Remark: the graph vertices must have the 'bipartite' attribute
set. Left vertices must have it set to 0 and the right ones to
1. A KeyException is raised otherwise.
"""
def var_name(p,h):
return 'p_{{{0},{1}}}'.format(p,h)
if functional:
if onto:
formula_name="Graph matching"
else:
formula_name="Graph functional pigeonhole principle"
else:
if onto:
formula_name="Graph onto pigeonhole principle"
else:
formula_name="Graph pigeonhole principle"
gphp=CNF()
gphp.header="{0} formula for graph {1}\n".format(formula_name,graph.name)
Left, Right = bipartite_sets(graph)
mapping = gphp.unary_mapping(Left,Right,
sparsity_pattern=graph,
var_name=var_name,
injective = True,
functional = functional,
surjective = onto)
for v in mapping.variables():
gphp.add_variable(v)
for c in mapping.clauses():
gphp.add_clause_unsafe(c)
return gphp
def PigeonholePrinciple(pigeons,holes,functional=False,onto=False):
"""Pigeonhole Principle CNF formula
The pigeonhole principle claims that no M pigeons can sit in N
pigeonholes without collision if M>N. The counterpositive CNF
formulation requires such mapping to be satisfied. There are
different variants of this formula, depending on the values of
`functional` and `onto` argument.
- PHP: pigeon can sit in multiple holes
- FPHP: each pigeon sits in exactly one hole
- onto-PHP: pigeon can sit in multiple holes, every hole must be
covered.
- Matching: one-to-one bijection between pigeons and holes.
Arguments:
- `pigeon`: number of pigeons
- `hole`: number of holes
- `functional`: add clauses to enforce at most one hole per pigeon
- `onto`: add clauses to enforce that any hole must have a pigeon
>>> print(PigeonholePrinciple(4,3).dimacs(export_header=False))
p cnf 12 22
1 2 3 0
4 5 6 0
7 8 9 0
10 11 12 0
-1 -4 0
-1 -7 0
-1 -10 0
-4 -7 0
-4 -10 0
-7 -10 0
-2 -5 0
-2 -8 0
-2 -11 0
-5 -8 0
-5 -11 0
-8 -11 0
-3 -6 0
-3 -9 0
-3 -12 0
-6 -9 0
-6 -12 0
-9 -12 0
"""
def var_name(p,h):
return 'p_{{{0},{1}}}'.format(p,h)
if functional:
if onto:
formula_name="Matching"
else:
formula_name="Functional pigeonhole principle"
else:
if onto:
formula_name="Onto pigeonhole principle"
else:
formula_name="Pigeonhole principle"
php=CNF()
php.header="{0} formula for {1} pigeons and {2} holes\n".format(formula_name,pigeons,holes)\
+ php.header
mapping=php.unary_mapping(
xrange(1,pigeons+1),
xrange(1,holes+1),
var_name=var_name,
injective = True,
functional = functional,
surjective = onto)
for v in mapping.variables():
php.add_variable(v)
for c in mapping.clauses():
php.add_clause_unsafe(c)
return php
def GraphPigeonholePrinciple(graph, functional=False, onto=False):
"""Graph Pigeonhole Principle CNF formula
The graph pigeonhole principle CNF formula, defined on a bipartite
graph G=(L,R,E), claims that there is a subset E' of the edges such that
every vertex on the left size L has at least one incident edge in E' and
every edge on the right side R has at most one incident edge in E'.
This is possible only if the graph has a matching of size |L|.
There are different variants of this formula, depending on the
values of `functional` and `onto` argument.
- PHP(G): each left vertex can be incident to multiple edges in E'
- FPHP(G): each left vertex must be incident to exaclty one edge in E'
- onto-PHP: all right vertices must be incident to some vertex
- matching: E' must be a perfect matching between L and R
Arguments:
- `graph` : bipartite graph
- `functional`: add clauses to enforce at most one edge per left vertex
- `onto`: add clauses to enforce that any right vertex has one incident edge
Remark: the graph vertices must have the 'bipartite' attribute
set. Left vertices must have it set to 0 and the right ones to
1. A KeyException is raised otherwise.
"""
def var_name(p, h):
return 'p_{{{0},{1}}}'.format(p, h)
if functional:
if onto:
formula_name = "Graph matching"
else:
formula_name = "Graph functional pigeonhole principle"
else:
if onto:
formula_name = "Graph onto pigeonhole principle"
else:
formula_name = "Graph pigeonhole principle"
gphp = CNF()
gphp.header = "{0} formula for graph {1}\n".format(formula_name,
graph.name)
Left, Right = bipartite_sets(graph)
mapping = gphp.unary_mapping(Left,
Right,
sparsity_pattern=graph,
var_name=var_name,
injective=True,
functional=functional,
surjective=onto)
for v in mapping.variables():
gphp.add_variable(v)
for c in mapping.clauses():
gphp.add_clause_unsafe(c)
return gphp
def PigeonholePrinciple(pigeons, holes, functional=False, onto=False):
"""Pigeonhole Principle CNF formula
The pigeonhole principle claims that no M pigeons can sit in N
pigeonholes without collision if M>N. The counterpositive CNF
formulation requires such mapping to be satisfied. There are
different variants of this formula, depending on the values of
`functional` and `onto` argument.
- PHP: pigeon can sit in multiple holes
- FPHP: each pigeon sits in exactly one hole
- onto-PHP: pigeon can sit in multiple holes, every hole must be
covered.
- Matching: one-to-one bijection between pigeons and holes.
Arguments:
- `pigeon`: number of pigeons
- `hole`: number of holes
- `functional`: add clauses to enforce at most one hole per pigeon
- `onto`: add clauses to enforce that any hole must have a pigeon
>>> print(PigeonholePrinciple(4,3).dimacs(export_header=False))
p cnf 12 22
1 2 3 0
4 5 6 0
7 8 9 0
10 11 12 0
-1 -4 0
-1 -7 0
-1 -10 0
-4 -7 0
-4 -10 0
-7 -10 0
-2 -5 0
-2 -8 0
-2 -11 0
-5 -8 0
-5 -11 0
-8 -11 0
-3 -6 0
-3 -9 0
-3 -12 0
-6 -9 0
-6 -12 0
-9 -12 0
"""
def var_name(p, h):
return 'p_{{{0},{1}}}'.format(p, h)
if functional:
if onto:
formula_name = "Matching"
else:
formula_name = "Functional pigeonhole principle"
else:
if onto:
formula_name = "Onto pigeonhole principle"
else:
formula_name = "Pigeonhole principle"
php = CNF()
php.header="{0} formula for {1} pigeons and {2} holes\n".format(formula_name,pigeons,holes)\
+ php.header
mapping = php.unary_mapping(xrange(1, pigeons + 1),
xrange(1, holes + 1),
var_name=var_name,
injective=True,
functional=functional,
surjective=onto)
for v in mapping.variables():
php.add_variable(v)
for c in mapping.clauses():
php.add_clause_unsafe(c)
return php