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python-algorithms/algorithms/graph/satisfiability.py at master · plumnx/python-algorithms · GitHub
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python-algorithms
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algorithms
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graph
/
satisfiability.py
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python-algorithms
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algorithms
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graph
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satisfiability.py
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"""
Given a formula in conjunctive normal form (2-CNF), finds a way to assign
True/False values to all variables to satisfy all clauses, or reports there
is no solution.
https://en.wikipedia.org/wiki/2-satisfiability
Format:
- each clause is a pair of literals
- each literal in the form (name, is_neg)
where name is an arbitrary identifier,
and is_neg is true if the literal is negated
"""
def
dfs_transposed
(
vertex
,
graph
,
order
,
visited
):
"""
Perform a depth first search traversal of the graph starting at the given vertex.
Stores the order in which nodes were visited to the list, in transposed order.
"""
visited
[
vertex
]
=
True
for
adjacent
in
graph
[
vertex
]:
if
not
visited
[
adjacent
]:
dfs_transposed
(
adjacent
,
graph
,
order
,
visited
)
order
.
append
(
vertex
)
def
dfs
(
vertex
,
current_comp
,
vertex_scc
,
graph
,
visited
):
"""
Perform a depth first search traversal of the graph starting at the given vertex.
Records all visited nodes as being of a certain strongly connected component.
"""
visited
[
vertex
]
=
True
vertex_scc
[
vertex
]
=
current_comp
for
adjacent
in
graph
[
vertex
]:
if
not
visited
[
adjacent
]:
dfs
(
adjacent
,
current_comp
,
vertex_scc
,
graph
,
visited
)
def
add_edge
(
graph
,
vertex_from
,
vertex_to
):
"""
Add a directed edge to the graph.
"""
if
vertex_from
not
in
graph
:
graph
[
vertex_from
]
=
[]
graph
[
vertex_from
].
append
(
vertex_to
)
def
scc
(
graph
):
''' Computes the strongly connected components of a graph '''
order
=
[]
visited
=
{
vertex
:
False
for
vertex
in
graph
}
graph_transposed
=
{
vertex
: []
for
vertex
in
graph
}
for
(
source
,
neighbours
)
in
graph
.
iteritems
():
for
target
in
neighbours
:
add_edge
(
graph_transposed
,
target
,
source
)
for
vertex
in
graph
:
if
not
visited
[
vertex
]:
dfs_transposed
(
vertex
,
graph_transposed
,
order
,
visited
)
visited
=
{
vertex
:
False
for
vertex
in
graph
}
vertex_scc
=
{}
current_comp
=
0
for
vertex
in
reversed
(
order
):
if
not
visited
[
vertex
]:
# Each dfs will visit exactly one component
dfs
(
vertex
,
current_comp
,
vertex_scc
,
graph
,
visited
)
current_comp
+=
1
return
vertex_scc
def
build_graph
(
formula
):
''' Builds the implication graph from the formula '''
graph
=
{}
for
clause
in
formula
:
for
(
lit
,
_
)
in
clause
:
for
neg
in
[
False
,
True
]:
graph
[(
lit
,
neg
)]
=
[]
for
((
a_lit
,
a_neg
), (
b_lit
,
b_neg
))
in
formula
:
add_edge
(
graph
, (
a_lit
,
a_neg
), (
b_lit
,
not
b_neg
))
add_edge
(
graph
, (
b_lit
,
b_neg
), (
a_lit
,
not
a_neg
))
return
graph
def
solve_sat
(
formula
):
"""
Solves the 2-SAT problem
"""
graph
=
build_graph
(
formula
)
vertex_scc
=
scc
(
graph
)
for
(
var
,
_
)
in
graph
:
if
vertex_scc
[(
var
,
False
)]
==
vertex_scc
[(
var
,
True
)]:
return
None
# The formula is contradictory
comp_repr
=
{}
# An arbitrary representant from each component
for
vertex
in
graph
:
if
not
vertex_scc
[
vertex
]
in
comp_repr
:
comp_repr
[
vertex_scc
[
vertex
]]
=
vertex
comp_value
=
{}
# True/False value for each strongly connected component
components
=
sorted
(
vertex_scc
.
values
())
for
comp
in
components
:
if
comp
not
in
comp_value
:
comp_value
[
comp
]
=
False
(
lit
,
neg
)
=
comp_repr
[
comp
]
comp_value
[
vertex_scc
[(
lit
,
not
neg
)]]
=
True
value
=
{
var
:
comp_value
[
vertex_scc
[(
var
,
False
)]]
for
(
var
,
_
)
in
graph
}
return
value
def
main
():
"""
Entry point for testing
"""
formula
=
[((
'x'
,
False
), (
'y'
,
False
)),
((
'y'
,
True
), (
'y'
,
True
)),
((
'a'
,
False
), (
'b'
,
False
)),
((
'a'
,
True
), (
'c'
,
True
)),
((
'c'
,
False
), (
'b'
,
True
))]
result
=
solve_sat
(
formula
)
for
(
variable
,
assign
)
in
result
.
items
():
print
(
f"
{
variable
}
:
{
assign
}
"
)
if
__name__
==
'__main__'
:
main
()
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