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"""
Representations and Inference for Logic. (Chapters 7-9, 12)
Covers both Propositional and First-Order Logic. First we have four
important data types:
KB Abstract class holds a knowledge base of logical expressions
KB_Agent Abstract class subclasses agents.Agent
Expr A logical expression, imported from utils.py
substitution Implemented as a dictionary of var:value pairs, {x:1, y:x}
Be careful: some functions take an Expr as argument, and some take a KB.
Logical expressions can be created with Expr or expr, imported from utils, TODO
or with expr, which adds the capability to write a string that uses
the connectives ==>, <==, <=>, or <=/=>. But be careful: these have the
operator precedence of commas; you may need to add parens to make precedence work.
See logic.ipynb for examples.
Then we implement various functions for doing logical inference:
pl_true Evaluate a propositional logical sentence in a model
tt_entails Say if a statement is entailed by a KB
pl_resolution Do resolution on propositional sentences
dpll_satisfiable See if a propositional sentence is satisfiable
WalkSAT Try to find a solution for a set of clauses
And a few other functions:
to_cnf Convert to conjunctive normal form
unify Do unification of two FOL sentences
diff, simp Symbolic differentiation and simplification
"""
import
heapq
import
itertools
import
random
from
collections
import
defaultdict
,
Counter
import
networkx
as
nx
from
agents
import
Agent
,
Glitter
,
Bump
,
Stench
,
Breeze
,
Scream
from
csp
import
parse_neighbors
,
UniversalDict
from
search
import
astar_search
,
PlanRoute
from
utils
import
remove_all
,
unique
,
first
,
probability
,
isnumber
,
issequence
,
Expr
,
expr
,
subexpressions
,
extend
class
KB
:
"""A knowledge base to which you can tell and ask sentences.
To create a KB, first subclass this class and implement
tell, ask_generator, and retract. Why ask_generator instead of ask?
The book is a bit vague on what ask means --
For a Propositional Logic KB, ask(P & Q) returns True or False, but for an
FOL KB, something like ask(Brother(x, y)) might return many substitutions
such as {x: Cain, y: Abel}, {x: Abel, y: Cain}, {x: George, y: Jeb}, etc.
So ask_generator generates these one at a time, and ask either returns the
first one or returns False."""
def
__init__
(
self
,
sentence
=
None
):
if
sentence
:
self
.
tell
(
sentence
)
def
tell
(
self
,
sentence
):
"""Add the sentence to the KB."""
raise
NotImplementedError
def
ask
(
self
,
query
):
"""Return a substitution that makes the query true, or, failing that, return False."""
return
first
(
self
.
ask_generator
(
query
),
default
=
False
)
def
ask_generator
(
self
,
query
):
"""Yield all the substitutions that make query true."""
raise
NotImplementedError
def
retract
(
self
,
sentence
):
"""Remove sentence from the KB."""
raise
NotImplementedError
class
PropKB
(
KB
):
"""A KB for propositional logic. Inefficient, with no indexing."""
def
__init__
(
self
,
sentence
=
None
):
super
().
__init__
(
sentence
)
self
.
clauses
=
[]
def
tell
(
self
,
sentence
):
"""Add the sentence's clauses to the KB."""
self
.
clauses
.
extend
(
conjuncts
(
to_cnf
(
sentence
)))
def
ask_generator
(
self
,
query
):
"""Yield the empty substitution {} if KB entails query; else no results."""
if
tt_entails
(
Expr
(
'&'
,
*
self
.
clauses
),
query
):
yield
{}
def
ask_if_true
(
self
,
query
):
"""Return True if the KB entails query, else return False."""
for
_
in
self
.
ask_generator
(
query
):
return
True
return
False
def
retract
(
self
,
sentence
):
"""Remove the sentence's clauses from the KB."""
for
c
in
conjuncts
(
to_cnf
(
sentence
)):
if
c
in
self
.
clauses
:
self
.
clauses
.
remove
(
c
)
# ______________________________________________________________________________
def
KBAgentProgram
(
kb
):
"""
[Figure 7.1]
A generic logical knowledge-based agent program.
"""
steps
=
itertools
.
count
()
def
program
(
percept
):
t
=
next
(
steps
)
kb
.
tell
(
make_percept_sentence
(
percept
,
t
))
action
=
kb
.
ask
(
make_action_query
(
t
))
kb
.
tell
(
make_action_sentence
(
action
,
t
))
return
action
def
make_percept_sentence
(
percept
,
t
):
return
Expr
(
'Percept'
)(
percept
,
t
)
def
make_action_query
(
t
):
return
expr
(
'ShouldDo(action, {})'
.
format
(
t
))
def
make_action_sentence
(
action
,
t
):
return
Expr
(
'Did'
)(
action
[
expr
(
'action'
)],
t
)
return
program
def
is_symbol
(
s
):
"""A string s is a symbol if it starts with an alphabetic char.
>>> is_symbol('R2D2')
True
"""
return
isinstance
(
s
,
str
)
and
s
[:
1
].
isalpha
()
def
is_var_symbol
(
s
):
"""A logic variable symbol is an initial-lowercase string.
>>> is_var_symbol('EXE')
False
"""
return
is_symbol
(
s
)
and
s
[
0
].
islower
()
def
is_prop_symbol
(
s
):
"""A proposition logic symbol is an initial-uppercase string.
>>> is_prop_symbol('exe')
False
"""
return
is_symbol
(
s
)
and
s
[
0
].
isupper
()
def
variables
(
s
):
"""Return a set of the variables in expression s.
>>> variables(expr('F(x, x) & G(x, y) & H(y, z) & R(A, z, 2)')) == {x, y, z}
True
"""
return
{
x
for
x
in
subexpressions
(
s
)
if
is_variable
(
x
)}
def
is_definite_clause
(
s
):
"""Returns True for exprs s of the form A & B & ... & C ==> D,
where all literals are positive. In clause form, this is
~A | ~B | ... | ~C | D, where exactly one clause is positive.
>>> is_definite_clause(expr('Farmer(Mac)'))
True
"""
if
is_symbol
(
s
.
op
):
return
True
elif
s
.
op
==
'==>'
:
antecedent
,
consequent
=
s
.
args
return
is_symbol
(
consequent
.
op
)
and
all
(
is_symbol
(
arg
.
op
)
for
arg
in
conjuncts
(
antecedent
))
else
:
return
False
def
parse_definite_clause
(
s
):
"""Return the antecedents and the consequent of a definite clause."""
assert
is_definite_clause
(
s
)
if
is_symbol
(
s
.
op
):
return
[],
s
else
:
antecedent
,
consequent
=
s
.
args
return
conjuncts
(
antecedent
),
consequent
# Useful constant Exprs used in examples and code:
A
,
B
,
C
,
D
,
E
,
F
,
G
,
P
,
Q
,
a
,
x
,
y
,
z
,
u
=
map
(
Expr
,
'ABCDEFGPQaxyzu'
)
# ______________________________________________________________________________
def
tt_entails
(
kb
,
alpha
):
"""
[Figure 7.10]
Does kb entail the sentence alpha? Use truth tables. For propositional
kb's and sentences. Note that the 'kb' should be an Expr which is a
conjunction of clauses.
>>> tt_entails(expr('P & Q'), expr('Q'))
True
"""
assert
not
variables
(
alpha
)
symbols
=
list
(
prop_symbols
(
kb
&
alpha
))
return
tt_check_all
(
kb
,
alpha
,
symbols
, {})
def
tt_check_all
(
kb
,
alpha
,
symbols
,
model
):
"""Auxiliary routine to implement tt_entails."""
if
not
symbols
:
if
pl_true
(
kb
,
model
):
result
=
pl_true
(
alpha
,
model
)
assert
result
in
(
True
,
False
)
return
result
else
:
return
True
else
:
P
,
rest
=
symbols
[
0
],
symbols
[
1
:]
return
(
tt_check_all
(
kb
,
alpha
,
rest
,
extend
(
model
,
P
,
True
))
and
tt_check_all
(
kb
,
alpha
,
rest
,
extend
(
model
,
P
,
False
)))
def
prop_symbols
(
x
):
"""Return the set of all propositional symbols in x."""
if
not
isinstance
(
x
,
Expr
):
return
set
()
elif
is_prop_symbol
(
x
.
op
):
return
{
x
}
else
:
return
{
symbol
for
arg
in
x
.
args
for
symbol
in
prop_symbols
(
arg
)}
def
constant_symbols
(
x
):
"""Return the set of all constant symbols in x."""
if
not
isinstance
(
x
,
Expr
):
return
set
()
elif
is_prop_symbol
(
x
.
op
)
and
not
x
.
args
:
return
{
x
}
else
:
return
{
symbol
for
arg
in
x
.
args
for
symbol
in
constant_symbols
(
arg
)}
def
predicate_symbols
(
x
):
"""Return a set of (symbol_name, arity) in x.
All symbols (even functional) with arity > 0 are considered."""
if
not
isinstance
(
x
,
Expr
)
or
not
x
.
args
:
return
set
()
pred_set
=
{(
x
.
op
,
len
(
x
.
args
))}
if
is_prop_symbol
(
x
.
op
)
else
set
()
pred_set
.
update
({
symbol
for
arg
in
x
.
args
for
symbol
in
predicate_symbols
(
arg
)})
return
pred_set
def
tt_true
(
s
):
"""Is a propositional sentence a tautology?
>>> tt_true('P | ~P')
True
"""
s
=
expr
(
s
)
return
tt_entails
(
True
,
s
)
def
pl_true
(
exp
,
model
=
{}):
"""Return True if the propositional logic expression is true in the model,
and False if it is false. If the model does not specify the value for
every proposition, this may return None to indicate 'not obvious';
this may happen even when the expression is tautological.
>>> pl_true(P, {}) is None
True
"""
if
exp
in
(
True
,
False
):
return
exp
op
,
args
=
exp
.
op
,
exp
.
args
if
is_prop_symbol
(
op
):
return
model
.
get
(
exp
)
elif
op
==
'~'
:
p
=
pl_true
(
args
[
0
],
model
)
if
p
is
None
:
return
None
else
:
return
not
p
elif
op
==
'|'
:
result
=
False
for
arg
in
args
:
p
=
pl_true
(
arg
,
model
)
if
p
is
True
:
return
True
if
p
is
None
:
result
=
None
return
result
elif
op
==
'&'
:
result
=
True
for
arg
in
args
:
p
=
pl_true
(
arg
,
model
)
if
p
is
False
:
return
False
if
p
is
None
:
result
=
None
return
result
p
,
q
=
args
if
op
==
'==>'
:
return
pl_true
(
~
p
|
q
,
model
)
elif
op
==
'<=='
:
return
pl_true
(
p
|
~
q
,
model
)
pt
=
pl_true
(
p
,
model
)
if
pt
is
None
:
return
None
qt
=
pl_true
(
q
,
model
)
if
qt
is
None
:
return
None
if
op
==
'<=>'
:
return
pt
==
qt
elif
op
==
'^'
:
# xor or 'not equivalent'
return
pt
!=
qt
else
:
raise
ValueError
(
'Illegal operator in logic expression'
+
str
(
exp
))
# ______________________________________________________________________________
# Convert to Conjunctive Normal Form (CNF)
def
to_cnf
(
s
):
"""
[Page 253]
Convert a propositional logical sentence to conjunctive normal form.
That is, to the form ((A | ~B | ...) & (B | C | ...) & ...)
>>> to_cnf('~(B | C)')
(~B & ~C)
"""
s
=
expr
(
s
)
if
isinstance
(
s
,
str
):
s
=
expr
(
s
)
s
=
eliminate_implications
(
s
)
# Steps 1, 2 from p. 253
s
=
move_not_inwards
(
s
)
# Step 3
return
distribute_and_over_or
(
s
)
# Step 4
def
eliminate_implications
(
s
):
"""Change implications into equivalent form with only &, |, and ~ as logical operators."""
s
=
expr
(
s
)
if
not
s
.
args
or
is_symbol
(
s
.
op
):
return
s
# Atoms are unchanged.
args
=
list
(
map
(
eliminate_implications
,
s
.
args
))
a
,
b
=
args
[
0
],
args
[
-
1
]
if
s
.
op
==
'==>'
:
return
b
|
~
a
elif
s
.
op
==
'<=='
:
return
a
|
~
b
elif
s
.
op
==
'<=>'
:
return
(
a
|
~
b
)
&
(
b
|
~
a
)
elif
s
.
op
==
'^'
:
assert
len
(
args
)
==
2
# TODO: relax this restriction
return
(
a
&
~
b
)
|
(
~
a
&
b
)
else
:
assert
s
.
op
in
(
'&'
,
'|'
,
'~'
)
return
Expr
(
s
.
op
,
*
args
)
def
move_not_inwards
(
s
):
"""Rewrite sentence s by moving negation sign inward.
>>> move_not_inwards(~(A | B))
(~A & ~B)
"""
s
=
expr
(
s
)
if
s
.
op
==
'~'
:
def
NOT
(
b
):
return
move_not_inwards
(
~
b
)
a
=
s
.
args
[
0
]
if
a
.
op
==
'~'
:
return
move_not_inwards
(
a
.
args
[
0
])
# ~~A ==> A
if
a
.
op
==
'&'
:
return
associate
(
'|'
,
list
(
map
(
NOT
,
a
.
args
)))
if
a
.
op
==
'|'
:
return
associate
(
'&'
,
list
(
map
(
NOT
,
a
.
args
)))
return
s
elif
is_symbol
(
s
.
op
)
or
not
s
.
args
:
return
s
else
:
return
Expr
(
s
.
op
,
*
list
(
map
(
move_not_inwards
,
s
.
args
)))
def
distribute_and_over_or
(
s
):
"""Given a sentence s consisting of conjunctions and disjunctions
of literals, return an equivalent sentence in CNF.
>>> distribute_and_over_or((A & B) | C)
((A | C) & (B | C))
"""
s
=
expr
(
s
)
if
s
.
op
==
'|'
:
s
=
associate
(
'|'
,
s
.
args
)
if
s
.
op
!=
'|'
:
return
distribute_and_over_or
(
s
)
if
len
(
s
.
args
)
==
0
:
return
False
if
len
(
s
.
args
)
==
1
:
return
distribute_and_over_or
(
s
.
args
[
0
])
conj
=
first
(
arg
for
arg
in
s
.
args
if
arg
.
op
==
'&'
)
if
not
conj
:
return
s
others
=
[
a
for
a
in
s
.
args
if
a
is
not
conj
]
rest
=
associate
(
'|'
,
others
)
return
associate
(
'&'
, [
distribute_and_over_or
(
c
|
rest
)
for
c
in
conj
.
args
])
elif
s
.
op
==
'&'
:
return
associate
(
'&'
,
list
(
map
(
distribute_and_over_or
,
s
.
args
)))
else
:
return
s
def
associate
(
op
,
args
):
"""Given an associative op, return an expression with the same
meaning as Expr(op, *args), but flattened -- that is, with nested
instances of the same op promoted to the top level.
>>> associate('&', [(A&B),(B|C),(B&C)])
(A & B & (B | C) & B & C)
>>> associate('|', [A|(B|(C|(A&B)))])
(A | B | C | (A & B))
"""
args
=
dissociate
(
op
,
args
)
if
len
(
args
)
==
0
:
return
_op_identity
[
op
]
elif
len
(
args
)
==
1
:
return
args
[
0
]
else
:
return
Expr
(
op
,
*
args
)
_op_identity
=
{
'&'
:
True
,
'|'
:
False
,
'+'
:
0
,
'*'
:
1
}
def
dissociate
(
op
,
args
):
"""Given an associative op, return a flattened list result such
that Expr(op, *result) means the same as Expr(op, *args).
>>> dissociate('&', [A & B])
[A, B]
"""
result
=
[]
def
collect
(
subargs
):
for
arg
in
subargs
:
if
arg
.
op
==
op
:
collect
(
arg
.
args
)
else
:
result
.
append
(
arg
)
collect
(
args
)
return
result
def
conjuncts
(
s
):
"""Return a list of the conjuncts in the sentence s.
>>> conjuncts(A & B)
[A, B]
>>> conjuncts(A | B)
[(A | B)]
"""
return
dissociate
(
'&'
, [
s
])
def
disjuncts
(
s
):
"""Return a list of the disjuncts in the sentence s.
>>> disjuncts(A | B)
[A, B]
>>> disjuncts(A & B)
[(A & B)]
"""
return
dissociate
(
'|'
, [
s
])
# ______________________________________________________________________________
def
pl_resolution
(
kb
,
alpha
):
"""
[Figure 7.12]
Propositional-logic resolution: say if alpha follows from KB.
>>> pl_resolution(horn_clauses_KB, A)
True
"""
clauses
=
kb
.
clauses
+
conjuncts
(
to_cnf
(
~
alpha
))
new
=
set
()
while
True
:
n
=
len
(
clauses
)
pairs
=
[(
clauses
[
i
],
clauses
[
j
])
for
i
in
range
(
n
)
for
j
in
range
(
i
+
1
,
n
)]
for
(
ci
,
cj
)
in
pairs
:
resolvents
=
pl_resolve
(
ci
,
cj
)
if
False
in
resolvents
:
return
True
new
=
new
.
union
(
set
(
resolvents
))
if
new
.
issubset
(
set
(
clauses
)):
return
False
for
c
in
new
:
if
c
not
in
clauses
:
clauses
.
append
(
c
)
def
pl_resolve
(
ci
,
cj
):
"""Return all clauses that can be obtained by resolving clauses ci and cj."""
clauses
=
[]
for
di
in
disjuncts
(
ci
):
for
dj
in
disjuncts
(
cj
):
if
di
==
~
dj
or
~
di
==
dj
:
clauses
.
append
(
associate
(
'|'
,
unique
(
remove_all
(
di
,
disjuncts
(
ci
))
+
remove_all
(
dj
,
disjuncts
(
cj
)))))
return
clauses
# ______________________________________________________________________________
class
PropDefiniteKB
(
PropKB
):
"""A KB of propositional definite clauses."""
def
tell
(
self
,
sentence
):
"""Add a definite clause to this KB."""
assert
is_definite_clause
(
sentence
),
"Must be definite clause"
self
.
clauses
.
append
(
sentence
)
def
ask_generator
(
self
,
query
):
"""Yield the empty substitution if KB implies query; else nothing."""
if
pl_fc_entails
(
self
.
clauses
,
query
):
yield
{}
def
retract
(
self
,
sentence
):
self
.
clauses
.
remove
(
sentence
)
def
clauses_with_premise
(
self
,
p
):
"""Return a list of the clauses in KB that have p in their premise.
This could be cached away for O(1) speed, but we'll recompute it."""
return
[
c
for
c
in
self
.
clauses
if
c
.
op
==
'==>'
and
p
in
conjuncts
(
c
.
args
[
0
])]
def
pl_fc_entails
(
kb
,
q
):
"""
[Figure 7.15]
Use forward chaining to see if a PropDefiniteKB entails symbol q.
>>> pl_fc_entails(horn_clauses_KB, expr('Q'))
True
"""
count
=
{
c
:
len
(
conjuncts
(
c
.
args
[
0
]))
for
c
in
kb
.
clauses
if
c
.
op
==
'==>'
}
inferred
=
defaultdict
(
bool
)
agenda
=
[
s
for
s
in
kb
.
clauses
if
is_prop_symbol
(
s
.
op
)]
while
agenda
:
p
=
agenda
.
pop
()
if
p
==
q
:
return
True
if
not
inferred
[
p
]:
inferred
[
p
]
=
True
for
c
in
kb
.
clauses_with_premise
(
p
):
count
[
c
]
-=
1
if
count
[
c
]
==
0
:
agenda
.
append
(
c
.
args
[
1
])
return
False
"""
[Figure 7.13]
Simple inference in a wumpus world example
"""
wumpus_world_inference
=
expr
(
'(B11 <=> (P12 | P21)) & ~B11'
)
"""
[Figure 7.16]
Propositional Logic Forward Chaining example
"""
horn_clauses_KB
=
PropDefiniteKB
()
for
clause
in
[
'P ==> Q'
,
'(L & M) ==> P'
,
'(B & L) ==> M'
,
'(A & P) ==> L'
,
'(A & B) ==> L'
,
'A'
,
'B'
]:
horn_clauses_KB
.
tell
(
expr
(
clause
))
"""
Definite clauses KB example
"""
definite_clauses_KB
=
PropDefiniteKB
()
for
clause
in
[
'(B & F) ==> E'
,
'(A & E & F) ==> G'
,
'(B & C) ==> F'
,
'(A & B) ==> D'
,
'(E & F) ==> H'
,
'(H & I) ==>J'
,
'A'
,
'B'
,
'C'
]:
definite_clauses_KB
.
tell
(
expr
(
clause
))
# ______________________________________________________________________________
# Heuristics for SAT Solvers
def
no_branching_heuristic
(
symbols
,
clauses
):
return
first
(
symbols
),
True
def
min_clauses
(
clauses
):
min_len
=
min
(
map
(
lambda
c
:
len
(
c
.
args
),
clauses
),
default
=
2
)
return
filter
(
lambda
c
:
len
(
c
.
args
)
==
(
min_len
if
min_len
>
1
else
2
),
clauses
)
def
moms
(
symbols
,
clauses
):
"""
MOMS (Maximum Occurrence in clauses of Minimum Size) heuristic
Returns the literal with the most occurrences in all clauses of minimum size
"""
scores
=
Counter
(
l
for
c
in
min_clauses
(
clauses
)
for
l
in
prop_symbols
(
c
))
return
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]),
True
def
momsf
(
symbols
,
clauses
,
k
=
0
):
"""
MOMS alternative heuristic
If f(x) the number of occurrences of the variable x in clauses with minimum size,
we choose the variable maximizing [f(x) + f(-x)] * 2^k + f(x) * f(-x)
Returns x if f(x) >= f(-x) otherwise -x
"""
scores
=
Counter
(
l
for
c
in
min_clauses
(
clauses
)
for
l
in
disjuncts
(
c
))
P
=
max
(
symbols
,
key
=
lambda
symbol
: (
scores
[
symbol
]
+
scores
[
~
symbol
])
*
pow
(
2
,
k
)
+
scores
[
symbol
]
*
scores
[
~
symbol
])
return
P
,
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
def
posit
(
symbols
,
clauses
):
"""
Freeman's POSIT version of MOMs
Counts the positive x and negative x for each variable x in clauses with minimum size
Returns x if f(x) >= f(-x) otherwise -x
"""
scores
=
Counter
(
l
for
c
in
min_clauses
(
clauses
)
for
l
in
disjuncts
(
c
))
P
=
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]
+
scores
[
~
symbol
])
return
P
,
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
def
zm
(
symbols
,
clauses
):
"""
Zabih and McAllester's version of MOMs
Counts the negative occurrences only of each variable x in clauses with minimum size
"""
scores
=
Counter
(
l
for
c
in
min_clauses
(
clauses
)
for
l
in
disjuncts
(
c
)
if
l
.
op
==
'~'
)
return
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
~
symbol
]),
True
def
dlis
(
symbols
,
clauses
):
"""
DLIS (Dynamic Largest Individual Sum) heuristic
Choose the variable and value that satisfies the maximum number of unsatisfied clauses
Like DLCS but we only consider the literal (thus Cp and Cn are individual)
"""
scores
=
Counter
(
l
for
c
in
clauses
for
l
in
disjuncts
(
c
))
P
=
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
])
return
P
,
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
def
dlcs
(
symbols
,
clauses
):
"""
DLCS (Dynamic Largest Combined Sum) heuristic
Cp the number of clauses containing literal x
Cn the number of clauses containing literal -x
Here we select the variable maximizing Cp + Cn
Returns x if Cp >= Cn otherwise -x
"""
scores
=
Counter
(
l
for
c
in
clauses
for
l
in
disjuncts
(
c
))
P
=
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]
+
scores
[
~
symbol
])
return
P
,
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
def
jw
(
symbols
,
clauses
):
"""
Jeroslow-Wang heuristic
For each literal compute J(l) = \sum{l in clause c} 2^{-|c|}
Return the literal maximizing J
"""
scores
=
Counter
()
for
c
in
clauses
:
for
l
in
prop_symbols
(
c
):
scores
[
l
]
+=
pow
(
2
,
-
len
(
c
.
args
))
return
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]),
True
def
jw2
(
symbols
,
clauses
):
"""
Two Sided Jeroslow-Wang heuristic
Compute J(l) also counts the negation of l = J(x) + J(-x)
Returns x if J(x) >= J(-x) otherwise -x
"""
scores
=
Counter
()
for
c
in
clauses
:
for
l
in
disjuncts
(
c
):
scores
[
l
]
+=
pow
(
2
,
-
len
(
c
.
args
))
P
=
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]
+
scores
[
~
symbol
])
return
P
,
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
# ______________________________________________________________________________
# DPLL-Satisfiable [Figure 7.17]
def
dpll_satisfiable
(
s
,
branching_heuristic
=
no_branching_heuristic
):
"""Check satisfiability of a propositional sentence.
This differs from the book code in two ways: (1) it returns a model
rather than True when it succeeds; this is more useful. (2) The
function find_pure_symbol is passed a list of unknown clauses, rather
than a list of all clauses and the model; this is more efficient.
>>> dpll_satisfiable(A |'<=>'| B) == {A: True, B: True}
True
"""
return
dpll
(
conjuncts
(
to_cnf
(
s
)),
prop_symbols
(
s
), {},
branching_heuristic
)
def
dpll
(
clauses
,
symbols
,
model
,
branching_heuristic
=
no_branching_heuristic
):
"""See if the clauses are true in a partial model."""
unknown_clauses
=
[]
# clauses with an unknown truth value
for
c
in
clauses
:
val
=
pl_true
(
c
,
model
)
if
val
is
False
:
return
False
if
val
is
None
:
unknown_clauses
.
append
(
c
)
if
not
unknown_clauses
:
return
model
P
,
value
=
find_pure_symbol
(
symbols
,
unknown_clauses
)
if
P
:
return
dpll
(
clauses
,
remove_all
(
P
,
symbols
),
extend
(
model
,
P
,
value
),
branching_heuristic
)
P
,
value
=
find_unit_clause
(
clauses
,
model
)
if
P
:
return
dpll
(
clauses
,
remove_all
(
P
,
symbols
),
extend
(
model
,
P
,
value
),
branching_heuristic
)
P
,
value
=
branching_heuristic
(
symbols
,
unknown_clauses
)
return
(
dpll
(
clauses
,
remove_all
(
P
,
symbols
),
extend
(
model
,
P
,
value
),
branching_heuristic
)
or
dpll
(
clauses
,
remove_all
(
P
,
symbols
),
extend
(
model
,
P
,
not
value
),
branching_heuristic
))
def
find_pure_symbol
(
symbols
,
clauses
):
"""Find a symbol and its value if it appears only as a positive literal
(or only as a negative) in clauses.
>>> find_pure_symbol([A, B, C], [A|~B,~B|~C,C|A])
(A, True)
"""
for
s
in
symbols
:
found_pos
,
found_neg
=
False
,
False
for
c
in
clauses
:
if
not
found_pos
and
s
in
disjuncts
(
c
):
found_pos
=
True
if
not
found_neg
and
~
s
in
disjuncts
(
c
):
found_neg
=
True
if
found_pos
!=
found_neg
:
return
s
,
found_pos
return
None
,
None
def
find_unit_clause
(
clauses
,
model
):
"""Find a forced assignment if possible from a clause with only 1
variable not bound in the model.
>>> find_unit_clause([A|B|C, B|~C, ~A|~B], {A:True})
(B, False)
"""
for
clause
in
clauses
:
P
,
value
=
unit_clause_assign
(
clause
,
model
)
if
P
:
return
P
,
value
return
None
,
None
def
unit_clause_assign
(
clause
,
model
):
"""Return a single variable/value pair that makes clause true in
the model, if possible.
>>> unit_clause_assign(A|B|C, {A:True})
(None, None)
>>> unit_clause_assign(B|~C, {A:True})
(None, None)
>>> unit_clause_assign(~A|~B, {A:True})
(B, False)
"""
P
,
value
=
None
,
None
for
literal
in
disjuncts
(
clause
):
sym
,
positive
=
inspect_literal
(
literal
)
if
sym
in
model
:
if
model
[
sym
]
==
positive
:
return
None
,
None
# clause already True
elif
P
:
return
None
,
None
# more than 1 unbound variable
else
:
P
,
value
=
sym
,
positive
return
P
,
value
def
inspect_literal
(
literal
):
"""The symbol in this literal, and the value it should take to
make the literal true.
>>> inspect_literal(P)
(P, True)
>>> inspect_literal(~P)
(P, False)
"""
if
literal
.
op
==
'~'
:
return
literal
.
args
[
0
],
False
else
:
return
literal
,
True
# ______________________________________________________________________________
# CDCL - Conflict-Driven Clause Learning with 1UIP Learning Scheme,
# 2WL Lazy Data Structure, VSIDS Branching Heuristic & Restarts
def
no_restart
(
conflicts
,
restarts
,
queue_lbd
,
sum_lbd
):
return
False
def
luby
(
conflicts
,
restarts
,
queue_lbd
,
sum_lbd
,
unit
=
512
):
# in the state-of-art tested with unit value 1, 2, 4, 6, 8, 12, 16, 32, 64, 128, 256 and 512
def
_luby
(
i
):
k
=
1
while
True
:
if
i
==
(
1
<<
k
)
-
1
:
return
1
<<
(
k
-
1
)
elif
(
1
<<
(
k
-
1
))
<=
i
<
(
1
<<
k
)
-
1
:
return
_luby
(
i
-
(
1
<<
(
k
-
1
))
+
1
)
k
+=
1
return
unit
*
_luby
(
restarts
)
==
len
(
queue_lbd
)
def
glucose
(
conflicts
,
restarts
,
queue_lbd
,
sum_lbd
,
x
=
100
,
k
=
0.7
):
# in the state-of-art tested with (x, k) as (50, 0.8) and (100, 0.7)
# if there were at least x conflicts since the last restart, and then the average LBD of the last
# x learnt clauses was at least k times higher than the average LBD of all learnt clauses
return
len
(
queue_lbd
)
>=
x
and
sum
(
queue_lbd
)
/
len
(
queue_lbd
)
*
k
>
sum_lbd
/
conflicts
def
cdcl_satisfiable
(
s
,
vsids_decay
=
0.95
,
restart_strategy
=
no_restart
):
"""
>>> cdcl_satisfiable(A |'<=>'| B) == {A: True, B: True}
True
"""
clauses
=
TwoWLClauseDatabase
(
conjuncts
(
to_cnf
(
s
)))
symbols
=
prop_symbols
(
s
)
scores
=
Counter
()
G
=
nx
.
DiGraph
()
model
=
{}
dl
=
0
conflicts
=
0
restarts
=
1
sum_lbd
=
0
queue_lbd
=
[]
while
True
:
conflict
=
unit_propagation
(
clauses
,
symbols
,
model
,
G
,
dl
)
if
conflict
:
if
dl
==
0
:
return
False
conflicts
+=
1
dl
,
learn
,
lbd
=
conflict_analysis
(
G
,
dl
)
queue_lbd
.
append
(
lbd
)
sum_lbd
+=
lbd
backjump
(
symbols
,
model
,
G
,
dl
)
clauses
.
add
(
learn
,
model
)
scores
.
update
(
l
for
l
in
disjuncts
(
learn
))
for
symbol
in
scores
:
scores
[
symbol
]
*=
vsids_decay
if
restart_strategy
(
conflicts
,
restarts
,
queue_lbd
,
sum_lbd
):
backjump
(
symbols
,
model
,
G
)
queue_lbd
.
clear
()
restarts
+=
1
else
:
if
not
symbols
:
return
model
dl
+=
1
assign_decision_literal
(
symbols
,
model
,
scores
,
G
,
dl
)
def
assign_decision_literal
(
symbols
,
model
,
scores
,
G
,
dl
):
P
=
max
(
symbols
,
key
=
lambda
symbol
:
scores
[
symbol
]
+
scores
[
~
symbol
])
value
=
True
if
scores
[
P
]
>=
scores
[
~
P
]
else
False
symbols
.
remove
(
P
)
model
[
P
]
=
value
G
.
add_node
(
P
,
val
=
value
,
dl
=
dl
)
def
unit_propagation
(
clauses
,
symbols
,
model
,
G
,
dl
):
def
check
(
c
):
if
not
model
or
clauses
.
get_first_watched
(
c
)
==
clauses
.
get_second_watched
(
c
):
return
True
w1
,
_
=
inspect_literal
(
clauses
.
get_first_watched
(
c
))
if
w1
in
model
:
return
c
in
(
clauses
.
get_neg_watched
(
w1
)
if
model
[
w1
]
else
clauses
.
get_pos_watched
(
w1
))
w2
,
_
=
inspect_literal
(
clauses
.
get_second_watched
(
c
))
if
w2
in
model
:
return
c
in
(
clauses
.
get_neg_watched
(
w2
)
if
model
[
w2
]
else
clauses
.
get_pos_watched
(
w2
))
def
unit_clause
(
watching
):
w
,
p
=
inspect_literal
(
watching
)
G
.
add_node
(
w
,
val
=
p
,
dl
=
dl
)
G
.
add_edges_from
(
zip
(
prop_symbols
(
c
)
-
{
w
},
itertools
.
cycle
([
w
])),
antecedent
=
c
)
symbols
.
remove
(
w
)
model
[
w
]
=
p
def
conflict_clause
(
c
):
G
.
add_edges_from
(
zip
(
prop_symbols
(
c
),
itertools
.
cycle
(
'K'
)),
antecedent
=
c
)
while
True
:
bcp
=
False
for
c
in
filter
(
check
,
clauses
.
get_clauses
()):
# we need only visit each clause when one of its two watched literals is assigned to 0 because, until
# this happens, we can guarantee that there cannot be more than n-2 literals in the clause assigned to 0
first_watched
=
pl_true
(
clauses
.
get_first_watched
(
c
),
model
)
second_watched
=
pl_true
(
clauses
.
get_second_watched
(
c
),
model
)
if
first_watched
is
None
and
clauses
.
get_first_watched
(
c
)
==
clauses
.
get_second_watched
(
c
):
unit_clause
(
clauses
.
get_first_watched
(
c
))
bcp
=
True
break
elif
first_watched
is
False
and
second_watched
is
not
True
:
if
clauses
.
update_second_watched
(
c
,
model
):
bcp
=
True
else
:
# if the only literal with a non-zero value is the other watched literal then
if
second_watched
is
None
:
# if it is free, then the clause is a unit clause
unit_clause
(
clauses
.
get_second_watched
(
c
))
bcp
=
True
break
else
:
# else (it is False) the clause is a conflict clause
conflict_clause
(
c
)
return
True
elif
second_watched
is
False
and
first_watched
is
not
True
:
if
clauses
.
update_first_watched
(
c
,
model
):
bcp
=
True
else
:
# if the only literal with a non-zero value is the other watched literal then
if
first_watched
is
None
:
# if it is free, then the clause is a unit clause
unit_clause
(
clauses
.
get_first_watched
(
c
))
bcp
=
True
break
else
:
# else (it is False) the clause is a conflict clause
conflict_clause
(
c
)
return
True
if
not
bcp
:
return
False
def
conflict_analysis
(
G
,
dl
):
conflict_clause
=
next
(
G
[
p
][
'K'
][
'antecedent'
]
for
p
in
G
.
pred
[
'K'
])
P
=
next
(
node
for
node
in
G
.
nodes
()
-
'K'
if
G
.
nodes
[
node
][
'dl'
]
==
dl
and
G
.
in_degree
(
node
)
==
0
)
first_uip
=
nx
.
immediate_dominators
(
G
,
P
)[
'K'
]
G
.
remove_node
(
'K'
)
conflict_side
=
nx
.
descendants
(
G
,
first_uip
)
while
True
:
for
l
in
prop_symbols
(
conflict_clause
).
intersection
(
conflict_side
):
antecedent
=
next
(
G
[
p
][
l
][
'antecedent'
]
for
p
in
G
.
pred
[
l
])
conflict_clause
=
pl_binary_resolution
(
conflict_clause
,
antecedent
)
# the literal block distance is calculated by taking the decision levels from variables of all
# literals in the clause, and counting how many different decision levels were in this set
lbd
=
[
G
.
nodes
[
l
][
'dl'
]
for
l
in
prop_symbols
(
conflict_clause
)]
if
lbd
.
count
(
dl
)
==
1
and
first_uip
in
prop_symbols
(
conflict_clause
):
return
0
if
len
(
lbd
)
==
1
else
heapq
.
nlargest
(
2
,
lbd
)[
-
1
],
conflict_clause
,
len
(
set
(
lbd
))
def
pl_binary_resolution
(
ci
,
cj
):
for
di
in
disjuncts
(
ci
):
for
dj
in
disjuncts
(
cj
):
if
di
==
~
dj
or
~
di
==
dj
:
return
pl_binary_resolution
(
associate
(
'|'
,
remove_all
(
di
,
disjuncts
(
ci
))),
associate
(
'|'
,
remove_all
(
dj
,
disjuncts
(
cj
))))
return
associate
(
'|'
,
unique
(
disjuncts
(
ci
)
+
disjuncts
(
cj
)))
def
backjump
(
symbols
,
model
,
G
,
dl
=
0
):
delete
=
{
node
for
node
in
G
.
nodes
()
if
G
.
nodes
[
node
][
'dl'
]
>
dl
}
G
.
remove_nodes_from
(
delete
)
for
node
in
delete
:
del
model
[
node
]
symbols
|=
delete
class
TwoWLClauseDatabase
:
def
__init__
(
self
,
clauses
):
self
.
__twl
=
{}
self
.
__watch_list
=
defaultdict
(
lambda
: [
set
(),
set
()])
for
c
in
clauses
:
self
.
add
(
c
,
None
)
def
get_clauses
(
self
):
return
self
.
__twl
.
keys
()
def
set_first_watched
(
self
,
clause
,
new_watching
):
if
len
(
clause
.
args
)
>
2
:
self
.
__twl
[
clause
][
0
]
=
new_watching
def
set_second_watched
(
self
,
clause
,
new_watching
):
if
len
(
clause
.
args
)
>
2
:
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