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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
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.
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 (not yet implemented)
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
itertools
,
re
import
agents
from
utils
import
*
#______________________________________________________________________________
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
):
abstract
def
tell
(
self
,
sentence
):
"Add the sentence to the KB."
abstract
def
ask
(
self
,
query
):
"""Return a substitution that makes the query true, or,
failing that, return False."""
for
result
in
self
.
ask_generator
(
query
):
return
result
return
False
def
ask_generator
(
self
,
query
):
"Yield all the substitutions that make query true."
abstract
def
retract
(
self
,
sentence
):
"Remove sentence from the KB."
abstract
class
PropKB
(
KB
):
"A KB for propositional logic. Inefficient, with no indexing."
def
__init__
(
self
,
sentence
=
None
):
self
.
clauses
=
[]
if
sentence
:
self
.
tell
(
sentence
)
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 implies query; else nothing."
if
tt_entails
(
Expr
(
'&'
,
*
self
.
clauses
),
query
):
yield
{}
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
KB_AgentProgram
(
KB
):
"""A generic logical knowledge-based agent program. [Fig. 7.1]"""
steps
=
itertools
.
count
()
def
program
(
percept
):
t
=
steps
.
next
()
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
(
self
,
percept
,
t
):
return
Expr
(
"Percept"
)(
percept
,
t
)
def
make_action_query
(
self
,
t
):
return
expr
(
"ShouldDo(action, %d)"
%
t
)
def
make_action_sentence
(
self
,
action
,
t
):
return
Expr
(
"Did"
)(
action
[
expr
(
'action'
)],
t
)
return
program
#______________________________________________________________________________
class
Expr
:
"""A symbolic mathematical expression. We use this class for logical
expressions, and for terms within logical expressions. In general, an
Expr has an op (operator) and a list of args. The op can be:
Null-ary (no args) op:
A number, representing the number itself. (e.g. Expr(42) => 42)
A symbol, representing a variable or constant (e.g. Expr('F') => F)
Unary (1 arg) op:
'~', '-', representing NOT, negation (e.g. Expr('~', Expr('P')) => ~P)
Binary (2 arg) op:
'>>', '<<', representing forward and backward implication
'+', '-', '*', '/', '**', representing arithmetic operators
'<', '>', '>=', '<=', representing comparison operators
'<=>', '^', representing logical equality and XOR
N-ary (0 or more args) op:
'&', '|', representing conjunction and disjunction
A symbol, representing a function term or FOL proposition
Exprs can be constructed with operator overloading: if x and y are Exprs,
then so are x + y and x & y, etc. Also, if F and x are Exprs, then so is
F(x); it works by overloading the __call__ method of the Expr F. Note
that in the Expr that is created by F(x), the op is the str 'F', not the
Expr F. See http://www.python.org/doc/current/ref/specialnames.html
to learn more about operator overloading in Python.
WARNING: x == y and x != y are NOT Exprs. The reason is that we want
to write code that tests 'if x == y:' and if x == y were the same
as Expr('==', x, y), then the result would always be true; not what a
programmer would expect. But we still need to form Exprs representing
equalities and disequalities. We concentrate on logical equality (or
equivalence) and logical disequality (or XOR). You have 3 choices:
(1) Expr('<=>', x, y) and Expr('^', x, y)
Note that ^ is bitwose XOR in Python (and Java and C++)
(2) expr('x <=> y') and expr('x =/= y').
See the doc string for the function expr.
(3) (x % y) and (x ^ y).
It is very ugly to have (x % y) mean (x <=> y), but we need
SOME operator to make (2) work, and this seems the best choice.
WARNING: if x is an Expr, then so is x + 1, because the int 1 gets
coerced to an Expr by the constructor. But 1 + x is an error, because
1 doesn't know how to add an Expr. (Adding an __radd__ method to Expr
wouldn't help, because int.__add__ is still called first.) Therefore,
you should use Expr(1) + x instead, or ONE + x, or expr('1 + x').
"""
def
__init__
(
self
,
op
,
*
args
):
"Op is a string or number; args are Exprs (or are coerced to Exprs)."
assert
isinstance
(
op
,
str
)
or
(
isnumber
(
op
)
and
not
args
)
self
.
op
=
num_or_str
(
op
)
self
.
args
=
map
(
expr
,
args
)
## Coerce args to Exprs
def
__call__
(
self
,
*
args
):
"""Self must be a symbol with no args, such as Expr('F'). Create a new
Expr with 'F' as op and the args as arguments."""
assert
is_symbol
(
self
.
op
)
and
not
self
.
args
return
Expr
(
self
.
op
,
*
args
)
def
__repr__
(
self
):
"Show something like 'P' or 'P(x, y)', or '~P' or '(P | Q | R)'"
if
not
self
.
args
:
# Constant or proposition with arity 0
return
str
(
self
.
op
)
elif
is_symbol
(
self
.
op
):
# Functional or propositional operator
return
'%s(%s)'
%
(
self
.
op
,
', '
.
join
(
map
(
repr
,
self
.
args
)))
elif
len
(
self
.
args
)
==
1
:
# Prefix operator
return
self
.
op
+
repr
(
self
.
args
[
0
])
else
:
# Infix operator
return
'(%s)'
%
(
' '
+
self
.
op
+
' '
).
join
(
map
(
repr
,
self
.
args
))
def
__eq__
(
self
,
other
):
"""x and y are equal iff their ops and args are equal."""
return
(
other
is
self
)
or
(
isinstance
(
other
,
Expr
)
and
self
.
op
==
other
.
op
and
self
.
args
==
other
.
args
)
def
__ne__
(
self
,
other
):
return
not
self
.
__eq__
(
other
)
def
__hash__
(
self
):
"Need a hash method so Exprs can live in dicts."
return
hash
(
self
.
op
)
^
hash
(
tuple
(
self
.
args
))
# See http://www.python.org/doc/current/lib/module-operator.html
# Not implemented: not, abs, pos, concat, contains, *item, *slice
def
__lt__
(
self
,
other
):
return
Expr
(
'<'
,
self
,
other
)
def
__le__
(
self
,
other
):
return
Expr
(
'<='
,
self
,
other
)
def
__ge__
(
self
,
other
):
return
Expr
(
'>='
,
self
,
other
)
def
__gt__
(
self
,
other
):
return
Expr
(
'>'
,
self
,
other
)
def
__add__
(
self
,
other
):
return
Expr
(
'+'
,
self
,
other
)
def
__sub__
(
self
,
other
):
return
Expr
(
'-'
,
self
,
other
)
def
__and__
(
self
,
other
):
return
Expr
(
'&'
,
self
,
other
)
def
__div__
(
self
,
other
):
return
Expr
(
'/'
,
self
,
other
)
def
__truediv__
(
self
,
other
):
return
Expr
(
'/'
,
self
,
other
)
def
__invert__
(
self
):
return
Expr
(
'~'
,
self
)
def
__lshift__
(
self
,
other
):
return
Expr
(
'<<'
,
self
,
other
)
def
__rshift__
(
self
,
other
):
return
Expr
(
'>>'
,
self
,
other
)
def
__mul__
(
self
,
other
):
return
Expr
(
'*'
,
self
,
other
)
def
__neg__
(
self
):
return
Expr
(
'-'
,
self
)
def
__or__
(
self
,
other
):
return
Expr
(
'|'
,
self
,
other
)
def
__pow__
(
self
,
other
):
return
Expr
(
'**'
,
self
,
other
)
def
__xor__
(
self
,
other
):
return
Expr
(
'^'
,
self
,
other
)
def
__mod__
(
self
,
other
):
return
Expr
(
'<=>'
,
self
,
other
)
def
expr
(
s
):
"""Create an Expr representing a logic expression by parsing the input
string. Symbols and numbers are automatically converted to Exprs.
In addition you can use alternative spellings of these operators:
'x ==> y' parses as (x >> y) # Implication
'x <== y' parses as (x << y) # Reverse implication
'x <=> y' parses as (x % y) # Logical equivalence
'x =/= y' parses as (x ^ y) # Logical disequality (xor)
But BE CAREFUL; precedence of implication is wrong. expr('P & Q ==> R & S')
is ((P & (Q >> R)) & S); so you must use expr('(P & Q) ==> (R & S)').
>>> expr('P <=> Q(1)')
(P <=> Q(1))
>>> expr('P & Q | ~R(x, F(x))')
((P & Q) | ~R(x, F(x)))
"""
if
isinstance
(
s
,
Expr
):
return
s
if
isnumber
(
s
):
return
Expr
(
s
)
## Replace the alternative spellings of operators with canonical spellings
s
=
s
.
replace
(
'==>'
,
'>>'
).
replace
(
'<=='
,
'<<'
)
s
=
s
.
replace
(
'<=>'
,
'%'
).
replace
(
'=/='
,
'^'
)
## Replace a symbol or number, such as 'P' with 'Expr("P")'
s
=
re
.
sub
(
r'([a-zA-Z0-9_.]+)'
,
r'Expr("\1")'
,
s
)
## Now eval the string. (A security hole; do not use with an adversary.)
return
eval
(
s
, {
'Expr'
:
Expr
})
def
is_symbol
(
s
):
"A string s is a symbol if it starts with an alphabetic char."
return
isinstance
(
s
,
str
)
and
s
[:
1
].
isalpha
()
def
is_var_symbol
(
s
):
"A logic variable symbol is an initial-lowercase string."
return
is_symbol
(
s
)
and
s
[
0
].
islower
()
def
is_prop_symbol
(
s
):
"""A proposition logic symbol is an initial-uppercase string other than
TRUE or FALSE."""
return
is_symbol
(
s
)
and
s
[
0
].
isupper
()
and
s
!=
'TRUE'
and
s
!=
'FALSE'
def
variables
(
s
):
"""Return a set of the variables in expression s.
>>> ppset(variables(F(x, A, y)))
set([x, y])
>>> ppset(variables(F(G(x), z)))
set([x, z])
>>> ppset(variables(expr('F(x, x) & G(x, y) & H(y, z) & R(A, z, z)')))
set([x, y, z])
"""
result
=
set
([])
def
walk
(
s
):
if
is_variable
(
s
):
result
.
add
(
s
)
else
:
for
arg
in
s
.
args
:
walk
(
arg
)
walk
(
s
)
return
result
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
>>> is_definite_clause(expr('~Farmer(Mac)'))
False
>>> is_definite_clause(expr('(Farmer(f) & Rabbit(r)) ==> Hates(f, r)'))
True
>>> is_definite_clause(expr('(Farmer(f) & ~Rabbit(r)) ==> Hates(f, r)'))
False
>>> is_definite_clause(expr('(Farmer(f) | Rabbit(r)) ==> Hates(f, r)'))
False
"""
if
is_symbol
(
s
.
op
):
return
True
elif
s
.
op
==
'>>'
:
antecedent
,
consequent
=
s
.
args
return
(
is_symbol
(
consequent
.
op
)
and
every
(
lambda
arg
:
is_symbol
(
arg
.
op
),
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:
TRUE
,
FALSE
,
ZERO
,
ONE
,
TWO
=
map
(
Expr
, [
'TRUE'
,
'FALSE'
,
0
,
1
,
2
])
A
,
B
,
C
,
D
,
E
,
F
,
G
,
P
,
Q
,
x
,
y
,
z
=
map
(
Expr
,
'ABCDEFGPQxyz'
)
#______________________________________________________________________________
def
tt_entails
(
kb
,
alpha
):
"""Does kb entail the sentence alpha? Use truth tables. For propositional
kb's and sentences. [Fig. 7.10]
>>> tt_entails(expr('P & Q'), expr('Q'))
True
"""
assert
not
variables
(
alpha
)
return
tt_check_all
(
kb
,
alpha
,
prop_symbols
(
kb
&
alpha
), {})
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 a list of all propositional symbols in x."
if
not
isinstance
(
x
,
Expr
):
return
[]
elif
is_prop_symbol
(
x
.
op
):
return
[
x
]
else
:
return
list
(
set
(
symbol
for
arg
in
x
.
args
for
symbol
in
prop_symbols
(
arg
)))
def
tt_true
(
alpha
):
"""Is the propositional sentence alpha a tautology? (alpha will be
coerced to an expr.)
>>> tt_true(expr("(P >> Q) <=> (~P | Q)"))
True
"""
return
tt_entails
(
TRUE
,
expr
(
alpha
))
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."""
op
,
args
=
exp
.
op
,
exp
.
args
if
exp
==
TRUE
:
return
True
elif
exp
==
FALSE
:
return
False
elif
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
==
'^'
:
return
pt
!=
qt
else
:
raise
ValueError
,
"illegal operator in logic expression"
+
str
(
exp
)
#______________________________________________________________________________
## Convert to Conjunctive Normal Form (CNF)
def
to_cnf
(
s
):
"""Convert a propositional logical sentence s to conjunctive normal form.
That is, to the form ((A | ~B | ...) & (B | C | ...) & ...) [p. 253]
>>> to_cnf("~(B|C)")
(~B & ~C)
>>> to_cnf("B <=> (P1|P2)")
((~P1 | B) & (~P2 | B) & (P1 | P2 | ~B))
>>> to_cnf("a | (b & c) | d")
((b | a | d) & (c | a | d))
>>> to_cnf("A & (B | (D & E))")
(A & (D | B) & (E | B))
>>> to_cnf("A | (B | (C | (D & E)))")
((D | A | B | C) & (E | A | B | C))
"""
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 >>, <<, and <=> into &, |, and ~. That is, return an Expr
that is equivalent to s, but has only &, |, and ~ as logical operators.
>>> eliminate_implications(A >> (~B << C))
((~B | ~C) | ~A)
>>> eliminate_implications(A ^ B)
((A & ~B) | (~A & B))
"""
if
not
s
.
args
or
is_symbol
(
s
.
op
):
return
s
## (Atoms are unchanged.)
args
=
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)
>>> move_not_inwards(~(A & B))
(~A | ~B)
>>> move_not_inwards(~(~(A | ~B) | ~~C))
((A | ~B) & ~C)
"""
if
s
.
op
==
'~'
:
NOT
=
lambda
b
:
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
(
'|'
,
map
(
NOT
,
a
.
args
))
if
a
.
op
==
'|'
:
return
associate
(
'&'
,
map
(
NOT
,
a
.
args
))
return
s
elif
is_symbol
(
s
.
op
)
or
not
s
.
args
:
return
s
else
:
return
Expr
(
s
.
op
,
*
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))
"""
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
=
find_if
((
lambda
d
:
d
.
op
==
'&'
),
s
.
args
)
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
(
'&'
,
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
,
'+'
:
ZERO
,
'*'
:
ONE
}
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)."""
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
):
"Propositional-logic resolution: say if alpha follows from KB. [Fig. 7.12]"
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.
>>> for res in pl_resolve(to_cnf(A|B|C), to_cnf(~B|~C|F)):
... ppset(disjuncts(res))
set([A, C, F, ~C])
set([A, B, F, ~B])
"""
clauses
=
[]
for
di
in
disjuncts
(
ci
):
for
dj
in
disjuncts
(
cj
):
if
di
==
~
dj
or
~
di
==
dj
:
dnew
=
unique
(
removeall
(
di
,
disjuncts
(
ci
))
+
removeall
(
dj
,
disjuncts
(
cj
)))
clauses
.
append
(
associate
(
'|'
,
dnew
))
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
):
"""Use forward chaining to see if a PropDefiniteKB entails symbol q.
[Fig. 7.15]
>>> pl_fc_entails(Fig[7,15], expr('Q'))
True
"""
count
=
dict
([(
c
,
len
(
conjuncts
(
c
.
args
[
0
])))
for
c
in
KB
.
clauses
if
c
.
op
==
'>>'
])
inferred
=
DefaultDict
(
False
)
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
## Wumpus World example [Fig. 7.13]
Fig
[
7
,
13
]
=
expr
(
"(B11 <=> (P12 | P21)) & ~B11"
)
## Propositional Logic Forward Chaining example [Fig. 7.16]
Fig
[
7
,
15
]
=
PropDefiniteKB
()
for
s
in
"P>>Q (L&M)>>P (B&L)>>M (A&P)>>L (A&B)>>L A B"
.
split
():
Fig
[
7
,
15
].
tell
(
expr
(
s
))
#______________________________________________________________________________
# DPLL-Satisfiable [Fig. 7.17]
def
dpll_satisfiable
(
s
):
"""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.
>>> ppsubst(dpll_satisfiable(A&~B))
{A: True, B: False}
>>> dpll_satisfiable(P&~P)
False
"""
clauses
=
conjuncts
(
to_cnf
(
s
))
symbols
=
prop_symbols
(
s
)
return
dpll
(
clauses
,
symbols
, {})
def
dpll
(
clauses
,
symbols
,
model
):
"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
==
False
:
return
False
if
val
!=
True
:
unknown_clauses
.
append
(
c
)
if
not
unknown_clauses
:
return
model
P
,
value
=
find_pure_symbol
(
symbols
,
unknown_clauses
)
if
P
:
return
dpll
(
clauses
,
removeall
(
P
,
symbols
),
extend
(
model
,
P
,
value
))
P
,
value
=
find_unit_clause
(
clauses
,
model
)
if
P
:
return
dpll
(
clauses
,
removeall
(
P
,
symbols
),
extend
(
model
,
P
,
value
))
P
,
symbols
=
symbols
[
0
],
symbols
[
1
:]
return
(
dpll
(
clauses
,
symbols
,
extend
(
model
,
P
,
True
))
or
dpll
(
clauses
,
symbols
,
extend
(
model
,
P
,
False
)))
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
#______________________________________________________________________________
# Walk-SAT [Fig. 7.18]
def
WalkSAT
(
clauses
,
p
=
0.5
,
max_flips
=
10000
):
## model is a random assignment of true/false to the symbols in clauses
## See ~/aima1e/print1/manual/knowledge+logic-answers.tex ???
model
=
dict
([(
s
,
random
.
choice
([
True
,
False
]))
for
s
in
prop_symbols
(
clauses
)])
for
i
in
range
(
max_flips
):
satisfied
,
unsatisfied
=
[], []
for
clause
in
clauses
:
if_
(
pl_true
(
clause
,
model
),
satisfied
,
unsatisfied
).
append
(
clause
)
if
not
unsatisfied
:
## if model satisfies all the clauses
return
model
clause
=
random
.
choice
(
unsatisfied
)
if
probability
(
p
):
sym
=
random
.
choice
(
prop_symbols
(
clause
))
else
:
## Flip the symbol in clause that maximizes number of sat. clauses
raise
NotImplementedError
model
[
sym
]
=
not
model
[
sym
]
#______________________________________________________________________________
class
HybridWumpusAgent
(
agents
.
Agent
):
"An agent for the wumpus world that does logical inference. [Fig. 7.19]"
""
def
__init__
(
self
):
unimplemented
()
def
plan_route
(
current
,
goals
,
allowed
):
unimplemented
()
#______________________________________________________________________________
def
SAT_plan
(
init
,
transition
,
goal
,
t_max
,
SAT_solver
=
dpll_satisfiable
):
"[Fig. 7.22]"
for
t
in
range
(
t_max
):
cnf
=
translate_to_SAT
(
init
,
transition
,
goal
,
t
)
model
=
SAT_solver
(
cnf
)
if
model
is
not
False
:
return
extract_solution
(
model
)
return
None
def
translate_to_SAT
(
init
,
transition
,
goal
,
t
):
unimplemented
()
def
extract_solution
(
model
):
unimplemented
()
#______________________________________________________________________________
def
unify
(
x
,
y
,
s
):
"""Unify expressions x,y with substitution s; return a substitution that
would make x,y equal, or None if x,y can not unify. x and y can be
variables (e.g. Expr('x')), constants, lists, or Exprs. [Fig. 9.1]
>>> ppsubst(unify(x + y, y + C, {}))
{x: y, y: C}
"""
if
s
is
None
:
return
None
elif
x
==
y
:
return
s
elif
is_variable
(
x
):
return
unify_var
(
x
,
y
,
s
)
elif
is_variable
(
y
):
return
unify_var
(
y
,
x
,
s
)
elif
isinstance
(
x
,
Expr
)
and
isinstance
(
y
,
Expr
):
return
unify
(
x
.
args
,
y
.
args
,
unify
(
x
.
op
,
y
.
op
,
s
))
elif
isinstance
(
x
,
str
)
or
isinstance
(
y
,
str
):
return
None
elif
issequence
(
x
)
and
issequence
(
y
)
and
len
(
x
)
==
len
(
y
):
if
not
x
:
return
s
return
unify
(
x
[
1
:],
y
[
1
:],
unify
(
x
[
0
],
y
[
0
],
s
))
else
:
return
None
def
is_variable
(
x
):
"A variable is an Expr with no args and a lowercase symbol as the op."
return
isinstance
(
x
,
Expr
)
and
not
x
.
args
and
is_var_symbol
(
x
.
op
)
def
unify_var
(
var
,
x
,
s
):
if
var
in
s
:
return
unify
(
s
[
var
],
x
,
s
)
elif
occur_check
(
var
,
x
,
s
):
return
None
else
:
return
extend
(
s
,
var
,
x
)
def
occur_check
(
var
,
x
,
s
):
"""Return true if variable var occurs anywhere in x
(or in subst(s, x), if s has a binding for x)."""
if
var
==
x
:
return
True
elif
is_variable
(
x
)
and
x
in
s
:
return
occur_check
(
var
,
s
[
x
],
s
)
elif
isinstance
(
x
,
Expr
):
return
(
occur_check
(
var
,
x
.
op
,
s
)
or
occur_check
(
var
,
x
.
args
,
s
))
elif
isinstance
(
x
, (
list
,
tuple
)):
return
some
(
lambda
element
:
occur_check
(
var
,
element
,
s
),
x
)
else
:
return
False
def
extend
(
s
,
var
,
val
):
"""Copy the substitution s and extend it by setting var to val;
return copy.
>>> ppsubst(extend({x: 1}, y, 2))
{x: 1, y: 2}
"""
s2
=
s
.
copy
()
s2
[
var
]
=
val
return
s2
def
subst
(
s
,
x
):
"""Substitute the substitution s into the expression x.
>>> subst({x: 42, y:0}, F(x) + y)
(F(42) + 0)
"""
if
isinstance
(
x
,
list
):
return
[
subst
(
s
,
xi
)
for
xi
in
x
]
elif
isinstance
(
x
,
tuple
):
return
tuple
([
subst
(
s
,
xi
)
for
xi
in
x
])
elif
not
isinstance
(
x
,
Expr
):
return
x
elif
is_var_symbol
(
x
.
op
):
return
s
.
get
(
x
,
x
)
else
:
return
Expr
(
x
.
op
,
*
[
subst
(
s
,
arg
)
for
arg
in
x
.
args
])
def
fol_fc_ask
(
KB
,
alpha
):
"""Inefficient forward chaining for first-order logic. [Fig. 9.3]
KB is a FolKB and alpha must be an atomic sentence."""
while
True
:
new
=
{}
for
r
in
KB
.
clauses
:
ps
,
q
=
parse_definite_clause
(
standardize_variables
(
r
))
raise
NotImplementedError
def
standardize_variables
(
sentence
,
dic
=
None
):
"""Replace all the variables in sentence with new variables.
>>> e = expr('F(a, b, c) & G(c, A, 23)')
>>> len(variables(standardize_variables(e)))
3
>>> variables(e).intersection(variables(standardize_variables(e)))
set([])
>>> is_variable(standardize_variables(expr('x')))
True
"""
if
dic
is
None
:
dic
=
{}
if
not
isinstance
(
sentence
,
Expr
):
return
sentence
elif
is_var_symbol
(
sentence
.
op
):
if
sentence
in
dic
:
return
dic
[
sentence
]
else
:
v
=
Expr
(
'v_%d'
%
standardize_variables
.
counter
.
next
())
dic
[
sentence
]
=
v
return
v
else
:
return
Expr
(
sentence
.
op
,
*
[
standardize_variables
(
a
,
dic
)
for
a
in
sentence
.
args
])
standardize_variables
.
counter
=
itertools
.
count
()
#______________________________________________________________________________
class
FolKB
(
KB
):
"""A knowledge base consisting of first-order definite clauses.
>>> kb0 = FolKB([expr('Farmer(Mac)'), expr('Rabbit(Pete)'),
... expr('(Rabbit(r) & Farmer(f)) ==> Hates(f, r)')])
>>> kb0.tell(expr('Rabbit(Flopsie)'))
>>> kb0.retract(expr('Rabbit(Pete)'))
>>> kb0.ask(expr('Hates(Mac, x)'))[x]
Flopsie
>>> kb0.ask(expr('Wife(Pete, x)'))
False
"""
def
__init__
(
self
,
initial_clauses
=
[]):
self
.
clauses
=
[]
# inefficient: no indexing
for
clause
in
initial_clauses
:
self
.
tell
(
clause
)
def
tell
(
self
,
sentence
):
if
is_definite_clause
(
sentence
):
self
.
clauses
.
append
(
sentence
)
else
:
raise
Exception
(
"Not a definite clause: %s"
%
sentence
)
def
ask_generator
(
self
,
query
):
return
fol_bc_ask
(
self
,
query
)
def
retract
(
self
,
sentence
):
self
.
clauses
.
remove
(
sentence
)
def
fetch_rules_for_goal
(
self
,
goal
):
return
self
.
clauses
def
test_ask
(
query
,
kb
=
None
):
q
=
expr
(
query
)
vars
=
variables
(
q
)
answers
=
fol_bc_ask
(
kb
or
test_kb
,
q
)
return
sorted
([
pretty
(
dict
((
x
,
v
)
for
x
,
v
in
a
.
items
()
if
x
in
vars
))
for
a
in
answers
],
key
=
repr
)
test_kb
=
FolKB
(
map
(
expr
, [
'Farmer(Mac)'
,
'Rabbit(Pete)'
,
'Mother(MrsMac, Mac)'
,
'Mother(MrsRabbit, Pete)'
,
'(Rabbit(r) & Farmer(f)) ==> Hates(f, r)'
,
'(Mother(m, c)) ==> Loves(m, c)'
,
'(Mother(m, r) & Rabbit(r)) ==> Rabbit(m)'
,
'(Farmer(f)) ==> Human(f)'
,
# Note that this order of conjuncts
# would result in infinite recursion:
#'(Human(h) & Mother(m, h)) ==> Human(m)'
'(Mother(m, h) & Human(h)) ==> Human(m)'
])
)
crime_kb
=
FolKB
(
map
(
expr
,
[
'(American(x) & Weapon(y) & Sells(x, y, z) & Hostile(z)) ==> Criminal(x)'
,
'Owns(Nono, M1)'
,
'Missile(M1)'
,
'(Missile(x) & Owns(Nono, x)) ==> Sells(West, x, Nono)'
,
'Missile(x) ==> Weapon(x)'
,
'Enemy(x, America) ==> Hostile(x)'
,
'American(West)'
,
'Enemy(Nono, America)'
])
)
def
fol_bc_ask
(
KB
,
query
):
"""A simple backward-chaining algorithm for first-order logic. [Fig. 9.6]
KB should be an instance of FolKB, and goals a list of literals.
>>> test_ask('Farmer(x)')
['{x: Mac}']
>>> test_ask('Human(x)')
['{x: Mac}', '{x: MrsMac}']
>>> test_ask('Hates(x, y)')
['{x: Mac, y: MrsRabbit}', '{x: Mac, y: Pete}']
>>> test_ask('Loves(x, y)')
['{x: MrsMac, y: Mac}', '{x: MrsRabbit, y: Pete}']
>>> test_ask('Rabbit(x)')
['{x: MrsRabbit}', '{x: Pete}']
>>> test_ask('Criminal(x)', crime_kb)
['{x: West}']
"""
return
fol_bc_or
(
KB
,
query
, {})
def
fol_bc_or
(
KB
,
goal
,
theta
):
for
rule
in
KB
.
fetch_rules_for_goal
(
goal
):
lhs
,
rhs
=
parse_definite_clause
(
standardize_variables
(
rule
))
for
theta1
in
fol_bc_and
(
KB
,
lhs
,
unify
(
rhs
,
goal
,
theta
)):
yield
theta1
def
fol_bc_and
(
KB
,
goals
,
theta
):
if
theta
is
None
:
pass
elif
not
goals
:
yield
theta
else
:
first
,
rest
=
goals
[
0
],
goals
[
1
:]
for
theta1
in
fol_bc_or
(
KB
,
subst
(
theta
,
first
),
theta
):
for
theta2
in
fol_bc_and
(
KB
,
rest
,
theta1
):
yield
theta2
#______________________________________________________________________________
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