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"""Knowledge in learning (Chapter 19)"""
from
collections
import
defaultdict
from
functools
import
partial
from
itertools
import
combinations
,
product
from
random
import
shuffle
import
numpy
as
np
from
logic
import
(
FolKB
,
constant_symbols
,
predicate_symbols
,
standardize_variables
,
variables
,
is_definite_clause
,
subst
,
expr
,
Expr
)
from
utils
import
power_set
def
current_best_learning
(
examples
,
h
,
examples_so_far
=
None
):
"""
[Figure 19.2]
The hypothesis is a list of dictionaries, with each dictionary representing
a disjunction.
"""
if
examples_so_far
is
None
:
examples_so_far
=
[]
if
not
examples
:
return
h
e
=
examples
[
0
]
if
is_consistent
(
e
,
h
):
return
current_best_learning
(
examples
[
1
:],
h
,
examples_so_far
+
[
e
])
elif
false_positive
(
e
,
h
):
for
h2
in
specializations
(
examples_so_far
+
[
e
],
h
):
h3
=
current_best_learning
(
examples
[
1
:],
h2
,
examples_so_far
+
[
e
])
if
h3
!=
'FAIL'
:
return
h3
elif
false_negative
(
e
,
h
):
for
h2
in
generalizations
(
examples_so_far
+
[
e
],
h
):
h3
=
current_best_learning
(
examples
[
1
:],
h2
,
examples_so_far
+
[
e
])
if
h3
!=
'FAIL'
:
return
h3
return
'FAIL'
def
specializations
(
examples_so_far
,
h
):
"""Specialize the hypothesis by adding AND operations to the disjunctions"""
hypotheses
=
[]
for
i
,
disj
in
enumerate
(
h
):
for
e
in
examples_so_far
:
for
k
,
v
in
e
.
items
():
if
k
in
disj
or
k
==
'GOAL'
:
continue
h2
=
h
[
i
].
copy
()
h2
[
k
]
=
'!'
+
v
h3
=
h
.
copy
()
h3
[
i
]
=
h2
if
check_all_consistency
(
examples_so_far
,
h3
):
hypotheses
.
append
(
h3
)
shuffle
(
hypotheses
)
return
hypotheses
def
generalizations
(
examples_so_far
,
h
):
"""Generalize the hypothesis. First delete operations
(including disjunctions) from the hypothesis. Then, add OR operations."""
hypotheses
=
[]
# Delete disjunctions
disj_powerset
=
power_set
(
range
(
len
(
h
)))
for
disjs
in
disj_powerset
:
h2
=
h
.
copy
()
for
d
in
reversed
(
list
(
disjs
)):
del
h2
[
d
]
if
check_all_consistency
(
examples_so_far
,
h2
):
hypotheses
+=
h2
# Delete AND operations in disjunctions
for
i
,
disj
in
enumerate
(
h
):
a_powerset
=
power_set
(
disj
.
keys
())
for
attrs
in
a_powerset
:
h2
=
h
[
i
].
copy
()
for
a
in
attrs
:
del
h2
[
a
]
if
check_all_consistency
(
examples_so_far
, [
h2
]):
h3
=
h
.
copy
()
h3
[
i
]
=
h2
.
copy
()
hypotheses
+=
h3
# Add OR operations
if
hypotheses
==
[]
or
hypotheses
==
[{}]:
hypotheses
=
add_or
(
examples_so_far
,
h
)
else
:
hypotheses
.
extend
(
add_or
(
examples_so_far
,
h
))
shuffle
(
hypotheses
)
return
hypotheses
def
add_or
(
examples_so_far
,
h
):
"""Add an OR operation to the hypothesis. The AND operations in the disjunction
are generated by the last example (which is the problematic one)."""
ors
=
[]
e
=
examples_so_far
[
-
1
]
attrs
=
{
k
:
v
for
k
,
v
in
e
.
items
()
if
k
!=
'GOAL'
}
a_powerset
=
power_set
(
attrs
.
keys
())
for
c
in
a_powerset
:
h2
=
{}
for
k
in
c
:
h2
[
k
]
=
attrs
[
k
]
if
check_negative_consistency
(
examples_so_far
,
h2
):
h3
=
h
.
copy
()
h3
.
append
(
h2
)
ors
.
append
(
h3
)
return
ors
# ______________________________________________________________________________
def
version_space_learning
(
examples
):
"""
[Figure 19.3]
The version space is a list of hypotheses, which in turn are a list
of dictionaries/disjunctions.
"""
V
=
all_hypotheses
(
examples
)
for
e
in
examples
:
if
V
:
V
=
version_space_update
(
V
,
e
)
return
V
def
version_space_update
(
V
,
e
):
return
[
h
for
h
in
V
if
is_consistent
(
e
,
h
)]
def
all_hypotheses
(
examples
):
"""Build a list of all the possible hypotheses"""
values
=
values_table
(
examples
)
h_powerset
=
power_set
(
values
.
keys
())
hypotheses
=
[]
for
s
in
h_powerset
:
hypotheses
.
extend
(
build_attr_combinations
(
s
,
values
))
hypotheses
.
extend
(
build_h_combinations
(
hypotheses
))
return
hypotheses
def
values_table
(
examples
):
"""Build a table with all the possible values for each attribute.
Returns a dictionary with keys the attribute names and values a list
with the possible values for the corresponding attribute."""
values
=
defaultdict
(
lambda
: [])
for
e
in
examples
:
for
k
,
v
in
e
.
items
():
if
k
==
'GOAL'
:
continue
mod
=
'!'
if
e
[
'GOAL'
]:
mod
=
''
if
mod
+
v
not
in
values
[
k
]:
values
[
k
].
append
(
mod
+
v
)
values
=
dict
(
values
)
return
values
def
build_attr_combinations
(
s
,
values
):
"""Given a set of attributes, builds all the combinations of values.
If the set holds more than one attribute, recursively builds the
combinations."""
if
len
(
s
)
==
1
:
# s holds just one attribute, return its list of values
k
=
values
[
s
[
0
]]
h
=
[[{
s
[
0
]:
v
}]
for
v
in
values
[
s
[
0
]]]
return
h
h
=
[]
for
i
,
a
in
enumerate
(
s
):
rest
=
build_attr_combinations
(
s
[
i
+
1
:],
values
)
for
v
in
values
[
a
]:
o
=
{
a
:
v
}
for
r
in
rest
:
t
=
o
.
copy
()
for
d
in
r
:
t
.
update
(
d
)
h
.
append
([
t
])
return
h
def
build_h_combinations
(
hypotheses
):
"""Given a set of hypotheses, builds and returns all the combinations of the
hypotheses."""
h
=
[]
h_powerset
=
power_set
(
range
(
len
(
hypotheses
)))
for
s
in
h_powerset
:
t
=
[]
for
i
in
s
:
t
.
extend
(
hypotheses
[
i
])
h
.
append
(
t
)
return
h
# ______________________________________________________________________________
def
minimal_consistent_det
(
E
,
A
):
"""Return a minimal set of attributes which give consistent determination"""
n
=
len
(
A
)
for
i
in
range
(
n
+
1
):
for
A_i
in
combinations
(
A
,
i
):
if
consistent_det
(
A_i
,
E
):
return
set
(
A_i
)
def
consistent_det
(
A
,
E
):
"""Check if the attributes(A) is consistent with the examples(E)"""
H
=
{}
for
e
in
E
:
attr_values
=
tuple
(
e
[
attr
]
for
attr
in
A
)
if
attr_values
in
H
and
H
[
attr_values
]
!=
e
[
'GOAL'
]:
return
False
H
[
attr_values
]
=
e
[
'GOAL'
]
return
True
# ______________________________________________________________________________
class
FOILContainer
(
FolKB
):
"""Hold the kb and other necessary elements required by FOIL."""
def
__init__
(
self
,
clauses
=
None
):
self
.
const_syms
=
set
()
self
.
pred_syms
=
set
()
super
().
__init__
(
clauses
)
def
tell
(
self
,
sentence
):
if
is_definite_clause
(
sentence
):
self
.
clauses
.
append
(
sentence
)
self
.
const_syms
.
update
(
constant_symbols
(
sentence
))
self
.
pred_syms
.
update
(
predicate_symbols
(
sentence
))
else
:
raise
Exception
(
'Not a definite clause: {}'
.
format
(
sentence
))
def
foil
(
self
,
examples
,
target
):
"""Learn a list of first-order horn clauses
'examples' is a tuple: (positive_examples, negative_examples).
positive_examples and negative_examples are both lists which contain substitutions."""
clauses
=
[]
pos_examples
=
examples
[
0
]
neg_examples
=
examples
[
1
]
while
pos_examples
:
clause
,
extended_pos_examples
=
self
.
new_clause
((
pos_examples
,
neg_examples
),
target
)
# remove positive examples covered by clause
pos_examples
=
self
.
update_examples
(
target
,
pos_examples
,
extended_pos_examples
)
clauses
.
append
(
clause
)
return
clauses
def
new_clause
(
self
,
examples
,
target
):
"""Find a horn clause which satisfies part of the positive
examples but none of the negative examples.
The horn clause is specified as [consequent, list of antecedents]
Return value is the tuple (horn_clause, extended_positive_examples)."""
clause
=
[
target
, []]
extended_examples
=
examples
while
extended_examples
[
1
]:
l
=
self
.
choose_literal
(
self
.
new_literals
(
clause
),
extended_examples
)
clause
[
1
].
append
(
l
)
extended_examples
=
[
sum
([
list
(
self
.
extend_example
(
example
,
l
))
for
example
in
extended_examples
[
i
]], [])
for
i
in
range
(
2
)]
return
clause
,
extended_examples
[
0
]
def
extend_example
(
self
,
example
,
literal
):
"""Generate extended examples which satisfy the literal."""
# find all substitutions that satisfy literal
for
s
in
self
.
ask_generator
(
subst
(
example
,
literal
)):
s
.
update
(
example
)
yield
s
def
new_literals
(
self
,
clause
):
"""Generate new literals based on known predicate symbols.
Generated literal must share at least one variable with clause"""
share_vars
=
variables
(
clause
[
0
])
for
l
in
clause
[
1
]:
share_vars
.
update
(
variables
(
l
))
for
pred
,
arity
in
self
.
pred_syms
:
new_vars
=
{
standardize_variables
(
expr
(
'x'
))
for
_
in
range
(
arity
-
1
)}
for
args
in
product
(
share_vars
.
union
(
new_vars
),
repeat
=
arity
):
if
any
(
var
in
share_vars
for
var
in
args
):
# make sure we don't return an existing rule
if
not
Expr
(
pred
,
args
)
in
clause
[
1
]:
yield
Expr
(
pred
,
*
[
var
for
var
in
args
])
def
choose_literal
(
self
,
literals
,
examples
):
"""Choose the best literal based on the information gain."""
return
max
(
literals
,
key
=
partial
(
self
.
gain
,
examples
=
examples
))
def
gain
(
self
,
l
,
examples
):
"""
Find the utility of each literal when added to the body of the clause.
Utility function is:
gain(R, l) = T * (log_2 (post_pos / (post_pos + post_neg)) - log_2 (pre_pos / (pre_pos + pre_neg)))
where:
pre_pos = number of possitive bindings of rule R (=current set of rules)
pre_neg = number of negative bindings of rule R
post_pos = number of possitive bindings of rule R' (= R U {l} )
post_neg = number of negative bindings of rule R'
T = number of possitive bindings of rule R that are still covered
after adding literal l
"""
pre_pos
=
len
(
examples
[
0
])
pre_neg
=
len
(
examples
[
1
])
post_pos
=
sum
([
list
(
self
.
extend_example
(
example
,
l
))
for
example
in
examples
[
0
]], [])
post_neg
=
sum
([
list
(
self
.
extend_example
(
example
,
l
))
for
example
in
examples
[
1
]], [])
if
pre_pos
+
pre_neg
==
0
or
len
(
post_pos
)
+
len
(
post_neg
)
==
0
:
return
-
1
# number of positive example that are represented in extended_examples
T
=
0
for
example
in
examples
[
0
]:
represents
=
lambda
d
:
all
(
d
[
x
]
==
example
[
x
]
for
x
in
example
)
if
any
(
represents
(
l_
)
for
l_
in
post_pos
):
T
+=
1
value
=
T
*
(
np
.
log2
(
len
(
post_pos
)
/
(
len
(
post_pos
)
+
len
(
post_neg
))
+
1e-12
)
-
np
.
log2
(
pre_pos
/
(
pre_pos
+
pre_neg
)))
return
value
def
update_examples
(
self
,
target
,
examples
,
extended_examples
):
"""Add to the kb those examples what are represented in extended_examples
List of omitted examples is returned."""
uncovered
=
[]
for
example
in
examples
:
represents
=
lambda
d
:
all
(
d
[
x
]
==
example
[
x
]
for
x
in
example
)
if
any
(
represents
(
l
)
for
l
in
extended_examples
):
self
.
tell
(
subst
(
example
,
target
))
else
:
uncovered
.
append
(
example
)
return
uncovered
# ______________________________________________________________________________
def
check_all_consistency
(
examples
,
h
):
"""Check for the consistency of all examples under h."""
for
e
in
examples
:
if
not
is_consistent
(
e
,
h
):
return
False
return
True
def
check_negative_consistency
(
examples
,
h
):
"""Check if the negative examples are consistent under h."""
for
e
in
examples
:
if
e
[
'GOAL'
]:
continue
if
not
is_consistent
(
e
, [
h
]):
return
False
return
True
def
disjunction_value
(
e
,
d
):
"""The value of example e under disjunction d."""
for
k
,
v
in
d
.
items
():
if
v
[
0
]
==
'!'
:
# v is a NOT expression
# e[k], thus, should not be equal to v
if
e
[
k
]
==
v
[
1
:]:
return
False
elif
e
[
k
]
!=
v
:
return
False
return
True
def
guess_value
(
e
,
h
):
"""Guess value of example e under hypothesis h."""
for
d
in
h
:
if
disjunction_value
(
e
,
d
):
return
True
return
False
def
is_consistent
(
e
,
h
):
return
e
[
'GOAL'
]
==
guess_value
(
e
,
h
)
def
false_positive
(
e
,
h
):
return
guess_value
(
e
,
h
)
and
not
e
[
'GOAL'
]
def
false_negative
(
e
,
h
):
return
e
[
'GOAL'
]
and
not
guess_value
(
e
,
h
)
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