You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
398 lines
14 KiB
Python
398 lines
14 KiB
Python
# Natural Language Toolkit: Logic
|
|
#
|
|
# Author: Peter Wang
|
|
# Updated by: Dan Garrette <dhgarrette@gmail.com>
|
|
#
|
|
# Copyright (C) 2001-2020 NLTK Project
|
|
# URL: <http://nltk.org>
|
|
# For license information, see LICENSE.TXT
|
|
|
|
"""
|
|
An implementation of the Hole Semantics model, following Blackburn and Bos,
|
|
Representation and Inference for Natural Language (CSLI, 2005).
|
|
|
|
The semantic representations are built by the grammar hole.fcfg.
|
|
This module contains driver code to read in sentences and parse them
|
|
according to a hole semantics grammar.
|
|
|
|
After parsing, the semantic representation is in the form of an underspecified
|
|
representation that is not easy to read. We use a "plugging" algorithm to
|
|
convert that representation into first-order logic formulas.
|
|
"""
|
|
|
|
from functools import reduce
|
|
|
|
from nltk.parse import load_parser
|
|
|
|
from nltk.sem.skolemize import skolemize
|
|
from nltk.sem.logic import (
|
|
AllExpression,
|
|
AndExpression,
|
|
ApplicationExpression,
|
|
ExistsExpression,
|
|
IffExpression,
|
|
ImpExpression,
|
|
LambdaExpression,
|
|
NegatedExpression,
|
|
OrExpression,
|
|
)
|
|
|
|
|
|
# Note that in this code there may be multiple types of trees being referred to:
|
|
#
|
|
# 1. parse trees
|
|
# 2. the underspecified representation
|
|
# 3. first-order logic formula trees
|
|
# 4. the search space when plugging (search tree)
|
|
#
|
|
|
|
|
|
class Constants(object):
|
|
ALL = "ALL"
|
|
EXISTS = "EXISTS"
|
|
NOT = "NOT"
|
|
AND = "AND"
|
|
OR = "OR"
|
|
IMP = "IMP"
|
|
IFF = "IFF"
|
|
PRED = "PRED"
|
|
LEQ = "LEQ"
|
|
HOLE = "HOLE"
|
|
LABEL = "LABEL"
|
|
|
|
MAP = {
|
|
ALL: lambda v, e: AllExpression(v.variable, e),
|
|
EXISTS: lambda v, e: ExistsExpression(v.variable, e),
|
|
NOT: NegatedExpression,
|
|
AND: AndExpression,
|
|
OR: OrExpression,
|
|
IMP: ImpExpression,
|
|
IFF: IffExpression,
|
|
PRED: ApplicationExpression,
|
|
}
|
|
|
|
|
|
class HoleSemantics(object):
|
|
"""
|
|
This class holds the broken-down components of a hole semantics, i.e. it
|
|
extracts the holes, labels, logic formula fragments and constraints out of
|
|
a big conjunction of such as produced by the hole semantics grammar. It
|
|
then provides some operations on the semantics dealing with holes, labels
|
|
and finding legal ways to plug holes with labels.
|
|
"""
|
|
|
|
def __init__(self, usr):
|
|
"""
|
|
Constructor. `usr' is a ``sem.Expression`` representing an
|
|
Underspecified Representation Structure (USR). A USR has the following
|
|
special predicates:
|
|
ALL(l,v,n),
|
|
EXISTS(l,v,n),
|
|
AND(l,n,n),
|
|
OR(l,n,n),
|
|
IMP(l,n,n),
|
|
IFF(l,n,n),
|
|
PRED(l,v,n,v[,v]*) where the brackets and star indicate zero or more repetitions,
|
|
LEQ(n,n),
|
|
HOLE(n),
|
|
LABEL(n)
|
|
where l is the label of the node described by the predicate, n is either
|
|
a label or a hole, and v is a variable.
|
|
"""
|
|
self.holes = set()
|
|
self.labels = set()
|
|
self.fragments = {} # mapping of label -> formula fragment
|
|
self.constraints = set() # set of Constraints
|
|
self._break_down(usr)
|
|
self.top_most_labels = self._find_top_most_labels()
|
|
self.top_hole = self._find_top_hole()
|
|
|
|
def is_node(self, x):
|
|
"""
|
|
Return true if x is a node (label or hole) in this semantic
|
|
representation.
|
|
"""
|
|
return x in (self.labels | self.holes)
|
|
|
|
def _break_down(self, usr):
|
|
"""
|
|
Extract holes, labels, formula fragments and constraints from the hole
|
|
semantics underspecified representation (USR).
|
|
"""
|
|
if isinstance(usr, AndExpression):
|
|
self._break_down(usr.first)
|
|
self._break_down(usr.second)
|
|
elif isinstance(usr, ApplicationExpression):
|
|
func, args = usr.uncurry()
|
|
if func.variable.name == Constants.LEQ:
|
|
self.constraints.add(Constraint(args[0], args[1]))
|
|
elif func.variable.name == Constants.HOLE:
|
|
self.holes.add(args[0])
|
|
elif func.variable.name == Constants.LABEL:
|
|
self.labels.add(args[0])
|
|
else:
|
|
label = args[0]
|
|
assert label not in self.fragments
|
|
self.fragments[label] = (func, args[1:])
|
|
else:
|
|
raise ValueError(usr.label())
|
|
|
|
def _find_top_nodes(self, node_list):
|
|
top_nodes = node_list.copy()
|
|
for f in self.fragments.values():
|
|
# the label is the first argument of the predicate
|
|
args = f[1]
|
|
for arg in args:
|
|
if arg in node_list:
|
|
top_nodes.discard(arg)
|
|
return top_nodes
|
|
|
|
def _find_top_most_labels(self):
|
|
"""
|
|
Return the set of labels which are not referenced directly as part of
|
|
another formula fragment. These will be the top-most labels for the
|
|
subtree that they are part of.
|
|
"""
|
|
return self._find_top_nodes(self.labels)
|
|
|
|
def _find_top_hole(self):
|
|
"""
|
|
Return the hole that will be the top of the formula tree.
|
|
"""
|
|
top_holes = self._find_top_nodes(self.holes)
|
|
assert len(top_holes) == 1 # it must be unique
|
|
return top_holes.pop()
|
|
|
|
def pluggings(self):
|
|
"""
|
|
Calculate and return all the legal pluggings (mappings of labels to
|
|
holes) of this semantics given the constraints.
|
|
"""
|
|
record = []
|
|
self._plug_nodes([(self.top_hole, [])], self.top_most_labels, {}, record)
|
|
return record
|
|
|
|
def _plug_nodes(self, queue, potential_labels, plug_acc, record):
|
|
"""
|
|
Plug the nodes in `queue' with the labels in `potential_labels'.
|
|
|
|
Each element of `queue' is a tuple of the node to plug and the list of
|
|
ancestor holes from the root of the graph to that node.
|
|
|
|
`potential_labels' is a set of the labels which are still available for
|
|
plugging.
|
|
|
|
`plug_acc' is the incomplete mapping of holes to labels made on the
|
|
current branch of the search tree so far.
|
|
|
|
`record' is a list of all the complete pluggings that we have found in
|
|
total so far. It is the only parameter that is destructively updated.
|
|
"""
|
|
if queue != []:
|
|
(node, ancestors) = queue[0]
|
|
if node in self.holes:
|
|
# The node is a hole, try to plug it.
|
|
self._plug_hole(
|
|
node, ancestors, queue[1:], potential_labels, plug_acc, record
|
|
)
|
|
else:
|
|
assert node in self.labels
|
|
# The node is a label. Replace it in the queue by the holes and
|
|
# labels in the formula fragment named by that label.
|
|
args = self.fragments[node][1]
|
|
head = [(a, ancestors) for a in args if self.is_node(a)]
|
|
self._plug_nodes(head + queue[1:], potential_labels, plug_acc, record)
|
|
else:
|
|
raise Exception("queue empty")
|
|
|
|
def _plug_hole(self, hole, ancestors0, queue, potential_labels0, plug_acc0, record):
|
|
"""
|
|
Try all possible ways of plugging a single hole.
|
|
See _plug_nodes for the meanings of the parameters.
|
|
"""
|
|
# Add the current hole we're trying to plug into the list of ancestors.
|
|
assert hole not in ancestors0
|
|
ancestors = [hole] + ancestors0
|
|
|
|
# Try each potential label in this hole in turn.
|
|
for l in potential_labels0:
|
|
# Is the label valid in this hole?
|
|
if self._violates_constraints(l, ancestors):
|
|
continue
|
|
|
|
plug_acc = plug_acc0.copy()
|
|
plug_acc[hole] = l
|
|
potential_labels = potential_labels0.copy()
|
|
potential_labels.remove(l)
|
|
|
|
if len(potential_labels) == 0:
|
|
# No more potential labels. That must mean all the holes have
|
|
# been filled so we have found a legal plugging so remember it.
|
|
#
|
|
# Note that the queue might not be empty because there might
|
|
# be labels on there that point to formula fragments with
|
|
# no holes in them. _sanity_check_plugging will make sure
|
|
# all holes are filled.
|
|
self._sanity_check_plugging(plug_acc, self.top_hole, [])
|
|
record.append(plug_acc)
|
|
else:
|
|
# Recursively try to fill in the rest of the holes in the
|
|
# queue. The label we just plugged into the hole could have
|
|
# holes of its own so at the end of the queue. Putting it on
|
|
# the end of the queue gives us a breadth-first search, so that
|
|
# all the holes at level i of the formula tree are filled
|
|
# before filling level i+1.
|
|
# A depth-first search would work as well since the trees must
|
|
# be finite but the bookkeeping would be harder.
|
|
self._plug_nodes(
|
|
queue + [(l, ancestors)], potential_labels, plug_acc, record
|
|
)
|
|
|
|
def _violates_constraints(self, label, ancestors):
|
|
"""
|
|
Return True if the `label' cannot be placed underneath the holes given
|
|
by the set `ancestors' because it would violate the constraints imposed
|
|
on it.
|
|
"""
|
|
for c in self.constraints:
|
|
if c.lhs == label:
|
|
if c.rhs not in ancestors:
|
|
return True
|
|
return False
|
|
|
|
def _sanity_check_plugging(self, plugging, node, ancestors):
|
|
"""
|
|
Make sure that a given plugging is legal. We recursively go through
|
|
each node and make sure that no constraints are violated.
|
|
We also check that all holes have been filled.
|
|
"""
|
|
if node in self.holes:
|
|
ancestors = [node] + ancestors
|
|
label = plugging[node]
|
|
else:
|
|
label = node
|
|
assert label in self.labels
|
|
for c in self.constraints:
|
|
if c.lhs == label:
|
|
assert c.rhs in ancestors
|
|
args = self.fragments[label][1]
|
|
for arg in args:
|
|
if self.is_node(arg):
|
|
self._sanity_check_plugging(plugging, arg, [label] + ancestors)
|
|
|
|
def formula_tree(self, plugging):
|
|
"""
|
|
Return the first-order logic formula tree for this underspecified
|
|
representation using the plugging given.
|
|
"""
|
|
return self._formula_tree(plugging, self.top_hole)
|
|
|
|
def _formula_tree(self, plugging, node):
|
|
if node in plugging:
|
|
return self._formula_tree(plugging, plugging[node])
|
|
elif node in self.fragments:
|
|
pred, args = self.fragments[node]
|
|
children = [self._formula_tree(plugging, arg) for arg in args]
|
|
return reduce(Constants.MAP[pred.variable.name], children)
|
|
else:
|
|
return node
|
|
|
|
|
|
class Constraint(object):
|
|
"""
|
|
This class represents a constraint of the form (L =< N),
|
|
where L is a label and N is a node (a label or a hole).
|
|
"""
|
|
|
|
def __init__(self, lhs, rhs):
|
|
self.lhs = lhs
|
|
self.rhs = rhs
|
|
|
|
def __eq__(self, other):
|
|
if self.__class__ == other.__class__:
|
|
return self.lhs == other.lhs and self.rhs == other.rhs
|
|
else:
|
|
return False
|
|
|
|
def __ne__(self, other):
|
|
return not (self == other)
|
|
|
|
def __hash__(self):
|
|
return hash(repr(self))
|
|
|
|
def __repr__(self):
|
|
return "(%s < %s)" % (self.lhs, self.rhs)
|
|
|
|
|
|
def hole_readings(sentence, grammar_filename=None, verbose=False):
|
|
if not grammar_filename:
|
|
grammar_filename = "grammars/sample_grammars/hole.fcfg"
|
|
|
|
if verbose:
|
|
print("Reading grammar file", grammar_filename)
|
|
|
|
parser = load_parser(grammar_filename)
|
|
|
|
# Parse the sentence.
|
|
tokens = sentence.split()
|
|
trees = list(parser.parse(tokens))
|
|
if verbose:
|
|
print("Got %d different parses" % len(trees))
|
|
|
|
all_readings = []
|
|
for tree in trees:
|
|
# Get the semantic feature from the top of the parse tree.
|
|
sem = tree.label()["SEM"].simplify()
|
|
|
|
# Print the raw semantic representation.
|
|
if verbose:
|
|
print("Raw: ", sem)
|
|
|
|
# Skolemize away all quantifiers. All variables become unique.
|
|
while isinstance(sem, LambdaExpression):
|
|
sem = sem.term
|
|
skolemized = skolemize(sem)
|
|
|
|
if verbose:
|
|
print("Skolemized:", skolemized)
|
|
|
|
# Break the hole semantics representation down into its components
|
|
# i.e. holes, labels, formula fragments and constraints.
|
|
hole_sem = HoleSemantics(skolemized)
|
|
|
|
# Maybe show the details of the semantic representation.
|
|
if verbose:
|
|
print("Holes: ", hole_sem.holes)
|
|
print("Labels: ", hole_sem.labels)
|
|
print("Constraints: ", hole_sem.constraints)
|
|
print("Top hole: ", hole_sem.top_hole)
|
|
print("Top labels: ", hole_sem.top_most_labels)
|
|
print("Fragments:")
|
|
for l, f in hole_sem.fragments.items():
|
|
print("\t%s: %s" % (l, f))
|
|
|
|
# Find all the possible ways to plug the formulas together.
|
|
pluggings = hole_sem.pluggings()
|
|
|
|
# Build FOL formula trees using the pluggings.
|
|
readings = list(map(hole_sem.formula_tree, pluggings))
|
|
|
|
# Print out the formulas in a textual format.
|
|
if verbose:
|
|
for i, r in enumerate(readings):
|
|
print()
|
|
print("%d. %s" % (i, r))
|
|
print()
|
|
|
|
all_readings.extend(readings)
|
|
|
|
return all_readings
|
|
|
|
|
|
if __name__ == "__main__":
|
|
for r in hole_readings("a dog barks"):
|
|
print(r)
|
|
print()
|
|
for r in hole_readings("every girl chases a dog"):
|
|
print(r)
|