400 lines
14 KiB
Python
400 lines
14 KiB
Python
# Natural Language Toolkit: Logic
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#
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# Author: Peter Wang
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# Updated by: Dan Garrette <dhgarrette@gmail.com>
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#
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# Copyright (C) 2001-2019 NLTK Project
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# URL: <http://nltk.org>
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# For license information, see LICENSE.TXT
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"""
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An implementation of the Hole Semantics model, following Blackburn and Bos,
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Representation and Inference for Natural Language (CSLI, 2005).
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The semantic representations are built by the grammar hole.fcfg.
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This module contains driver code to read in sentences and parse them
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according to a hole semantics grammar.
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After parsing, the semantic representation is in the form of an underspecified
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representation that is not easy to read. We use a "plugging" algorithm to
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convert that representation into first-order logic formulas.
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"""
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from functools import reduce
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from six import itervalues
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from nltk.parse import load_parser
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from nltk.sem.skolemize import skolemize
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from nltk.sem.logic import (
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AllExpression,
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AndExpression,
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ApplicationExpression,
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ExistsExpression,
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IffExpression,
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ImpExpression,
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LambdaExpression,
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NegatedExpression,
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OrExpression,
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)
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# Note that in this code there may be multiple types of trees being referred to:
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#
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# 1. parse trees
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# 2. the underspecified representation
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# 3. first-order logic formula trees
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# 4. the search space when plugging (search tree)
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#
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class Constants(object):
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ALL = "ALL"
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EXISTS = "EXISTS"
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NOT = "NOT"
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AND = "AND"
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OR = "OR"
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IMP = "IMP"
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IFF = "IFF"
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PRED = "PRED"
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LEQ = "LEQ"
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HOLE = "HOLE"
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LABEL = "LABEL"
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MAP = {
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ALL: lambda v, e: AllExpression(v.variable, e),
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EXISTS: lambda v, e: ExistsExpression(v.variable, e),
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NOT: NegatedExpression,
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AND: AndExpression,
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OR: OrExpression,
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IMP: ImpExpression,
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IFF: IffExpression,
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PRED: ApplicationExpression,
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}
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class HoleSemantics(object):
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"""
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This class holds the broken-down components of a hole semantics, i.e. it
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extracts the holes, labels, logic formula fragments and constraints out of
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a big conjunction of such as produced by the hole semantics grammar. It
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then provides some operations on the semantics dealing with holes, labels
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and finding legal ways to plug holes with labels.
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"""
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def __init__(self, usr):
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"""
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Constructor. `usr' is a ``sem.Expression`` representing an
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Underspecified Representation Structure (USR). A USR has the following
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special predicates:
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ALL(l,v,n),
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EXISTS(l,v,n),
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AND(l,n,n),
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OR(l,n,n),
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IMP(l,n,n),
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IFF(l,n,n),
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PRED(l,v,n,v[,v]*) where the brackets and star indicate zero or more repetitions,
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LEQ(n,n),
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HOLE(n),
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LABEL(n)
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where l is the label of the node described by the predicate, n is either
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a label or a hole, and v is a variable.
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"""
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self.holes = set()
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self.labels = set()
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self.fragments = {} # mapping of label -> formula fragment
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self.constraints = set() # set of Constraints
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self._break_down(usr)
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self.top_most_labels = self._find_top_most_labels()
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self.top_hole = self._find_top_hole()
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def is_node(self, x):
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"""
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Return true if x is a node (label or hole) in this semantic
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representation.
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"""
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return x in (self.labels | self.holes)
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def _break_down(self, usr):
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"""
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Extract holes, labels, formula fragments and constraints from the hole
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semantics underspecified representation (USR).
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"""
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if isinstance(usr, AndExpression):
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self._break_down(usr.first)
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self._break_down(usr.second)
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elif isinstance(usr, ApplicationExpression):
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func, args = usr.uncurry()
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if func.variable.name == Constants.LEQ:
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self.constraints.add(Constraint(args[0], args[1]))
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elif func.variable.name == Constants.HOLE:
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self.holes.add(args[0])
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elif func.variable.name == Constants.LABEL:
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self.labels.add(args[0])
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else:
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label = args[0]
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assert label not in self.fragments
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self.fragments[label] = (func, args[1:])
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else:
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raise ValueError(usr.label())
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def _find_top_nodes(self, node_list):
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top_nodes = node_list.copy()
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for f in itervalues(self.fragments):
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# the label is the first argument of the predicate
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args = f[1]
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for arg in args:
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if arg in node_list:
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top_nodes.discard(arg)
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return top_nodes
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def _find_top_most_labels(self):
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"""
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Return the set of labels which are not referenced directly as part of
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another formula fragment. These will be the top-most labels for the
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subtree that they are part of.
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"""
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return self._find_top_nodes(self.labels)
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def _find_top_hole(self):
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"""
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Return the hole that will be the top of the formula tree.
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"""
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top_holes = self._find_top_nodes(self.holes)
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assert len(top_holes) == 1 # it must be unique
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return top_holes.pop()
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def pluggings(self):
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"""
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Calculate and return all the legal pluggings (mappings of labels to
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holes) of this semantics given the constraints.
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"""
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record = []
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self._plug_nodes([(self.top_hole, [])], self.top_most_labels, {}, record)
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return record
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def _plug_nodes(self, queue, potential_labels, plug_acc, record):
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"""
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Plug the nodes in `queue' with the labels in `potential_labels'.
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Each element of `queue' is a tuple of the node to plug and the list of
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ancestor holes from the root of the graph to that node.
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`potential_labels' is a set of the labels which are still available for
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plugging.
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`plug_acc' is the incomplete mapping of holes to labels made on the
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current branch of the search tree so far.
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`record' is a list of all the complete pluggings that we have found in
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total so far. It is the only parameter that is destructively updated.
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"""
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if queue != []:
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(node, ancestors) = queue[0]
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if node in self.holes:
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# The node is a hole, try to plug it.
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self._plug_hole(
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node, ancestors, queue[1:], potential_labels, plug_acc, record
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)
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else:
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assert node in self.labels
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# The node is a label. Replace it in the queue by the holes and
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# labels in the formula fragment named by that label.
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args = self.fragments[node][1]
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head = [(a, ancestors) for a in args if self.is_node(a)]
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self._plug_nodes(head + queue[1:], potential_labels, plug_acc, record)
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else:
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raise Exception("queue empty")
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def _plug_hole(self, hole, ancestors0, queue, potential_labels0, plug_acc0, record):
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"""
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Try all possible ways of plugging a single hole.
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See _plug_nodes for the meanings of the parameters.
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"""
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# Add the current hole we're trying to plug into the list of ancestors.
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assert hole not in ancestors0
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ancestors = [hole] + ancestors0
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# Try each potential label in this hole in turn.
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for l in potential_labels0:
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# Is the label valid in this hole?
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if self._violates_constraints(l, ancestors):
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continue
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plug_acc = plug_acc0.copy()
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plug_acc[hole] = l
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potential_labels = potential_labels0.copy()
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potential_labels.remove(l)
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if len(potential_labels) == 0:
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# No more potential labels. That must mean all the holes have
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# been filled so we have found a legal plugging so remember it.
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#
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# Note that the queue might not be empty because there might
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# be labels on there that point to formula fragments with
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# no holes in them. _sanity_check_plugging will make sure
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# all holes are filled.
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self._sanity_check_plugging(plug_acc, self.top_hole, [])
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record.append(plug_acc)
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else:
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# Recursively try to fill in the rest of the holes in the
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# queue. The label we just plugged into the hole could have
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# holes of its own so at the end of the queue. Putting it on
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# the end of the queue gives us a breadth-first search, so that
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# all the holes at level i of the formula tree are filled
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# before filling level i+1.
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# A depth-first search would work as well since the trees must
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# be finite but the bookkeeping would be harder.
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self._plug_nodes(
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queue + [(l, ancestors)], potential_labels, plug_acc, record
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)
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def _violates_constraints(self, label, ancestors):
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"""
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Return True if the `label' cannot be placed underneath the holes given
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by the set `ancestors' because it would violate the constraints imposed
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on it.
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"""
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for c in self.constraints:
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if c.lhs == label:
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if c.rhs not in ancestors:
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return True
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return False
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def _sanity_check_plugging(self, plugging, node, ancestors):
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"""
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Make sure that a given plugging is legal. We recursively go through
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each node and make sure that no constraints are violated.
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We also check that all holes have been filled.
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"""
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if node in self.holes:
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ancestors = [node] + ancestors
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label = plugging[node]
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else:
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label = node
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assert label in self.labels
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for c in self.constraints:
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if c.lhs == label:
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assert c.rhs in ancestors
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args = self.fragments[label][1]
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for arg in args:
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if self.is_node(arg):
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self._sanity_check_plugging(plugging, arg, [label] + ancestors)
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def formula_tree(self, plugging):
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"""
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Return the first-order logic formula tree for this underspecified
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representation using the plugging given.
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"""
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return self._formula_tree(plugging, self.top_hole)
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def _formula_tree(self, plugging, node):
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if node in plugging:
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return self._formula_tree(plugging, plugging[node])
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elif node in self.fragments:
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pred, args = self.fragments[node]
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children = [self._formula_tree(plugging, arg) for arg in args]
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return reduce(Constants.MAP[pred.variable.name], children)
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else:
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return node
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class Constraint(object):
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"""
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This class represents a constraint of the form (L =< N),
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where L is a label and N is a node (a label or a hole).
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"""
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def __init__(self, lhs, rhs):
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self.lhs = lhs
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self.rhs = rhs
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def __eq__(self, other):
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if self.__class__ == other.__class__:
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return self.lhs == other.lhs and self.rhs == other.rhs
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else:
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return False
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def __ne__(self, other):
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return not (self == other)
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def __hash__(self):
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return hash(repr(self))
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def __repr__(self):
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return "(%s < %s)" % (self.lhs, self.rhs)
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def hole_readings(sentence, grammar_filename=None, verbose=False):
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if not grammar_filename:
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grammar_filename = "grammars/sample_grammars/hole.fcfg"
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if verbose:
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print("Reading grammar file", grammar_filename)
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parser = load_parser(grammar_filename)
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# Parse the sentence.
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tokens = sentence.split()
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trees = list(parser.parse(tokens))
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if verbose:
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print("Got %d different parses" % len(trees))
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all_readings = []
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for tree in trees:
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# Get the semantic feature from the top of the parse tree.
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sem = tree.label()["SEM"].simplify()
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# Print the raw semantic representation.
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if verbose:
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print("Raw: ", sem)
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# Skolemize away all quantifiers. All variables become unique.
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while isinstance(sem, LambdaExpression):
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sem = sem.term
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skolemized = skolemize(sem)
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if verbose:
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print("Skolemized:", skolemized)
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# Break the hole semantics representation down into its components
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# i.e. holes, labels, formula fragments and constraints.
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hole_sem = HoleSemantics(skolemized)
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# Maybe show the details of the semantic representation.
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if verbose:
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print("Holes: ", hole_sem.holes)
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print("Labels: ", hole_sem.labels)
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print("Constraints: ", hole_sem.constraints)
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print("Top hole: ", hole_sem.top_hole)
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print("Top labels: ", hole_sem.top_most_labels)
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print("Fragments:")
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for l, f in hole_sem.fragments.items():
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print("\t%s: %s" % (l, f))
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# Find all the possible ways to plug the formulas together.
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pluggings = hole_sem.pluggings()
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# Build FOL formula trees using the pluggings.
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readings = list(map(hole_sem.formula_tree, pluggings))
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# Print out the formulas in a textual format.
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if verbose:
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for i, r in enumerate(readings):
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print()
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print("%d. %s" % (i, r))
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print()
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all_readings.extend(readings)
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return all_readings
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if __name__ == "__main__":
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for r in hole_readings("a dog barks"):
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print(r)
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print()
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for r in hole_readings("every girl chases a dog"):
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print(r)
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