Discourse Representation Theory

>>> from nltk.sem import logic
>>> from nltk.inference import TableauProver

Overview

A DRS can be created with the DRS() constructor. This takes two arguments: a list of discourse referents and list of conditions. .

>>> from nltk.sem.drt import *
>>> dexpr = DrtExpression.fromstring
>>> man_x = dexpr('man(x)')
>>> walk_x = dexpr('walk(x)')
>>> x = dexpr('x')
>>> print(DRS([x], [man_x, walk_x]))
([x],[man(x), walk(x)])

The parse() method can also be applied directly to DRS expressions, which allows them to be specified more easily.

>>> drs1 = dexpr('([x],[man(x),walk(x)])')
>>> print(drs1)
([x],[man(x), walk(x)])

DRSs can be merged using the + operator.

>>> drs2 = dexpr('([y],[woman(y),stop(y)])')
>>> drs3 = drs1 + drs2
>>> print(drs3)
(([x],[man(x), walk(x)]) + ([y],[woman(y), stop(y)]))
>>> print(drs3.simplify())
([x,y],[man(x), walk(x), woman(y), stop(y)])

We can embed DRSs as components of an implies condition.

>>> s = '([], [(%s -> %s)])' % (drs1, drs2)
>>> print(dexpr(s))
([],[(([x],[man(x), walk(x)]) -> ([y],[woman(y), stop(y)]))])

The fol() method converts DRSs into FOL formulae.

>>> print(dexpr(r'([x],[man(x), walks(x)])').fol())
exists x.(man(x) & walks(x))
>>> print(dexpr(r'([],[(([x],[man(x)]) -> ([],[walks(x)]))])').fol())
all x.(man(x) -> walks(x))

In order to visualize a DRS, the pretty_format() method can be used.

>>> print(drs3.pretty_format())
  _________     __________
 | x       |   | y        |
(|---------| + |----------|)
 | man(x)  |   | woman(y) |
 | walk(x) |   | stop(y)  |
 |_________|   |__________|

Parse to semantics

DRSs can be used for building compositional semantics in a feature based grammar. To specify that we want to use DRSs, the appropriate logic parser needs be passed as a parameter to load_earley()

>>> from nltk.parse import load_parser
>>> from nltk.sem.drt import DrtParser
>>> parser = load_parser('grammars/book_grammars/drt.fcfg', trace=0, logic_parser=DrtParser())
>>> for tree in parser.parse('a dog barks'.split()):
...     print(tree.label()['SEM'].simplify())
...
([x],[dog(x), bark(x)])

Alternatively, a FeatStructReader can be passed with the logic_parser set on it

>>> from nltk.featstruct import FeatStructReader
>>> from nltk.grammar import FeatStructNonterminal
>>> parser = load_parser('grammars/book_grammars/drt.fcfg', trace=0, fstruct_reader=FeatStructReader(fdict_class=FeatStructNonterminal, logic_parser=DrtParser()))
>>> for tree in parser.parse('every girl chases a dog'.split()):
...     print(tree.label()['SEM'].simplify().normalize())
...
([],[(([z1],[girl(z1)]) -> ([z2],[dog(z2), chase(z1,z2)]))])

Unit Tests

Parser

>>> print(dexpr(r'([x,y],[sees(x,y)])'))
([x,y],[sees(x,y)])
>>> print(dexpr(r'([x],[man(x), walks(x)])'))
([x],[man(x), walks(x)])
>>> print(dexpr(r'\x.([],[man(x), walks(x)])'))
\x.([],[man(x), walks(x)])
>>> print(dexpr(r'\x.\y.([],[sees(x,y)])'))
\x y.([],[sees(x,y)])

>>> print(dexpr(r'([x,y],[(x = y)])'))
([x,y],[(x = y)])
>>> print(dexpr(r'([x,y],[(x != y)])'))
([x,y],[-(x = y)])

>>> print(dexpr(r'\x.([],[walks(x)])(john)'))
(\x.([],[walks(x)]))(john)
>>> print(dexpr(r'\R.\x.([],[big(x,R)])(\y.([],[mouse(y)]))'))
(\R x.([],[big(x,R)]))(\y.([],[mouse(y)]))

>>> print(dexpr(r'(([x],[walks(x)]) + ([y],[runs(y)]))'))
(([x],[walks(x)]) + ([y],[runs(y)]))
>>> print(dexpr(r'(([x,y],[walks(x), jumps(y)]) + (([z],[twos(z)]) + ([w],[runs(w)])))'))
(([x,y],[walks(x), jumps(y)]) + ([z],[twos(z)]) + ([w],[runs(w)]))
>>> print(dexpr(r'((([],[walks(x)]) + ([],[twos(x)])) + ([],[runs(x)]))'))
(([],[walks(x)]) + ([],[twos(x)]) + ([],[runs(x)]))
>>> print(dexpr(r'((([],[walks(x)]) + ([],[runs(x)])) + (([],[threes(x)]) + ([],[fours(x)])))'))
(([],[walks(x)]) + ([],[runs(x)]) + ([],[threes(x)]) + ([],[fours(x)]))

>>> print(dexpr(r'(([],[walks(x)]) -> ([],[runs(x)]))'))
(([],[walks(x)]) -> ([],[runs(x)]))

>>> print(dexpr(r'([x],[PRO(x), sees(John,x)])'))
([x],[PRO(x), sees(John,x)])
>>> print(dexpr(r'([x],[man(x), -([],[walks(x)])])'))
([x],[man(x), -([],[walks(x)])])
>>> print(dexpr(r'([],[(([x],[man(x)]) -> ([],[walks(x)]))])'))
([],[(([x],[man(x)]) -> ([],[walks(x)]))])

>>> print(dexpr(r'DRS([x],[walk(x)])'))
([x],[walk(x)])
>>> print(dexpr(r'DRS([x][walk(x)])'))
([x],[walk(x)])
>>> print(dexpr(r'([x][walk(x)])'))
([x],[walk(x)])

simplify()

>>> print(dexpr(r'\x.([],[man(x), walks(x)])(john)').simplify())
([],[man(john), walks(john)])
>>> print(dexpr(r'\x.\y.([z],[dog(z),sees(x,y)])(john)(mary)').simplify())
([z],[dog(z), sees(john,mary)])
>>> print(dexpr(r'\R x.([],[big(x,R)])(\y.([],[mouse(y)]))').simplify())
\x.([],[big(x,\y.([],[mouse(y)]))])

>>> print(dexpr(r'(([x],[walks(x)]) + ([y],[runs(y)]))').simplify())
([x,y],[walks(x), runs(y)])
>>> print(dexpr(r'(([x,y],[walks(x), jumps(y)]) + (([z],[twos(z)]) + ([w],[runs(w)])))').simplify())
([w,x,y,z],[walks(x), jumps(y), twos(z), runs(w)])
>>> print(dexpr(r'((([],[walks(x)]) + ([],[runs(x)]) + ([],[threes(x)]) + ([],[fours(x)])))').simplify())
([],[walks(x), runs(x), threes(x), fours(x)])
>>> dexpr(r'([x],[man(x)])+([x],[walks(x)])').simplify() == \
... dexpr(r'([x,z1],[man(x), walks(z1)])')
True
>>> dexpr(r'([y],[boy(y), (([x],[dog(x)]) -> ([],[chase(x,y)]))])+([x],[run(x)])').simplify() == \
... dexpr(r'([y,z1],[boy(y), (([x],[dog(x)]) -> ([],[chase(x,y)])), run(z1)])')
True

>>> dexpr(r'\Q.(([x],[john(x),walks(x)]) + Q)(([x],[PRO(x),leaves(x)]))').simplify() == \
... dexpr(r'([x,z1],[john(x), walks(x), PRO(z1), leaves(z1)])')
True

>>> logic._counter._value = 0
>>> print(dexpr('([],[(([x],[dog(x)]) -> ([e,y],[boy(y), chase(e), subj(e,x), obj(e,y)]))])+([e,x],[PRO(x), run(e), subj(e,x)])').simplify().normalize().normalize())
([e02,z5],[(([z3],[dog(z3)]) -> ([e01,z4],[boy(z4), chase(e01), subj(e01,z3), obj(e01,z4)])), PRO(z5), run(e02), subj(e02,z5)])

fol()

>>> print(dexpr(r'([x,y],[sees(x,y)])').fol())
exists x y.sees(x,y)
>>> print(dexpr(r'([x],[man(x), walks(x)])').fol())
exists x.(man(x) & walks(x))
>>> print(dexpr(r'\x.([],[man(x), walks(x)])').fol())
\x.(man(x) & walks(x))
>>> print(dexpr(r'\x y.([],[sees(x,y)])').fol())
\x y.sees(x,y)

>>> print(dexpr(r'\x.([],[walks(x)])(john)').fol())
\x.walks(x)(john)
>>> print(dexpr(r'\R x.([],[big(x,R)])(\y.([],[mouse(y)]))').fol())
(\R x.big(x,R))(\y.mouse(y))

>>> print(dexpr(r'(([x],[walks(x)]) + ([y],[runs(y)]))').fol())
(exists x.walks(x) & exists y.runs(y))

>>> print(dexpr(r'(([],[walks(x)]) -> ([],[runs(x)]))').fol())
(walks(x) -> runs(x))

>>> print(dexpr(r'([x],[PRO(x), sees(John,x)])').fol())
exists x.(PRO(x) & sees(John,x))
>>> print(dexpr(r'([x],[man(x), -([],[walks(x)])])').fol())
exists x.(man(x) & -walks(x))
>>> print(dexpr(r'([],[(([x],[man(x)]) -> ([],[walks(x)]))])').fol())
all x.(man(x) -> walks(x))

>>> print(dexpr(r'([x],[man(x) | walks(x)])').fol())
exists x.(man(x) | walks(x))
>>> print(dexpr(r'P(x) + ([x],[walks(x)])').fol())
(P(x) & exists x.walks(x))

resolve_anaphora()

>>> from nltk.sem.drt import AnaphoraResolutionException

>>> print(resolve_anaphora(dexpr(r'([x,y,z],[dog(x), cat(y), walks(z), PRO(z)])')))
([x,y,z],[dog(x), cat(y), walks(z), (z = [x,y])])
>>> print(resolve_anaphora(dexpr(r'([],[(([x],[dog(x)]) -> ([y],[walks(y), PRO(y)]))])')))
([],[(([x],[dog(x)]) -> ([y],[walks(y), (y = x)]))])
>>> print(resolve_anaphora(dexpr(r'(([x,y],[]) + ([],[PRO(x)]))')).simplify())
([x,y],[(x = y)])
>>> try: print(resolve_anaphora(dexpr(r'([x],[walks(x), PRO(x)])')))
... except AnaphoraResolutionException as e: print(e)
Variable 'x' does not resolve to anything.
>>> print(resolve_anaphora(dexpr('([e01,z6,z7],[boy(z6), PRO(z7), run(e01), subj(e01,z7)])')))
([e01,z6,z7],[boy(z6), (z7 = z6), run(e01), subj(e01,z7)])

equiv():

>>> a = dexpr(r'([x],[man(x), walks(x)])')
>>> b = dexpr(r'([x],[walks(x), man(x)])')
>>> print(a.equiv(b, TableauProver()))
True

replace():

>>> a = dexpr(r'a')
>>> w = dexpr(r'w')
>>> x = dexpr(r'x')
>>> y = dexpr(r'y')
>>> z = dexpr(r'z')

replace bound

>>> print(dexpr(r'([x],[give(x,y,z)])').replace(x.variable, a, False))
([x],[give(x,y,z)])
>>> print(dexpr(r'([x],[give(x,y,z)])').replace(x.variable, a, True))
([a],[give(a,y,z)])

replace unbound

>>> print(dexpr(r'([x],[give(x,y,z)])').replace(y.variable, a, False))
([x],[give(x,a,z)])
>>> print(dexpr(r'([x],[give(x,y,z)])').replace(y.variable, a, True))
([x],[give(x,a,z)])

replace unbound with bound

>>> dexpr(r'([x],[give(x,y,z)])').replace(y.variable, x, False) == \
... dexpr('([z1],[give(z1,x,z)])')
True
>>> dexpr(r'([x],[give(x,y,z)])').replace(y.variable, x, True) == \
... dexpr('([z1],[give(z1,x,z)])')
True

replace unbound with unbound

>>> print(dexpr(r'([x],[give(x,y,z)])').replace(y.variable, z, False))
([x],[give(x,z,z)])
>>> print(dexpr(r'([x],[give(x,y,z)])').replace(y.variable, z, True))
([x],[give(x,z,z)])

replace unbound

>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(z.variable, a, False))
(([x],[P(x,y,a)]) + ([y],[Q(x,y,a)]))
>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(z.variable, a, True))
(([x],[P(x,y,a)]) + ([y],[Q(x,y,a)]))

replace bound

>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(x.variable, a, False))
(([x],[P(x,y,z)]) + ([y],[Q(x,y,z)]))
>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(x.variable, a, True))
(([a],[P(a,y,z)]) + ([y],[Q(a,y,z)]))

replace unbound with unbound

>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(z.variable, a, False))
(([x],[P(x,y,a)]) + ([y],[Q(x,y,a)]))
>>> print(dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,z)])').replace(z.variable, a, True))
(([x],[P(x,y,a)]) + ([y],[Q(x,y,a)]))

replace unbound with bound on same side

>>> dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,w)])').replace(z.variable, x, False) == \
... dexpr(r'(([z1],[P(z1,y,x)]) + ([y],[Q(z1,y,w)]))')
True
>>> dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,w)])').replace(z.variable, x, True) == \
... dexpr(r'(([z1],[P(z1,y,x)]) + ([y],[Q(z1,y,w)]))')
True

replace unbound with bound on other side

>>> dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,w)])').replace(w.variable, x, False) == \
... dexpr(r'(([z1],[P(z1,y,z)]) + ([y],[Q(z1,y,x)]))')
True
>>> dexpr(r'([x],[P(x,y,z)])+([y],[Q(x,y,w)])').replace(w.variable, x, True) == \
... dexpr(r'(([z1],[P(z1,y,z)]) + ([y],[Q(z1,y,x)]))')
True

replace unbound with double bound

>>> dexpr(r'([x],[P(x,y,z)])+([x],[Q(x,y,w)])').replace(z.variable, x, False) == \
... dexpr(r'(([z1],[P(z1,y,x)]) + ([z1],[Q(z1,y,w)]))')
True
>>> dexpr(r'([x],[P(x,y,z)])+([x],[Q(x,y,w)])').replace(z.variable, x, True) == \
... dexpr(r'(([z1],[P(z1,y,x)]) + ([z1],[Q(z1,y,w)]))')
True

regression tests

>>> d = dexpr('([x],[A(c), ([y],[B(x,y,z,a)])->([z],[C(x,y,z,a)])])')
>>> print(d)
([x],[A(c), (([y],[B(x,y,z,a)]) -> ([z],[C(x,y,z,a)]))])
>>> print(d.pretty_format())
 ____________________________________
| x                                  |
|------------------------------------|
| A(c)                               |
|   ____________      ____________   |
|  | y          |    | z          |  |
| (|------------| -> |------------|) |
|  | B(x,y,z,a) |    | C(x,y,z,a) |  |
|  |____________|    |____________|  |
|____________________________________|
>>> print(str(d))
([x],[A(c), (([y],[B(x,y,z,a)]) -> ([z],[C(x,y,z,a)]))])
>>> print(d.fol())
exists x.(A(c) & all y.(B(x,y,z,a) -> exists z.C(x,y,z,a)))
>>> print(d.replace(Variable('a'), DrtVariableExpression(Variable('r'))))
([x],[A(c), (([y],[B(x,y,z,r)]) -> ([z],[C(x,y,z,r)]))])
>>> print(d.replace(Variable('x'), DrtVariableExpression(Variable('r'))))
([x],[A(c), (([y],[B(x,y,z,a)]) -> ([z],[C(x,y,z,a)]))])
>>> print(d.replace(Variable('y'), DrtVariableExpression(Variable('r'))))
([x],[A(c), (([y],[B(x,y,z,a)]) -> ([z],[C(x,y,z,a)]))])
>>> print(d.replace(Variable('z'), DrtVariableExpression(Variable('r'))))
([x],[A(c), (([y],[B(x,y,r,a)]) -> ([z],[C(x,y,z,a)]))])
>>> print(d.replace(Variable('x'), DrtVariableExpression(Variable('r')), True))
([r],[A(c), (([y],[B(r,y,z,a)]) -> ([z],[C(r,y,z,a)]))])
>>> print(d.replace(Variable('y'), DrtVariableExpression(Variable('r')), True))
([x],[A(c), (([r],[B(x,r,z,a)]) -> ([z],[C(x,r,z,a)]))])
>>> print(d.replace(Variable('z'), DrtVariableExpression(Variable('r')), True))
([x],[A(c), (([y],[B(x,y,r,a)]) -> ([r],[C(x,y,r,a)]))])
>>> print(d == dexpr('([l],[A(c), ([m],[B(l,m,z,a)])->([n],[C(l,m,n,a)])])'))
True
>>> d = dexpr('([],[([x,y],[B(x,y,h), ([a,b],[dee(x,a,g)])])->([z,w],[cee(x,y,f), ([c,d],[E(x,c,d,e)])])])')
>>> sorted(d.free())
[Variable('B'), Variable('E'), Variable('e'), Variable('f'), Variable('g'), Variable('h')]
>>> sorted(d.variables())
[Variable('B'), Variable('E'), Variable('e'), Variable('f'), Variable('g'), Variable('h')]
>>> sorted(d.get_refs(True))
[Variable('a'), Variable('b'), Variable('c'), Variable('d'), Variable('w'), Variable('x'), Variable('y'), Variable('z')]
>>> sorted(d.conds[0].get_refs(False))
[Variable('x'), Variable('y')]
>>> print(dexpr('([x,y],[A(x,y), (x=y), ([],[B(x,y)])->([],[C(x,y)]), ([x,y],[D(x,y)])->([],[E(x,y)]), ([],[F(x,y)])->([x,y],[G(x,y)])])').eliminate_equality())
([x],[A(x,x), (([],[B(x,x)]) -> ([],[C(x,x)])), (([x,y],[D(x,y)]) -> ([],[E(x,y)])), (([],[F(x,x)]) -> ([x,y],[G(x,y)]))])
>>> print(dexpr('([x,y],[A(x,y), (x=y)]) -> ([],[B(x,y)])').eliminate_equality())
(([x],[A(x,x)]) -> ([],[B(x,x)]))
>>> print(dexpr('([x,y],[A(x,y)]) -> ([],[B(x,y), (x=y)])').eliminate_equality())
(([x,y],[A(x,y)]) -> ([],[B(x,x)]))
>>> print(dexpr('([x,y],[A(x,y), (x=y), ([],[B(x,y)])])').eliminate_equality())
([x],[A(x,x), ([],[B(x,x)])])
>>> print(dexpr('([x,y],[A(x,y), ([],[B(x,y), (x=y)])])').eliminate_equality())
([x,y],[A(x,y), ([],[B(x,x)])])
>>> print(dexpr('([z8 z9 z10],[A(z8), z8=z10, z9=z10, B(z9), C(z10), D(z10)])').eliminate_equality())
([z9],[A(z9), B(z9), C(z9), D(z9)])

>>> print(dexpr('([x,y],[A(x,y), (x=y), ([],[B(x,y)]), ([x,y],[C(x,y)])])').eliminate_equality())
([x],[A(x,x), ([],[B(x,x)]), ([x,y],[C(x,y)])])
>>> print(dexpr('([x,y],[A(x,y)]) + ([],[B(x,y), (x=y)]) + ([],[C(x,y)])').eliminate_equality())
([x],[A(x,x), B(x,x), C(x,x)])
>>> print(dexpr('([x,y],[B(x,y)])+([x,y],[C(x,y)])').replace(Variable('y'), DrtVariableExpression(Variable('x'))))
(([x,y],[B(x,y)]) + ([x,y],[C(x,y)]))
>>> print(dexpr('(([x,y],[B(x,y)])+([],[C(x,y)]))+([],[D(x,y)])').replace(Variable('y'), DrtVariableExpression(Variable('x'))))
(([x,y],[B(x,y)]) + ([],[C(x,y)]) + ([],[D(x,y)]))
>>> print(dexpr('(([],[B(x,y)])+([],[C(x,y)]))+([],[D(x,y)])').replace(Variable('y'), DrtVariableExpression(Variable('x'))))
(([],[B(x,x)]) + ([],[C(x,x)]) + ([],[D(x,x)]))
>>> print(dexpr('(([],[B(x,y), ([x,y],[A(x,y)])])+([],[C(x,y)]))+([],[D(x,y)])').replace(Variable('y'), DrtVariableExpression(Variable('x'))).normalize())
(([],[B(z3,z1), ([z2,z3],[A(z3,z2)])]) + ([],[C(z3,z1)]) + ([],[D(z3,z1)]))

Parse errors

>>> def parse_error(drtstring):
...     try: dexpr(drtstring)
...     except logic.LogicalExpressionException as e: print(e)

>>> parse_error(r'')
End of input found.  Expression expected.
<BLANKLINE>
^
>>> parse_error(r'(')
End of input found.  Expression expected.
(
 ^
>>> parse_error(r'()')
Unexpected token: ')'.  Expression expected.
()
 ^
>>> parse_error(r'([')
End of input found.  Expected token ']'.
([
  ^
>>> parse_error(r'([,')
',' is an illegal variable name.  Constants may not be quantified.
([,
  ^
>>> parse_error(r'([x,')
End of input found.  Variable expected.
([x,
    ^
>>> parse_error(r'([]')
End of input found.  Expected token '['.
([]
   ^
>>> parse_error(r'([][')
End of input found.  Expected token ']'.
([][
    ^
>>> parse_error(r'([][,')
Unexpected token: ','.  Expression expected.
([][,
    ^
>>> parse_error(r'([][]')
End of input found.  Expected token ')'.
([][]
     ^
>>> parse_error(r'([x][man(x)]) |')
End of input found.  Expression expected.
([x][man(x)]) |
               ^

Pretty Printing

>>> dexpr(r"([],[])").pretty_print()
 __
|  |
|--|
|__|

>>> dexpr(r"([],[([x],[big(x), dog(x)]) -> ([],[bark(x)]) -([x],[walk(x)])])").pretty_print()
 _____________________________
|                             |
|-----------------------------|
|   ________      _________   |
|  | x      |    |         |  |
| (|--------| -> |---------|) |
|  | big(x) |    | bark(x) |  |
|  | dog(x) |    |_________|  |
|  |________|                 |
|      _________              |
|     | x       |             |
| __  |---------|             |
|   | | walk(x) |             |
|     |_________|             |
|_____________________________|

>>> dexpr(r"([x,y],[x=y]) + ([z],[dog(z), walk(z)])").pretty_print()
  _________     _________
 | x y     |   | z       |
(|---------| + |---------|)
 | (x = y) |   | dog(z)  |
 |_________|   | walk(z) |
               |_________|

>>> dexpr(r"([],[([x],[]) | ([y],[]) | ([z],[dog(z), walk(z)])])").pretty_print()
 _______________________________
|                               |
|-------------------------------|
|   ___     ___     _________   |
|  | x |   | y |   | z       |  |
| (|---| | |---| | |---------|) |
|  |___|   |___|   | dog(z)  |  |
|                  | walk(z) |  |
|                  |_________|  |
|_______________________________|

>>> dexpr(r"\P.\Q.(([x],[]) + P(x) + Q(x))(\x.([],[dog(x)]))").pretty_print()
          ___                        ________
 \       | x |                 \    |        |
 /\ P Q.(|---| + P(x) + Q(x))( /\ x.|--------|)
         |___|                      | dog(x) |
                                    |________|