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385 lines
12 KiB
Plaintext
.. Copyright (C) 2001-2020 NLTK Project
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.. For license information, see LICENSE.TXT
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==============================================================================
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Glue Semantics
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==============================================================================
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.. include:: ../../../nltk_book/definitions.rst
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======================
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Linear logic
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======================
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>>> from nltk.sem import logic
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>>> from nltk.sem.glue import *
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>>> from nltk.sem.linearlogic import *
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>>> from nltk.sem.linearlogic import Expression
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>>> read_expr = Expression.fromstring
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Parser
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>>> print(read_expr(r'f'))
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f
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>>> print(read_expr(r'(g -o f)'))
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(g -o f)
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>>> print(read_expr(r'(g -o (h -o f))'))
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(g -o (h -o f))
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>>> print(read_expr(r'((g -o G) -o G)'))
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((g -o G) -o G)
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>>> print(read_expr(r'(g -o f)(g)'))
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(g -o f)(g)
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>>> print(read_expr(r'((g -o G) -o G)((g -o f))'))
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((g -o G) -o G)((g -o f))
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Simplify
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>>> print(read_expr(r'f').simplify())
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f
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>>> print(read_expr(r'(g -o f)').simplify())
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(g -o f)
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>>> print(read_expr(r'((g -o G) -o G)').simplify())
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((g -o G) -o G)
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>>> print(read_expr(r'(g -o f)(g)').simplify())
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f
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>>> try: read_expr(r'(g -o f)(f)').simplify()
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... except LinearLogicApplicationException as e: print(e)
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...
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Cannot apply (g -o f) to f. Cannot unify g with f given {}
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>>> print(read_expr(r'(G -o f)(g)').simplify())
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f
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>>> print(read_expr(r'((g -o G) -o G)((g -o f))').simplify())
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f
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Test BindingDict
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>>> h = ConstantExpression('h')
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>>> g = ConstantExpression('g')
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>>> f = ConstantExpression('f')
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>>> H = VariableExpression('H')
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>>> G = VariableExpression('G')
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>>> F = VariableExpression('F')
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>>> d1 = BindingDict({H: h})
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>>> d2 = BindingDict({F: f, G: F})
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>>> d12 = d1 + d2
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>>> all12 = ['%s: %s' % (v, d12[v]) for v in d12.d]
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>>> all12.sort()
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>>> print(all12)
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['F: f', 'G: f', 'H: h']
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>>> BindingDict([(F,f),(G,g),(H,h)]) == BindingDict({F:f, G:g, H:h})
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True
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>>> d4 = BindingDict({F: f})
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>>> try: d4[F] = g
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... except VariableBindingException as e: print(e)
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Variable F already bound to another value
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Test Unify
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>>> try: f.unify(g, BindingDict())
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... except UnificationException as e: print(e)
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...
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Cannot unify f with g given {}
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>>> f.unify(G, BindingDict()) == BindingDict({G: f})
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True
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>>> try: f.unify(G, BindingDict({G: h}))
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... except UnificationException as e: print(e)
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...
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Cannot unify f with G given {G: h}
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>>> f.unify(G, BindingDict({G: f})) == BindingDict({G: f})
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True
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>>> f.unify(G, BindingDict({H: f})) == BindingDict({G: f, H: f})
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True
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>>> G.unify(f, BindingDict()) == BindingDict({G: f})
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True
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>>> try: G.unify(f, BindingDict({G: h}))
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... except UnificationException as e: print(e)
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...
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Cannot unify G with f given {G: h}
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>>> G.unify(f, BindingDict({G: f})) == BindingDict({G: f})
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True
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>>> G.unify(f, BindingDict({H: f})) == BindingDict({G: f, H: f})
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True
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>>> G.unify(F, BindingDict()) == BindingDict({G: F})
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True
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>>> try: G.unify(F, BindingDict({G: H}))
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... except UnificationException as e: print(e)
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...
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Cannot unify G with F given {G: H}
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>>> G.unify(F, BindingDict({G: F})) == BindingDict({G: F})
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True
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>>> G.unify(F, BindingDict({H: F})) == BindingDict({G: F, H: F})
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True
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Test Compile
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>>> print(read_expr('g').compile_pos(Counter(), GlueFormula))
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(<ConstantExpression g>, [])
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>>> print(read_expr('(g -o f)').compile_pos(Counter(), GlueFormula))
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(<ImpExpression (g -o f)>, [])
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>>> print(read_expr('(g -o (h -o f))').compile_pos(Counter(), GlueFormula))
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(<ImpExpression (g -o (h -o f))>, [])
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======================
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Glue
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======================
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Demo of "John walks"
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--------------------
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>>> john = GlueFormula("John", "g")
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>>> print(john)
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John : g
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>>> walks = GlueFormula(r"\x.walks(x)", "(g -o f)")
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>>> print(walks)
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\x.walks(x) : (g -o f)
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>>> print(walks.applyto(john))
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\x.walks(x)(John) : (g -o f)(g)
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>>> print(walks.applyto(john).simplify())
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walks(John) : f
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Demo of "A dog walks"
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---------------------
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>>> a = GlueFormula("\P Q.some x.(P(x) and Q(x))", "((gv -o gr) -o ((g -o G) -o G))")
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>>> print(a)
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\P Q.exists x.(P(x) & Q(x)) : ((gv -o gr) -o ((g -o G) -o G))
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>>> man = GlueFormula(r"\x.man(x)", "(gv -o gr)")
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>>> print(man)
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\x.man(x) : (gv -o gr)
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>>> walks = GlueFormula(r"\x.walks(x)", "(g -o f)")
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>>> print(walks)
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\x.walks(x) : (g -o f)
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>>> a_man = a.applyto(man)
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>>> print(a_man.simplify())
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\Q.exists x.(man(x) & Q(x)) : ((g -o G) -o G)
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>>> a_man_walks = a_man.applyto(walks)
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>>> print(a_man_walks.simplify())
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exists x.(man(x) & walks(x)) : f
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Demo of 'every girl chases a dog'
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---------------------------------
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Individual words:
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>>> every = GlueFormula("\P Q.all x.(P(x) -> Q(x))", "((gv -o gr) -o ((g -o G) -o G))")
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>>> print(every)
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\P Q.all x.(P(x) -> Q(x)) : ((gv -o gr) -o ((g -o G) -o G))
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>>> girl = GlueFormula(r"\x.girl(x)", "(gv -o gr)")
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>>> print(girl)
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\x.girl(x) : (gv -o gr)
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>>> chases = GlueFormula(r"\x y.chases(x,y)", "(g -o (h -o f))")
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>>> print(chases)
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\x y.chases(x,y) : (g -o (h -o f))
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>>> a = GlueFormula("\P Q.some x.(P(x) and Q(x))", "((hv -o hr) -o ((h -o H) -o H))")
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>>> print(a)
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\P Q.exists x.(P(x) & Q(x)) : ((hv -o hr) -o ((h -o H) -o H))
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>>> dog = GlueFormula(r"\x.dog(x)", "(hv -o hr)")
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>>> print(dog)
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\x.dog(x) : (hv -o hr)
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Noun Quantification can only be done one way:
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>>> every_girl = every.applyto(girl)
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>>> print(every_girl.simplify())
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\Q.all x.(girl(x) -> Q(x)) : ((g -o G) -o G)
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>>> a_dog = a.applyto(dog)
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>>> print(a_dog.simplify())
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\Q.exists x.(dog(x) & Q(x)) : ((h -o H) -o H)
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The first reading is achieved by combining 'chases' with 'a dog' first.
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Since 'a girl' requires something of the form '(h -o H)' we must
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get rid of the 'g' in the glue of 'see'. We will do this with
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the '-o elimination' rule. So, x1 will be our subject placeholder.
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>>> xPrime = GlueFormula("x1", "g")
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>>> print(xPrime)
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x1 : g
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>>> xPrime_chases = chases.applyto(xPrime)
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>>> print(xPrime_chases.simplify())
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\y.chases(x1,y) : (h -o f)
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>>> xPrime_chases_a_dog = a_dog.applyto(xPrime_chases)
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>>> print(xPrime_chases_a_dog.simplify())
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exists x.(dog(x) & chases(x1,x)) : f
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Now we can retract our subject placeholder using lambda-abstraction and
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combine with the true subject.
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>>> chases_a_dog = xPrime_chases_a_dog.lambda_abstract(xPrime)
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>>> print(chases_a_dog.simplify())
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\x1.exists x.(dog(x) & chases(x1,x)) : (g -o f)
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>>> every_girl_chases_a_dog = every_girl.applyto(chases_a_dog)
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>>> r1 = every_girl_chases_a_dog.simplify()
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>>> r2 = GlueFormula(r'all x.(girl(x) -> exists z1.(dog(z1) & chases(x,z1)))', 'f')
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>>> r1 == r2
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True
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The second reading is achieved by combining 'every girl' with 'chases' first.
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>>> xPrime = GlueFormula("x1", "g")
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>>> print(xPrime)
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x1 : g
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>>> xPrime_chases = chases.applyto(xPrime)
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>>> print(xPrime_chases.simplify())
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\y.chases(x1,y) : (h -o f)
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>>> yPrime = GlueFormula("x2", "h")
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>>> print(yPrime)
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x2 : h
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>>> xPrime_chases_yPrime = xPrime_chases.applyto(yPrime)
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>>> print(xPrime_chases_yPrime.simplify())
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chases(x1,x2) : f
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>>> chases_yPrime = xPrime_chases_yPrime.lambda_abstract(xPrime)
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>>> print(chases_yPrime.simplify())
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\x1.chases(x1,x2) : (g -o f)
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>>> every_girl_chases_yPrime = every_girl.applyto(chases_yPrime)
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>>> print(every_girl_chases_yPrime.simplify())
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all x.(girl(x) -> chases(x,x2)) : f
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>>> every_girl_chases = every_girl_chases_yPrime.lambda_abstract(yPrime)
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>>> print(every_girl_chases.simplify())
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\x2.all x.(girl(x) -> chases(x,x2)) : (h -o f)
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>>> every_girl_chases_a_dog = a_dog.applyto(every_girl_chases)
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>>> r1 = every_girl_chases_a_dog.simplify()
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>>> r2 = GlueFormula(r'exists x.(dog(x) & all z2.(girl(z2) -> chases(z2,x)))', 'f')
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>>> r1 == r2
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True
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Compilation
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-----------
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>>> for cp in GlueFormula('m', '(b -o a)').compile(Counter()): print(cp)
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m : (b -o a) : {1}
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>>> for cp in GlueFormula('m', '((c -o b) -o a)').compile(Counter()): print(cp)
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v1 : c : {1}
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m : (b[1] -o a) : {2}
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>>> for cp in GlueFormula('m', '((d -o (c -o b)) -o a)').compile(Counter()): print(cp)
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v1 : c : {1}
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v2 : d : {2}
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m : (b[1, 2] -o a) : {3}
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>>> for cp in GlueFormula('m', '((d -o e) -o ((c -o b) -o a))').compile(Counter()): print(cp)
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v1 : d : {1}
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v2 : c : {2}
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m : (e[1] -o (b[2] -o a)) : {3}
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>>> for cp in GlueFormula('m', '(((d -o c) -o b) -o a)').compile(Counter()): print(cp)
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v1 : (d -o c) : {1}
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m : (b[1] -o a) : {2}
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>>> for cp in GlueFormula('m', '((((e -o d) -o c) -o b) -o a)').compile(Counter()): print(cp)
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v1 : e : {1}
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v2 : (d[1] -o c) : {2}
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m : (b[2] -o a) : {3}
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Demo of 'a man walks' using Compilation
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---------------------------------------
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Premises
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>>> a = GlueFormula('\\P Q.some x.(P(x) and Q(x))', '((gv -o gr) -o ((g -o G) -o G))')
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>>> print(a)
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\P Q.exists x.(P(x) & Q(x)) : ((gv -o gr) -o ((g -o G) -o G))
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>>> man = GlueFormula('\\x.man(x)', '(gv -o gr)')
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>>> print(man)
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\x.man(x) : (gv -o gr)
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>>> walks = GlueFormula('\\x.walks(x)', '(g -o f)')
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>>> print(walks)
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\x.walks(x) : (g -o f)
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Compiled Premises:
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>>> counter = Counter()
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>>> ahc = a.compile(counter)
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>>> g1 = ahc[0]
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>>> print(g1)
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v1 : gv : {1}
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>>> g2 = ahc[1]
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>>> print(g2)
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v2 : g : {2}
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>>> g3 = ahc[2]
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>>> print(g3)
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\P Q.exists x.(P(x) & Q(x)) : (gr[1] -o (G[2] -o G)) : {3}
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>>> g4 = man.compile(counter)[0]
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>>> print(g4)
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\x.man(x) : (gv -o gr) : {4}
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>>> g5 = walks.compile(counter)[0]
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>>> print(g5)
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\x.walks(x) : (g -o f) : {5}
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Derivation:
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>>> g14 = g4.applyto(g1)
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>>> print(g14.simplify())
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man(v1) : gr : {1, 4}
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>>> g134 = g3.applyto(g14)
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>>> print(g134.simplify())
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\Q.exists x.(man(x) & Q(x)) : (G[2] -o G) : {1, 3, 4}
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>>> g25 = g5.applyto(g2)
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>>> print(g25.simplify())
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walks(v2) : f : {2, 5}
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>>> g12345 = g134.applyto(g25)
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>>> print(g12345.simplify())
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exists x.(man(x) & walks(x)) : f : {1, 2, 3, 4, 5}
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---------------------------------
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Dependency Graph to Glue Formulas
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---------------------------------
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>>> from nltk.corpus.reader.dependency import DependencyGraph
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>>> depgraph = DependencyGraph("""1 John _ NNP NNP _ 2 SUBJ _ _
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... 2 sees _ VB VB _ 0 ROOT _ _
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... 3 a _ ex_quant ex_quant _ 4 SPEC _ _
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... 4 dog _ NN NN _ 2 OBJ _ _
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... """)
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>>> gfl = GlueDict('nltk:grammars/sample_grammars/glue.semtype').to_glueformula_list(depgraph)
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>>> print(gfl) # doctest: +SKIP
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[\x y.sees(x,y) : (f -o (i -o g)),
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\x.dog(x) : (iv -o ir),
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\P Q.exists x.(P(x) & Q(x)) : ((iv -o ir) -o ((i -o I3) -o I3)),
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\P Q.exists x.(P(x) & Q(x)) : ((fv -o fr) -o ((f -o F4) -o F4)),
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\x.John(x) : (fv -o fr)]
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>>> glue = Glue()
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>>> for r in sorted([r.simplify().normalize() for r in glue.get_readings(glue.gfl_to_compiled(gfl))], key=str):
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... print(r)
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exists z1.(John(z1) & exists z2.(dog(z2) & sees(z1,z2)))
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exists z1.(dog(z1) & exists z2.(John(z2) & sees(z2,z1)))
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-----------------------------------
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Dependency Graph to LFG f-structure
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-----------------------------------
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>>> from nltk.sem.lfg import FStructure
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>>> fstruct = FStructure.read_depgraph(depgraph)
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>>> print(fstruct) # doctest: +SKIP
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f:[pred 'sees'
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obj h:[pred 'dog'
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spec 'a']
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subj g:[pred 'John']]
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>>> fstruct.to_depgraph().tree().pprint()
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(sees (dog a) John)
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---------------------------------
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LFG f-structure to Glue
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---------------------------------
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>>> fstruct.to_glueformula_list(GlueDict('nltk:grammars/sample_grammars/glue.semtype')) # doctest: +SKIP
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[\x y.sees(x,y) : (i -o (g -o f)),
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\x.dog(x) : (gv -o gr),
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\P Q.exists x.(P(x) & Q(x)) : ((gv -o gr) -o ((g -o G3) -o G3)),
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\P Q.exists x.(P(x) & Q(x)) : ((iv -o ir) -o ((i -o I4) -o I4)),
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\x.John(x) : (iv -o ir)]
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.. see gluesemantics_malt.doctest for more
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