LING 388 Language and Computers Lecture 16 10/23/03 Sandiway FONG

Administrivia

Review Last time, we discussed how to implement the filter: Last time, we discussed how to implement the filter:  All variables must be bound by an operator ( x)  To block examples like:  *John hits(np(john),vp(v(hit),np(x))) Implementation: Implementation:  Based on a tree-walker (disjunctive form) that searched and returned true if a variable np(x) and an accompanying operator x could be found in phrase structure

Review: Disjunctive Tree-Walker  variable(np(x)).  variable(X) :-  X =.. [F,A1,A2],  variable(A1).  variable(X) :-  X =.. [F,A1,A2],  variable(A2).  variable(X) :-  X =.. [F,A],  variable(A). Given some tree structure X: variable(X) holds if there is some np(x) in X Clause by clause: 1.Matches np(x) (and stops) 2.Recursively calls variable(A1) when X is of form F(A1,A2) 3.Recursively calls variable(A2) when X is of form F(A1,A2) 4.Recursively calls variable(A) when X is of form F(A)

Review: Disjunctive Tree-Walker  variable(np(x)).  variable(X) :-  X =.. [F,A1,A2],  variable(A1).  variable(X) :-  X =.. [F,A1,A2],  variable(A2).  variable(X) :-  X =.. [F,A],  variable(A). Notes: 1. Every clause for variable/1 except for the first recursively calls variable/1 on a subtree 2.Only way variable/1 can terminate and succeed is if it terminates in the 1st clause

Review: Disjunctive Tree-Walker Worked example for *John hit [ NP e] Worked example for *John hit [ NP e] Prolog query: Prolog query:  ?- variable(s(np(john),vp(v(hit),np(x)))). s npvp vnp x hit john

Review: Disjunctive Tree-Walker  variable(np(x)).  variable(X) :-  X =.. [F,A1,A2],  variable(A1).  variable(X) :-  X =.. [F,A1,A2],  variable(A2).  variable(X) :-  X =.. [F,A],  variable(A). Example of derivation: ?- variable(s(np(john),vp(v(hit),np(x)))). ?- variable(np(john)) ?- variable(john). FAILS ?- variable(vp(v(hit),np(x))). ?- variable(v(hit)). ?- variable(hit). FAILS ?- variable(np(x)). SUCCEEDS Note: ?- variable(s(np(john),vp(v(hit),np(mary)))). FAILS ?-variable ( np(np(det(the),n(cat)),lambda(x,s(np(i),vp( v(saw),np(x)))))). SUCCEEDS

Review: Disjunctive Tree-Walker Operator/1 is a x, i.e. lambda(x,_), in the phrase structure Operator/1 is a x, i.e. lambda(x,_), in the phrase structure  operator(lambda(x,_)).  operator(X) :-  X =.. [F,A1,A2],  operator(A1).  operator(X) :-  X =.. [F,A1,A2],  operator(A2).  operator(X) :-  X =.. [F,A],  operator(A). Examples: ?- operator(s(np(john),vp(v(hit),np(x)))).?- operator(s(np(john),vp(v(hit),np(x)))). NoNo ?- operator(s(np(john),vp(v(hit),np(mary)))).?- operator(s(np(john),vp(v(hit),np(mary)))). NoNo ?-operator ( np(np(det(the),n(cat)),?-operator ( np(np(det(the),n(cat)),lambda(x,s(np(i),vp(v(saw),np(x)))))). YesYes

Review: Filter Implementation Filter: Filter:  All variables must be bound by an operator ( x) Implementation (Initial try) : Implementation (Initial try) :  Using predicates:  variable(X)np(x) finder  operator(X)lambda(x, … ) finder  Define:  filter(X) :- variable(X) -> operator(X).  To be read as: filter(X) holds if operator(X) holds when variable(X) holdsfilter(X) holds if operator(X) holds when variable(X) holds  Note: -> is the Prolog built-in “if … then” construct

Review: Filter Implementation Use : Use :  ?- s(X,Sentence,[]), filter(X).  X is the phrase structure returned by the DCG  Sentence is the input sentence encoded as a list Problem with definition: Problem with definition:  filter(X) :- variable(X) -> operator(X).  Semantics:  Holds if there is some variable that has an accompanying operator  Question:  What happens if we have two variables only one of which is properly bound?  Example: filter(X) fails to exclude …  *the cat that I saw hit *the cat that I saw [ NP e] hit [ NP e] s(np(np(det(the),n(cat)),lambda(x,s(np(i),vp(v(saw),np(x))))),vp(v(hit),np(x))) s(np(np(det(the),n(cat)),lambda(x,s(np(i),vp(v(saw),np(x))))),vp(v(hit),np(x))) ?

Filter Implementation: Part 2 A better (i.e. more faithful) implementation… A better (i.e. more faithful) implementation…  When we find np(x), we check nodes dominating np(x) for signs of x If we find lambda, we know np(x) is bound by that operator Implementation Question: Implementation Question:  We know predicate variable/1 finds np(x)  How do we modify it to report when it finds lambda as well? lambda(x, … ) np(x) s lambda x variable/1 true

Filter Implementation: Part 2 Idea: Idea:  Modify predicate variable/1 to produce a sign of some sort if it sees a lambda on the way down to np(x)  Note: if lambda doesn’t dominate np(x), we don’t want a sign  Use the extra argument mechanism from previous lectures for feature passing (person/number agreement)  We’ll pass an arbitrary feature, call it f, if we see an appropriate lambda lambda(x, … ) np(x) s lambda x variable(X,F) true F unbound F = f

Filter Implementation: Part 2  variable(np(x)).  variable(X) :-  X =.. [F,A1,A2],  variable(A1).  variable(X) :-  X =.. [F,A1,A2],  variable(A2).  variable(X) :-  X =.. [F,A],  variable(A).  variable(np(x),_).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A1,F).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F). We need to modify variable/1 so that we can pass information indicating the presence or absence of lambda as we traverse the tree We do this by first adding a 2nd argument F to variable/1

Filter Implementation: Part 2  variable(np(x),_).  variable(lambda(x,Y),f) :-  variable(Y,_).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A1,F).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F). step 2: add a clause to detect and report the presence of lambda  variable(np(x),_).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A1,F).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F).

Filter Implementation: Part 2  variable(np(x),_).  variable(lambda(x,Y),f) :-  variable(Y,_).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A1,F).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F). Notes: 1.Every clause for variable/2 except for the first one recursively calls variable/2 on a subtree Only way variable/2 can succeed is if it terminates in the 1st clause 2.When variable/2 holds, we can inspect the 2nd argument to see if along the way it found a lambda expression as well This is signaled by the f feature The 2nd clause exclusively sets the 2nd argument

Filter Implementation: Part 2 np(x) s lambda x x  variable(np(x),_).  variable(lambda(x,Y),f) :-  variable(Y,_).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A1,F).  variable(X,F) :-  X =.. [_,A1,A2],  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F). Underscore (don’t care variable) because we’re not interested in reporting multiple lambdas dominating np(x) sets 2nd argument to be f

Filter Implementation: Part 2  variable(np(x),_).  variable(lambda(x,Y),f) :-  variable(Y,_).  variable(X,F) :-  X =.. [Y,A1,A2],  \+ Y = lambda,  variable(A1,F).  variable(X,F) :-  X =.. [Y,A1,A2],  \+ Y = lambda,  variable(A2,F).  variable(X,F) :-  X =.. [_,A],  variable(A,F). Since clause 2 must set f exclusively, we must prevent lambda from matching these two clauses

Filter Implementation: Part 2 Use: Use:  filter(X) :- variable(X,F), F == f. Note: Note:  Filter/1 holds if variable/2 holds and the value reported in the 2nd argument F is identical to f  We use identity (==) instead of unification (=) because we’re testing whether F has been bound to a value

Identity (==) == ==  is a Prolog built-in taking two arguments  compares but does not perform unification  Note: = attempts to unify its arguments= attempts to unify its arguments Hence, neither of the clausesHence, neither of the clauses –filter(X) :- variable(X,F), F = f. –filter(X) :- variable(X,f). will work for our purposes  Examples:  ?- abc == abc.Yes?- abc = abc.Yes  ?- X == abc.No?- X = abc.Yes  ?- X = abc, X == abc.Yes?- X = abc, X = abc. Yes  ?- X == Y.No?- X == Y.Yes

Filter Implementation: Part 3 Definition: Definition:  filter(X) :- variable(X,F), F == f. Is still not quite right … Is still not quite right …  filter(X) holds if there exists one valid variable/lambda pair in X However … However …  This fails to block the case where there may be more than one variable, only one of which has a valid accompanying lambda expression, as in:  *the cat that I saw hit  s(np(np(det(the),n(cat)),lambda(x,s(np(i),vp(v(saw),np(x))))), vp(v(hit),np(x))) vp(v(hit),np(x))) In other words, variable/1 is true for the whole expression because it finds the lambda/np(x) pair

Filter Implementation: Part 3 Solution: Solution:  Want filter/1 to hold if there are no configurations for which variable(X,F) holds and F remains unbound New Definition: New Definition:  filter(X) :- \+ ( variable(X,F), var(F) ).  to be read as:  filter(X) holds if it’s not the case that when variable(X,F) succeeds and F is uninstantiated Note: Note:  var(F) is a Prolog built-in that holds if F is uninstantiated, i.e. a Prolog variable

var/1 var(X) var(X)  is a Prolog built-in that takes one argument  holds if X is a variable at the time of the call to var/1 Examples: Examples:  ?- var(X).Yes  ?- var(f).No  ?- X=f, var(X).No  ?- var(X), X = 2.Yes … fact that X is no longer a variable after unifying with 2 doesn’t change the semantics of var/1

Summary: Filter Implementation Summarizing: Summarizing:  filter(X) :- \+ ( variable(X,F), var(F) ) implements filter:  All variables must be bound by an operator ( x) variable(X,F) variable(X,F)  is a disjunctive tree-walker that looks for np(x)  also looks for lambda(x,Y) where Y contains a variable np(x)  returns f as the value of F if it finds the lambda above variable(X,F), var(F) variable(X,F), var(F)  will succeed if there is an np(x) without an accompanying lambda expression \+ ( variable(X,F), var(F) ) \+ ( variable(X,F), var(F) )  will succeed if there aren’t any np(x)s without an accompanying lambda expression

Next Time … We’ll look at how to implement the other constraint mentioned in Lecture 14: We’ll look at how to implement the other constraint mentioned in Lecture 14:  All variables must be bound by an operator ( x)  Operator ( x) can only bind one variable