1. LING 388 Language and Computers Lecture 15 10/21/03 Sandiway FONG

2. Administrivia Homework Assignment 3 Homework Assignment 3  Due today  Need help? Error in Lecture 10 slide for a n b n c n fixed … Error in Lecture 10 slide for a n b n c n fixed …  s --> [a],t(1),[c].  t(N) --> [a],{M is N+1},t(M),[c].  t(N) --> u(N).  u(N) --> {N>1}, [b],{M is N-1},u(M).  u(1) --> [b].

3. -Calculus Lambda ( ) calculus: Lambda ( ) calculus:  Concerned with the definition of anonymous functions and their application  Invented by Alonzo Church to explore computability of functions Example definition: Example definition:  x.x+3“is a function that adds 3 to x”  Note : we did not give this function a name - hence the term “anonymous”we did not give this function a name - hence the term “anonymous”

4. -Calculus Example of application: Example of application:   x.x+3) 2“apply lambda function to 2” (  -reduction)   x+3) [2/x]“substitute 2 for x”  2+3“evaluate function”  5 There is much more to -calculus There is much more to -calculus  But this is all we need …

5. Last Time … We looked at the parsing of empty NPs in relative clause constructions We looked at the parsing of empty NPs in relative clause constructions  Example:  the cat that I saw  the cat i that I saw [ NP e i ]  (the cat)( x.I saw x) By analogy:  that I saw [ NP e i ] has the semantics of x.I saw x  This function returns x

6. Last Time … The rule that generated the empty category [ NP e i ] allowed for considerable overgeneration The rule that generated the empty category [ NP e i ] allowed for considerable overgeneration  Examples:  *Hit the ball[ NP e i ] hit the ball  *John hitJohn hit [ NP e i ]  *Hit [ NP e i ] hit [ NP e i ]  … all of which can be accepted as complete sentences by the DCG

7. Last Time … To control the overgeneration, we needed some condition on representation … To control the overgeneration, we needed some condition on representation …  Filter:  All variables must be bound by an operator ( x)  Example:  *John hits(np(john),vp(v(hit),np(x))) Today’s Lecture: Today’s Lecture:  How to implement this filter

8. Condition on Representation Basic Idea: Basic Idea:  Scan phrase structure looking for a variable and operator  Succeed only if matching operator found Scan Operation: Scan Operation:  Tree-walker  Walk through a tree node-by-node, testing each node to see if we have found what we’re looking for

9. Condition on Representation Also make use of two elements that we’ve already seen: Also make use of two elements that we’ve already seen:  =.. (univ) man(X) =.. [man,X]  to deconstruct structures  Feature propagation  to send information up the tree s np vp v np detn theman likes [3,sg] John [3,sg] P,N

10. Tree-Walker Question: Question:  How to scan phrase structure tree? Answer: Answer:  Write a recursive predicate, a tree- walker, to visit every node of a tree

11. Tree-Walker Assuming phrase structure trees are binary branching at the most… Assuming phrase structure trees are binary branching at the most… Tree-Walker must deal with three cases: Tree-Walker must deal with three cases: A. Binary branching visit both subtrees B. Unary branching visit child subtree C. Terminal node done np detn theman vp ran v A B C

12. Tree-Walker Example Prolog code: Example Prolog code: Define a predicate visit/1 taking as its argument a phrase structure tree  visit(X) :- % Case A  X =.. [F,A1,A2],  visit(A1),  visit(A2).  visit(X) :-% Case B  X =.. [F,A],  visit(A).  visit(X) :-% Case C  atom(X).atom/1 built-in np detn theman vp ran v A B C X F=np A1 A2 X F=vp A

13. Tree-Walker visit/1 example: visit/1 example:  Inorder traversal s npvp vnp detn theman saw john 1 11 10 9 8 7 6 5 4 3 2

14. Tree-Walker visit/1 doesn’t really do anything yet visit/1 doesn’t really do anything yet  Need to add a specific test for np(x)  Example:  *John hits(np(john),vp(v(hit),np(x)))  visit(X) :-  X =.. [F,A1,A2],  visit(A1),  visit(A2).  visit(X) :-  X =.. [F,A],  visit(A).  visit(X) :- atom(X). visit(np(x)).

15. Tree-Walker We can write visit/1 to succeed only if it finds np(x) We can write visit/1 to succeed only if it finds np(x)  visit(np(x)).  visit(X) :-  X =.. [F,A1,A2],  visit(A1),  visit(A2).  visit(X) :-  X =.. [F,A],  visit(A).  visit(X) :- atom(X).  visit(np(x)).  visit(X) :-  X =.. [F,A1,A2],  visit(A1).  visit(X) :-  X =.. [F,A1,A2],  visit(A2).  visit(X) :-  X =.. [F,A],  visit(A). Succeeds always Succeeds only if np(x) exists

16. Tree-Walker New visit/1 example for *John hit [ NP e]: New visit/1 example for *John hit [ NP e]: s npvp vnp x hit john visit(X) succeeds visit(X) fails

17. Tree-Walker Call the first form a conjunctive tree-walker, the 2nd a disjunctive tree-walker Call the first form a conjunctive tree-walker, the 2nd a disjunctive tree-walker  visit(np(x)).  visit(X) :-  X =.. [F,A1,A2],  visit(A1),  visit(A2).  visit(X) :-  X =.. [F,A],  visit(A).  visit(X) :- atom(X).  visit(np(x)).  visit(X) :-  X =.. [F,A1,A2],  visit(A1).  visit(X) :-  X =.. [F,A1,A2],  visit(A2).  visit(X) :-  X =.. [F,A],  visit(A). Either visit A1 or A2 Visit A1 and A2

18. Tree-Walker Let’s rename our modified version of visit/1 to reflect the new semantics Let’s rename our modified version of visit/1 to reflect the new semantics  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/1 holds only if there exists np(x) in the phrase structure variable/1 holds only if there exists np(x) in the phrase structure Examples: ?- variable(s(np(john),vp(v(hit),np(x)))).?- variable(s(np(john),vp(v(hit),np(x)))). YesYes ?- variable(s(np(john),vp(v(hit),np(mary)))).?- variable(s(np(john),vp(v(hit),np(mary)))). NoNo ?-variable ( np(np(det(the),n(cat)),?-variable ( np(np(det(the),n(cat)),lambda(x,s(np(i),vp(v(saw),np(x)))))). YesYes

19. Tree-Walker Let’s build another modified version of visit/1 that succeeds only if there exists an operator  x, i.e. lambda(x,_), in the phrase structure Let’s build another modified version of visit/1 that succeeds only if there exists an operator  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

20. Filter Implementation Filter: Filter:  All variables must be bound by an operator ( x) Implementation (Initial try) : Implementation (Initial try) :  filter(X) :- variable(X), operator(X).  filter(X) :- \+ variable(X).  succeeds either when there is no variable in X or when a variable and an operator co-occur in X 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  filter/1 is a condition on representation

21. Filter Implementation Filter: Filter:  All variables must be bound by an operator ( x) Implementation : Implementation :  filter(X) :- variable(X), operator(X).  filter(X) :- \+ variable(X). Description: Description:  filter(X) succeeds either when there is no variable in X or when a variable and an operator co-occur in X

22. Filter Implementation A simpler rendering of filter/1 : A simpler rendering of filter/1 :  filter(X) :- variable(X) -> operator(X) ; true.  Advantage: one clause implementation instead of two … Note: Note:  -> is the “then” operator in Prolog  compare with the DCG operator -->  Semantics:  X -> Y ; Z if X is true then Y must be true if X is not true, Z must be true if X is not true, Z must be true

23. Filter Implementation Note: Note:  filter(X) :- variable(X) -> operator(X) ; true. is only a first approximation implementation of  All variables must be bound by an operator ( x)  Example:  *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))) filter/1 (incorrectly) holds for the above example because:  filter/1 looks only for one variable, and  filter/1 looks for an operator anywhere in the phrase structure