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LING 388 Language and Computers

LING 388 Language and Computers. Lecture 15 10/21 /03 Sandiway FONG. Administrivia. Homework Assignment 3 Due today Need help? 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).

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LING 388 Language and Computers

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

  2. Administrivia • Homework Assignment 3 • Due today • Need help? • Error in Lecture 10 slide for anbncn 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. l-Calculus • Lambda (l) calculus: • Concerned with the definition of anonymous functions and their application • Invented by Alonzo Church to explore computability of functions • Example definition: • lx.x+3 “is a function that adds 3 to x” • Note : • we did not give this function a name - hence the term “anonymous”

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

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

  6. Last Time … • The rule that generated the empty category [NP ei] allowed for considerable overgeneration • Examples: • *Hit the ball [NP ei] hit the ball • *John hit John hit [NP ei] • *Hit [NP ei] hit [NP ei] • … 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 … • Filter: • All variables must be bound by an operator (lx) • Example: • *John hit s(np(john),vp(v(hit),np(x))) • Today’s Lecture: • How to implement this filter

  8. Condition on Representation • Basic Idea: • Scan phrase structure looking for a variable and operator • Succeed only if matching operator found • 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. s vp np np v John [3,sg] det n likes [3,sg] the man Condition on Representation • 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 P,N

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

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

  12. A vp B v np ran det n C the man Tree-Walker X F=np A1 A2 • 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 X F=vp A

  13. s np vp v np john det n saw the man Tree-Walker • visit/1 example: • Inorder traversal 1 2 4 5 3 7 6 8 10 9 11

  14. Tree-Walker • visit/1 doesn’t really do anything yet • Need to add a specific test for np(x) • Example: • *John hit s(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) • 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). • visit(np(x)). • visit(X) :- • X =.. [F,A1,A2], • visit(A1), • visit(A2). • visit(X) :- • X =.. [F,A], • visit(A). • visit(X) :- atom(X). • Succeeds always • Succeeds only if np(x) exists

  16. Tree-Walker • New visit/1 example for *John hit [NPe]: s visit(X) fails visit(X) succeeds np vp v np john hit x

  17. Visit A1 and A2 Either visit A1 or A2 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(X) :- • X =.. [F,A1,A2], • visit(A2). • visit(X) :- • X =.. [F,A], • visit(A). • visit(np(x)). • visit(X) :- • X =.. [F,A1,A2], • visit(A1), • visit(A2). • visit(X) :- • X =.. [F,A], • visit(A). • visit(X) :- atom(X).

  18. Tree-Walker • 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 • Examples: • ?- variable(s(np(john),vp(v(hit),np(x)))). • Yes • ?- variable(s(np(john),vp(v(hit),np(mary)))). • No • ?-variable(np(np(det(the),n(cat)), • lambda(x,s(np(i),vp(v(saw),np(x)))))). • Yes

  19. Tree-Walker • Let’s build another modified version of visit/1 that succeeds only if there exists an operator lx, 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)))). • No • ?- operator(s(np(john),vp(v(hit),np(mary)))). • No • ?-operator(np(np(det(the),n(cat)), • lambda(x,s(np(i),vp(v(saw),np(x)))))). • Yes

  20. Filter Implementation • Filter: • All variables must be bound by an operator (lx) • 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: • ?- 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: • All variables must be bound by an operator (lx) • Implementation: • filter(X) :- variable(X), operator(X). • filter(X) :- \+ variable(X). • 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: • filter(X) :- variable(X) -> operator(X) ; true. • Advantage: one clause implementation instead of two … • 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

  23. Filter Implementation • Note: • filter(X) :- variable(X) -> operator(X) ; true. is only a first approximation implementation of • All variables must be bound by an operator (lx) • 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))) 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

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