Continuations in Type Logical Grammar. Grafting Trees: (with Chris Barker, UCSD) NJPLS, 27 February Harvard University

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1 Grafting Trees: Continuations in Type Logical Grammar Chung-chieh Shan Harvard University (with Chris Barker, UCSD) NJPLS, 27 February 2004

2 Computational Linguistics 1

3 What a linguist cares about Entailment Nobody liked any course Nobody liked any computer science course Somebody liked a course Somebody liked a computer science course Truth conditions are part of sentence meanings. 2

4 What a linguist cares about Entailment Nobody liked any course Nobody liked any computer science course Somebody liked a course Somebody liked a computer science course Truth conditions are part of sentence meanings. Ambiguity Nobody liked a course. Nobody liked any course. linear scope x. y. liked(x, y) y. x. liked(x, y) inverse scope You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time. Abraham Lincoln 2

5 What a linguist cares about Entailment Nobody liked any course Nobody liked any computer science course Somebody liked a course Somebody liked a computer science course Truth conditions are part of sentence meanings. Ambiguity Nobody liked a course. Nobody liked any course. linear scope x. y. liked(x, y) y. x. liked(x, y) inverse scope You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time. Abraham Lincoln Acceptability No student liked any course *Every student liked any course Few students liked any course *Any student liked no course 2

6 What a linguist cares about Entailment Nobody liked any course Nobody liked any computer science course Somebody liked a course Somebody liked a computer science course Truth conditions are part of sentence meanings. Ambiguity Nobody liked a course. Nobody liked any course. linear scope x. y. liked(x, y) y. x. liked(x, y) inverse scope You can fool some of the people all of the time, and all of the people some of the time, but you can not fool all of the people all of the time. Abraham Lincoln Acceptability No student liked any course *Every student liked any course Few students liked any course *Any student liked no course This talk deals with English, but the approach hopefully extends to other languages. 2

7 The Curry-Howard isomorphism Syntax : Deduction rules for proving grammaticality Semantics : Translation to a logical metalanguage Acceptability Ambiguity Entailment 3

8 Outline OLD Multiplicative linear logic for linguistics Alice liked Bob. NEW Delimited continuations for quantification Alice liked everybody s mother. OLD Unary modalities for polarity sensitivity *Alice liked anybody s mother. Nobody liked anybody s mother. NEW Evaluation order for linear precedence *Anybody liked nobody s mother. NEW Staging for scope ambiguities Somebody liked everybody s mother. Payoffs For linguistics: Cover more empirical data. Relate (denotational) semantics to (operational) psycholinguistics? For computer science: Understand delimited continuations geometrically and logically. Staging side effects? 4

9 Linear logic for linguistics Alice liked Bob. *Alice liked. *Alice liked Bob NJPLS. *Alice Bob liked. liked (np\s)/np Bob np /E Alice np liked Bob np\s \E Alice (liked Bob) s Alice liked A sequent is complete iff its antecedent is built up using the connective only and its conclusion is s ( sentence ). Bob 5

10 Linear logic for linguistics Alice liked Bob. *Alice liked. *Alice liked Bob NJPLS. *Alice Bob liked. liked (np\s)/np Bob np /E Alice np liked Bob np\s \E Alice (liked Bob) s Alice liked A sequent is complete iff its antecedent is built up using the connective only and its conclusion is s ( sentence ). Bob Natural-deduction rules for categorial grammar (a kind of type logical grammar) Γ B C I Γ B C C A /I A/C B A \I B\A B C Γ{B C} A E Γ{ } A A/C Γ C /E Γ A Γ B B\A \E Γ A A A Axiom Tensor B C means B followed by C (non-commutative; non-associative) Implications A/C means makes A before C B\A means makes A after B 5

11 In-situ quantification Everybody liked Bob. Somebody liked Bob. Nobody liked Bob. liked (np\s)/np Bob np /E nobody s/(np\s) liked Bob np\s /E nobody (liked Bob) s nobody liked Bob 6

12 In-situ quantification Everybody liked Bob. Somebody liked Bob. Nobody liked Bob. liked (np\s)/np Bob np /E nobody s/(np\s) liked Bob np\s /E nobody (liked Bob) s nobody liked Bob Alice liked [nobody] s mother. [Nobody] s mother liked Bob. s mother nobody Bob Alice nobody liked s mother liked 6

13 Delimited continuations Alice ( liked (nobody s mother) ) nobody ( s mother ((1 Alice) liked) ) s mother 1 s mother Alice nobody Alice nobody liked liked 7

14 Delimited continuations Alice ( liked (nobody s mother) ) nobody ( s mother ((1 Alice) liked) ) s mother 1 s mother Alice nobody Alice nobody liked liked liked nobody Alice s mother 1 7

15 Delimited continuations Alice ( liked (nobody s mother) ) nobody ( s mother ((1 Alice) liked) ) s mother 1 s mother Alice nobody Alice nobody liked liked Additional natural-deduction rules for multimodal categorial grammar Γ B C I Γ B C B C Γ{B C} A E Γ{ } A C A I A C B A I B A A C Γ C E Γ A Γ B B A E Γ A Coming up: nobody s (np s) Tensor B C means B plugged into C (non-commutative; non-associative) Implications A C means makes A inside C B A means makes A outside B 7

16 Delimited continuations: structural postulates A Γ S A ============== Root A 1 1 Γ S A A (B C) C Γ S A B ==================== Left (A B) C C Γ S A B ==================== Right B (C A) C Γ S A B 8

17 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S 9

18 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S Γ { Alice (liked (nobody s mother)) } S ========================================= Root Γ {( Alice (liked (nobody s mother)) ) 1 } S ========================================= Right Γ {( liked (nobody s mother) ) (1 Alice) } S ========================================= Right Γ { (nobody s mother) ( (1 Alice) liked )} S ========================================= Left Γ { nobody ( s mother ((1 Alice) liked) )} S s mother Alice liked nobody 9

19 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S Γ { Alice (liked (nobody s mother)) } S ========================================= Root Γ {( Alice (liked (nobody s mother)) ) 1 } S ========================================= Right Γ {( liked (nobody s mother) ) (1 Alice) } S ========================================= Right Γ { (nobody s mother) ( (1 Alice) liked )} S ========================================= Left Γ { nobody ( s mother ((1 Alice) liked) )} S 1 s mother Alice liked nobody 9

20 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S Γ { Alice (liked (nobody s mother)) } S ========================================= Root Γ {( Alice (liked (nobody s mother)) ) 1 } S ========================================= Right Γ {( liked (nobody s mother) ) (1 Alice) } S ========================================= Right Γ { (nobody s mother) ( (1 Alice) liked )} S ========================================= Left Γ { nobody ( s mother ((1 Alice) liked) )} S 1 s mother Alice nobody liked 9

21 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S Γ { Alice (liked (nobody s mother)) } S ========================================= Root Γ {( Alice (liked (nobody s mother)) ) 1 } S ========================================= Right Γ {( liked (nobody s mother) ) (1 Alice) } S ========================================= Right Γ { (nobody s mother) ( (1 Alice) liked )} S ========================================= Left Γ { nobody ( s mother ((1 Alice) liked) )} S 1 s mother Alice nobody liked 9

22 Delimited continuations: structural postulates Γ{A} S ========== Root Γ{A 1} S Γ{A (B C)} S =============== Left Γ{(A B) C} S =============== Right Γ{B (C A)} S Γ { Alice (liked (nobody s mother)) } S ========================================= Root Γ {( Alice (liked (nobody s mother)) ) 1 } S ========================================= Right Γ {( liked (nobody s mother) ) (1 Alice) } S ========================================= Right Γ { (nobody s mother) ( (1 Alice) liked )} S ========================================= Left Γ { nobody ( s mother ((1 Alice) liked) )} S 1 s mother Alice nobody liked 9

23 Delimited continuations for in-situ quantification Axiom np np s mother np\np \E liked (np\s)/np np s mother np /E Alice np liked (np s mother) np\s \E Alice ( liked (np s mother) ) s Root,Right,Right,Left np ( s mother ((1 Alice) liked) ) s I nobody s (np s) s mother ( (1 Alice) liked ) np s E nobody ( s mother ((1 Alice) liked) ) s Left,Right,Right,Root Alice ( liked (nobody s mother) ) s Ambiguity in delimitation: Alice told Bob to criticize nobody s mother. 10

24 Delimited continuations for in-situ quantifications np ( liked (np s mother) ) s Root,Right,Right,Left np ( s mother ((1 np) liked) ) s I nobody s (np s) s mother ( (1 np) liked ) np s E nobody ( s mother ((1 np) liked) ) s Left,Right,Right,Left np ( (liked (nobody s mother)) 1 ) s I somebody s (np s) (liked (nobody s mother)) 1 np s E somebody ( (liked (nobody s mother)) 1 ) s Left,Root somebody ( liked (nobody s mother) ) s The quantifier evaluated earlier takes wider scope in the syntactic proof and the truth-conditional meaning. 11

25 Polarity sensitivity The quantifiers a, some, and any all look existential: Did a student call? Did some student call? Did any student call? x. student(x) called(x) But do not behave the same: No student liked some course. (unambiguous ) No student liked a course. (ambiguous, ) No student liked any course. (unambiguous ) Some student liked no course. (unambiguous ) A student liked no course. (ambiguous, ) *Any student liked no course. (unacceptable) Any is a negative polarity item: Very roughly, it requires negative contexts, such as in the scope of no. Some is a positive polarity item: Very roughly, it is allergic to negative contexts. Meaning affects ambiguity and acceptability! But linear order matters too. 12

26 Unary polarities Define three types for sentences of varying polarity: s = r s (verbs now return this type) s + = r [ p] p s s = [ p] p r s Unary type constructors like p and [ p] come in pairs. Each pair is an adjunction. Additional natural-deduction rules for unary connectives (focus on polarity) Γ A p Γ p A p I p A Γ{ p A} B p E Γ{ } B p Γ A [ p]i Γ [ p]a A handy lemma: Γ [ p]a [ p]e p Γ A Axiom p A p A [ p]i A [ p] p A Hence s s + and s s. Possibility p A means A for some p-child q A means A for some q-child r A means A for some r-child Necessity [ p]a means A for every p-parent [ q]a means A for every q-parent [ r]a means A for every r-parent 13

27 Unary modalities for polarity sensitivity np ( liked (np s mother) ) s p I p ( np (liked (np s mother)) ) p s [ p]i np ( liked (np s mother) ) s Root,Right,Right,Left np ( s mother ((1 np) liked) ) s I anybody s (np s ) s mother ( (1 np) liked ) np s E anybody ( s mother ((1 np) liked) ) s Left,Right,Right,Left np ( (liked (anybody s mother)) 1 ) s I nobody s (np s ) (liked (anybody s mother)) 1 np s E nobody ( (liked (anybody s mother)) 1 ) s Left,Root nobody ( liked (anybody s mother) ) s 14

28 Unary modalities for polarity sensitivity, cont d A sequent is complete iff its antecedent is built up using the connective only and its conclusion is of the form r A (that is, either s or s + ). Alice ( liked (np s mother) ) s p I p ( Alice (liked (np s mother)) ) p s [ p]i Alice ( liked (np s mother) ) s Root,Right,Right,Left np ( s mother ((1 Alice) liked) ) s I anybody s (np s ) s mother ( (1 Alice) liked ) np s E anybody ( s mother ((1 Alice) liked) ) s *Alice liked anybody s mother. Nobody said that Alice liked anybody s mother. 15

29 Unary modalities for polarity sensitivities s s + some any s s nobody s (np s ) anybody s (np s ) somebody s + (np s + ) a woman s (np s ) s + ε ε s s no a No student liked some course. (unambiguous ) No student liked a course. (ambiguous, ) Prediction: the quantifiers in a sentence must No student liked any course. (unambiguous ) form a valid transition Some student liked no course. (unambiguous ) sequence, from widest A student liked no course. (ambiguous, ) to narrowest scope. *Any student liked no course. (unacceptable) 16

30 Evaluation order These structural postulates in multimodal categorial grammar: Γ{A} S ========== Root Γ{A 1} S Γ{(A B) C} S =============== Left Γ{A (B C)} S Γ{(A B) C} S =============== Right Γ{B (C A)} S encode this recursive definition of evaluation contexts: C ::= { } C { { } B } C { A { } }. Modify the Right rule: Γ{A} S ========== Root Γ{A 1} S Γ{(A B) C} S =============== Left Γ{A (B C)} S Γ{( q A B) C} S ================= Right Γ{B (C q A)} S to enforce left-to-right evaluation in evaluation contexts: C ::= { } C { { } B } C { q A { } }. Here we mark pure values with the unary prefix q. 17

31 Staging The q modality stands for quotation of programs. Any program can be quoted: Γ{ q A} S Quote (T) Γ{A} S And any two programs can be concatenated: Γ{ q (A B)} S Concat (K ) Γ{ q A q B} S But only complete programs (with types of the form r A) can be unquoted (run): Γ{ r A} S Run Γ{ q r A} S These stipulations interact to make correct linguistic predictions. Nobody liked a woman s mother (ambiguous, ) *Anybody liked nobody s mother (unacceptable) 18

32 Evaluation order and staging for linear scope First of all, nobody likes a woman s mother can take linear scope. This is easy under left-to-right evaluation, because quantifiers evaluated earlier scope wider. np ( liked (np s mother) ) s q I q ( np (liked (np s mother)) ) q s Axiom s s Run q s s q E q ( np (liked (np s mother)) ) s Concat 2 q np ( q liked q (np s mother) ) s Quote q np ( q liked (np s mother) ) s Root,Right,Right,Left, I, E,Left,Right,Right,Root q np ( q liked (a woman s mother) ) s p I p ( q np ( q liked (a woman s mother)) ) p s [ p]i q np ( q liked (a woman s mother) ) s Quote 2 np ( liked (a woman s mother) ) s Root,Left, I, E,Left,Root nobody ( liked (a woman s mother) ) s 19

33 Evaluation order and staging for inverse scope Moreover, nobody likes a woman s mother can also take inverse scope, because the answer type s = r s returned by nobody can be unquoted. np ( liked (np s mother) ) s p I p ( np (liked (np s mother)) ) p s [ p]i np ( liked (np s mother) ) s Root,Left, I, E,Left,Root nobody ( liked (np s mother) ) s q I q ( nobody (liked (np s mother)) ) q s Axiom s s Run q s s q E q ( nobody (liked (np s mother)) ) s Concat 2 q nobody ( q liked q (np s mother) ) s Quote q nobody ( q liked (np s mother) ) s Root,Right,Right,Left, I, E,Left,Right,Right,Root q nobody ( q liked (a woman s mother) ) s Quote 2 nobody ( liked (a woman s mother)) s 20

34 Evaluation order and staging for polarity sensitivity Nevertheless, anybody likes nobody s mother cannot take inverse scope, because the answer type s = [ p] p r s returned by anybody cannot be unquoted. np ( liked (np s mother) ) s p I p ( np (liked (np s mother)) ) p s [ p]i np ( liked (np s mother) ) s Root,Left, I, E,Left,Root anybody ( liked (np s mother) ) s q I q ( anybody (liked (np s mother)) ) s Axiom s q s q s Run s q E q ( anybody (liked (np s mother)) ) s Concat 2 q anybody ( q liked q (np s mother) ) s Quote q anybody ( q liked (np s mother) ) s Root,Right,Right,Left, I, E,Left,Right,Right,Root q anybody ( q liked (nobody s mother) ) s Quote 2 anybody ( liked (nobody s mother)) s 21

35 An old puzzle solved s s + some s s nobody s (np s ) anybody s (np s ) somebody s + (np s + ) a woman s (np s ) s + ε No student liked some course. (unambiguous ) No student liked a course. (ambiguous, ) No student liked any course. (unambiguous ) Some student liked no course. (unambiguous ) A student liked no course. (ambiguous, ) *Any student liked no course. (unacceptable) any ε s s no a Prediction: the quantifiers in a sentence must form a valid transition sequence, from widest to narrowest scope. Also, inverse-scope transitions must pass through a start state. 22

36 An old puzzle solved, and a new one s s + some any s s nobody s (np s ) anybody s (np s ) somebody s + (np s + ) a woman s (np s ) everybody??? s + ε ε s s no a What is the type of everybody? Hint: A woman introduced everybody to somebody : linear scope ok. 23

37 An old puzzle solved, and a new one s s + some s s nobody s (np s ) anybody s (np s ) somebody s + (np s + ) a woman s (np s ) everybody s (np s + ) s + ε every ε any s s no a What is the type of everybody? Hint: A woman introduced everybody to somebody : linear scope ok. So every must turn the output of some into the input of a. 23

38 An old puzzle solved, and a new one s s + some s s nobody s (np s ) anybody s (np s ) somebody s + (np s + ) a woman s (np s ) everybody s (np s + ) s + ε every ε any s s no a What is the type of everybody? Hint: A woman introduced everybody to somebody : linear scope ok. So every must turn the output of some into the input of a. Confirmation: Nobody introduced Alice to somebody : linear scope bad. Nobody introduced everybody to somebody : linear scope ok again! 23

39 Outline OLD Multiplicative linear logic for linguistics Alice liked Bob. NEW Delimited continuations for quantification Alice liked everybody s mother. OLD Unary modalities for polarity sensitivity *Alice liked anybody s mother. Nobody liked anybody s mother. NEW Evaluation order for linear precedence *Anybody liked nobody s mother. NEW Staging for scope ambiguities Somebody liked everybody s mother. Payoffs For linguistics: Cover more empirical data. Relate (denotational) semantics to (operational) psycholinguistics? For computer science: Understand delimited continuations geometrically and logically. Staging side effects? 24