Chiastic Lambda-Calculi

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1 Chiastic Lambda-Calculi wren ng thornton Cognitive Science & Computational Linguistics Indiana University, Bloomington NLCS, 28 June 2013 wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

2 Examples and Motivation Associative λ-calculi Chiastic λ-calculi λ χl in action wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

3 Examples and Motivation Associative λ-calculi Chiastic λ-calculi λ χl in action wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

4 Scrambling in Japanese Tarou -ga hon -wo Nom book Acc Examples yon-da read-perf hon -wo Tarou -ga book Acc Nom Taro read the book. yon-da read-perf Keyword arguments yonda(wo= hon, ga= Tarou ) yonda(ga= Tarou, wo= hon ) Shorthands in category theory (FG)X = F (G X) (ηf )X = η FX (F η)x = F (η X ) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

5 What do these have in common? Juxtaposition is associative f (g x) (f g) x Application is commutative f x y f y x Our Goal convert those into = wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

6 Scrambling in Japanese Many languages have free word order Tarou -ga Nom hon -wo book Acc yon-da read-perf hon -wo Tarou -ga book Acc Nom Taro read the book. yon-da read-perf Both orders are normal and natural Both have the same propositional content Though, information structure may differ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

7 Arguments: Chomskian-style accounts Tarou N -ga NP nom\n NP nom hon.... N V -wo NP acc\n NP acc yon V \NP nom\np acc V \NP nom S -da..... S\V hon.... N -wo NP acc\n NP acc Tarou N -ga NP nom\n NP nom T V /(V \NP nom) V V \NP acc yon V \NP nom\np acc Bx S -da..... S\V wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

8 Adjuncts: Radical neo-davidsonian accounts Tarou -ga N S/S\N S/S hon.... N S -wo S/S\N S/S S yon-.... V -da..... S\V S hon.... N -wo S/S\N S/S Tarou -ga N S/S\N S/S S S yon-.... V -da..... S\V S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

9 Adjuncts: Radical neo-davidsonian accounts Tarou -ga N S/S\N S/S S/S hon.... N -wo S/S\N S/S B S yon-.... V -da..... S\V S hon.... N -wo S/S\N S/S S/S Tarou N -ga S/S\N S/S B S yon-.... V -da..... S\V S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

10 Why prefer adjuncts? Arguments vs Adjuncts Avoids the need for T and Bx (they re dangerous together) Syntax matches morphology/prosody Same parse tree for different word orders Online and partial parsing is easy (commutativity) (associativity) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

11 Why prefer adjuncts? Arguments vs Adjuncts Avoids the need for T and Bx (they re dangerous together) Syntax matches morphology/prosody Same parse tree for different word orders Online and partial parsing is easy (commutativity) (associativity) Only moves the problem from syntax to semantics! Also true of other CCG approaches to scrambling wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

12 Why prefer adjuncts? Arguments vs Adjuncts Avoids the need for T and Bx (they re dangerous together) Syntax matches morphology/prosody Same parse tree for different word orders Online and partial parsing is easy (commutativity) (associativity) Only moves the problem from syntax to semantics! Also true of other CCG approaches to scrambling Chiastic λ-calculi solve the problem (in the semantics) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

13 Examples and Motivation Associative λ-calculi Chiastic λ-calculi λ χl in action wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

14 What are functions? Traditional λ-calculi intentionally confuse two ideas Procedures operations mapping values to values Data values representing procedures Category theory keeps them distinct Morphisms functions as procedures Exponentials functions as data For associativity, we must keep them distinct too (λx. e) Unbracketed abstractions are procedures λx. e Bracketed abstractions are values wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

15 Associative λ-calculi: λ Variables x, y, z,... Terms e, f, g,... ::= x variables (λx. e) abstraction e bracketing e f juxtaposition (λx. f ) e {x e}f Beta (e f ) g e (f g) Assoc wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

16 What does juxtaposition mean? Application (λx. e) f Composition (λx. e) (λy. f ) ( ) ( ) (λx. e) (λy. f ) g (λx. e) (λy. f ) g Tupling f g (λx.λy. e) ( ) f g ( ) (λx.λy. e) f g wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

17 How powerful is it? L is at least as powerful as L Every L -term has an evaluation-equivalent L -term x = x (λx. e) = (λx. e ). e f = e f (e) = e L can be more expressive than L λ has tuples, but they can t be encoded in λ Then again, almost everything stronger than λ has tuples wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

18 Examples and Motivation Associative λ-calculi Chiastic λ-calculi λ χl in action wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

19 Chiastic λ-calculi The term level Syntax Two flavors of chiasmus Equivalence Reduction Sanity check The type level Syntax Equivalence Reduction Sanity check Well-formed types wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

20 Formalizing restricted commutativity Actually we don t want full commutativity Tarou -ga kuruma Nom car Taro has a car. % kuruma -ga Tarou car Nom The car has a Taro. -ga Nom -ga Nom ar-u have-npst ar-u have-npst Let a dimension denote a class of non-commuting elements Elements along different dimensions don t interfere with one another wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

21 Chiastic λ-calculi: λ χ Variables x, y, z,... Dimensions A, B, C,... Terms e, f, g,... ::= x variables (λ A x. e) abstraction e A bracketing e f juxtaposition Choose one A B (λ A x. λ B y. e) (λ B y. λ A x. e) Chi L A B e A f B f B e A Chi R wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

22 Chiastic λ-calculi: λ χ Variables x, y, z,... Dimensions A, B, C,... Terms e, f, g,... ::= x variables (λ A x. e) abstraction e A bracketing e f juxtaposition For this talk, we ll only consider Chi L A B (λ A x. λ B y. e) (λ B y. λ A x. e) Chi L A B e A f B f B e A Chi R wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

23 Chi L vs Chi R Should we accept terms like this? (λ A x. λ B y. e) (λ C z. a A ) b B c C Should we accept sentences like this? [sono hon -wo Hanako -ga Tarou -ga kat-ta] -to that book Acc Nom Nom buy-perf Comp omot-te iru think.prog Hanako thinks that Taro bought that book. wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

24 Term equivalence for λ χl e f A B (λ A x. λ B y. e) (λ B y. λ A x. e) e (f g) (e f ) g e e (λ A x. e) (λ A x. e ) e e e e f f e A e A e f e f e e f e e f e f f g e g wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

25 Term reduction for λ χl e e h (λ A x. f ) h e A {x e}f e e (λ A x. e) (λ A x. e ) e e e e e A e A e f e f f f e f e f e f h e (f g) h g f g h (e f ) g e h wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

26 Do our terms make sense? Theorem Term reduction is weak Church Rosser. Proof There are no critical pairs. Corollary Term reduction is Church Rosser. Proof Supposing we can prove strong normalization, then just use Newman s lemma. Conjecture Term reduction (for λ χl ) is strongly normalizing. Remark This is suspiciously difficult to prove. wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

27 The intrinsic view (à la Church) Types are manifest in terms Terms can have only one type Ill-typed terms don t exist What are types? The extrinsic view (à la Curry) Types characterize properties of terms Terms could have multiple types All terms exist, but we only care about the well-typed ones wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

28 The intrinsic view (à la Church) Types are manifest in terms Terms can have only one type Ill-typed terms don t exist What are types? The extrinsic view (à la Curry) Types characterize properties of terms Terms could have multiple types All terms exist, but we only care about the well-typed ones Our view Types give abstract interpretations of terms Γ e τ τ τ e. e e Γ e τ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

29 Simply-typed left-chiastic λ-calculus: λ χl Types σ, τ, υ,... ::= T primitive types σ A τ arrow types τ A bracketed types σ τ juxtaposition Γ e τ ctx Γ Γ(x) τ Γ x τ Γ, x :σ e τ Γ (λ A x. e) σ A τ Γ e τ Γ e σ Γ f τ Γ e A τ A Γ e f σ τ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

30 Type equivalence for λ χl τ σ A B σ A τ B υ τ B σ A υ σ (τ υ) (σ τ) υ σ σ τ τ σ A τ σ A τ τ τ σ σ τ τ τ A τ A σ τ σ τ τ τ σ τ τ σ σ τ τ υ σ υ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

31 Type reduction for λ χl τ τ ρ σ A τ ρ σ A τ σ σ τ τ σ A τ σ A τ σ A τ σ A τ τ τ σ σ τ A τ A σ τ σ τ τ τ σ τ σ τ σ τ ρ σ (τ υ) ρ υ τ υ ρ (σ τ) υ τ ρ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

32 Do our types make sense? Every type has a normal form Theorem Type reduction for λ χl is strongly normalizing Proof Theorem Type reduction for λ χl is Church Rosser Proof So we can define Γ e τ 0 NF(τ 0 ) τ type τ Γ e : τ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

33 Do our types make sense? Every type has a normal form But, what does unresolved type juxtaposition mean? wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

34 Do our types make sense? Every type has a normal form But, what does unresolved type juxtaposition mean? Good σ A τ B wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

35 Do our types make sense? Every type has a normal form But, what does unresolved type juxtaposition mean? Good σ A τ B Bad (σ A τ B ) υ C where A C (σ A τ) υ A where σ υ wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

36 Do our types make sense? Every type has a normal form But, what does unresolved type juxtaposition mean? Good σ A τ B Bad (σ A τ B ) υ C where A C (σ A τ) υ A where σ υ Ugly σ A (τ B υ) (ρ A σ) (τ B υ) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

37 Do our types make sense? Every type has a normal form But, what does unresolved type juxtaposition mean? Good σ A τ B Bad (σ A τ B ) υ C where A C (σ A τ) υ A where σ υ Ugly σ A (τ B υ) (ρ A σ) (τ B υ) If type τ doesn t accept ugly terms, then it doesn t have the subterm property. wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

38 Examples and Motivation Associative λ-calculi Chiastic λ-calculi λ χl in action wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

39 Using λ χl to describe Japanese Noun phrase scrambling Tarou-ga hon-wo yonda Hon-wo Tarou-ga yonda Verbal morphology Paradoxical behavior Resolving the paradox wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

40 Semantic analysis of Tarou-ga Tarou... N -ga... S/S\N S/S β S/S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

41 Semantic analysis of Tarou-ga Tarou... Taro N : N -ga... (λ N n. λ S s. s n nom S ) : S/S\N (λ N n. λ S s. s n nom S ) Taro N : S/S β (λ S s. s Taro nom S ) : S/S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

42 Semantic analysis of Tarou-ga... Tarou -ga... Taro N (λ N n. λ S s. s n nom S ) (λ N n. λ S s. s n nom S ) Taro N β (λ S s. s Taro nom S ) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

43 Semantic analysis of Tarou-ga... Tarou -ga... Taro N (λ N n. λ S s. s n nom S ) (λ N n. λ S s. s n nom S ) Taro N β (λ S s. s Taro nom S ) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

44 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

45 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

46 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

47 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

48 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

49 Semantic analysis of Tarou-ga hon-wo yonda hon-wo (λ S s. s book acc S ) yonda λ acc a. λ nom n. n read a S Tarou-ga (λ S s. s Taro nom S ) (λ S s. s book acc S ) λ acc a. λ nom n. n read a S β (λ acc a. λ nom n. n read a) book acc S β λ nom n. n read book S (λ S s. s Taro nom S ) λ nom n. n read book S β (λ nom n. n read book ) Taro nom S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

50 Semantic analysis of hon-wo Tarou-ga yonda Tarou-ga (λ S s. s Taro nom S ) yonda λ acc a. λ nom n. n read a S (λ S s. s Taro nom S ) λ acc a. λ nom n. n read a S β hon-wo (λ S s. s book acc S ) (λ acc a. λ nom n. n read a) Taro nom S β(χl ) λ acc a. Taro read a S (λ S s. s book acc S ) λ acc a. Taro read a S β (λ acc a. Taro read a) book acc S β Taro read book S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

51 Paradoxical verbal morphology Causative and passive verb forms tabe-ru to eat tabe-sase-ru to cause to eat tabe-rare-ru to be made to eat Paradoxical behavior of causative and passive Morpho-phonologically behaves as a single word Semantically behaves as if involving complementation E.g., adverb scope ambiguity But this paradox is due to traditional notions of constituency Kubota 2008 vs GB, LFG, HPSG wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

52 General scheme for verbal morphology Let the dimension E denote eventualities Verbal roots have types of the general form τ E V Verbal inflections use multicomposition (λ V v. (λ E e. f ) v S ) v can have any arity The semantic content f, has access to the whole eventuality e So if e is a compound eventuality, f can affect all or part of it wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

53 Lexical entries for a few verbal inflections Form = Semantics Non-past -(r)u = λ V v. (λ E e. e Tense(e)=Npst) v S Perfect -ta = λ V v. (λ E e. e Tense(e)=Perf) v S Causative -(s)ase- = λ V v. λ nom n. λ dat d. (λ E e. e Cause(e)=n E ) v d nom V Passive -(r)are- = λ V v. λ nom n. λ dat d. (λ E e. e Exper(e)=n E ) v d nom V wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

54 Semantic analysis for yonda yon da λ acc a. λ nom n. n reads a E V λ V v. (λ E e. e Tense(e) = Perf) v S (λ V v. (λ E e. e Tense(e) = Perf) v S ) λ acc a. λ nom n. n reads a E V β (λ E e. e Tense(e) = Perf) (λ acc a. λ nom n. n reads a E ) S η λ acc a. λ nom n. (λ E e. e Tense(e) = Perf) n reads a E S β λ acc a. λ nom n. n reads a Tense(n reads a) = Perf E S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

55 Semantic analysis for yonda yon da λ acc a. λ nom n. n reads a E V λ V v. (λ E e. e Tense(e) = Perf) v S (λ V v. (λ E e. e Tense(e) = Perf) v S ) λ acc a. λ nom n. n reads a E V β (λ E e. e Tense(e) = Perf) (λ acc a. λ nom n. n reads a E ) S η λ acc a. λ nom n. (λ E e. e Tense(e) = Perf) n reads a E S β λ acc a. λ nom n. n reads a Tense(n reads a) = Perf E S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

56 Semantic analysis for yonda yon da λ acc a. λ nom n. n reads a E V λ V v. (λ E e. e Tense(e) = Perf) v S (λ V v. (λ E e. e Tense(e) = Perf) v S ) λ acc a. λ nom n. n reads a E V β (λ E e. e Tense(e) = Perf) (λ acc a. λ nom n. n reads a E ) S η λ acc a. λ nom n. (λ E e. e Tense(e) = Perf) n reads a E S β λ acc a. λ nom n. n reads a Tense(n reads a) = Perf E S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

57 Semantic analysis for yonda yon da λ acc a. λ nom n. n reads a E V λ V v. (λ E e. e Tense(e) = Perf) v S (λ V v. (λ E e. e Tense(e) = Perf) v S ) λ acc a. λ nom n. n reads a E V β (λ E e. e Tense(e) = Perf) (λ acc a. λ nom n. n reads a E ) S η λ acc a. λ nom n. (λ E e. e Tense(e) = Perf) n reads a E S β λ acc a. λ nom n. n reads a Tense(n reads a) = Perf E S wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

58 Semantic analysis for yonda yon da λ acc a. λ nom n. n reads a E V λ V v. (λ E e. e Tense(e) = Perf) v S (λ V v. (λ E e. e Tense(e) = Perf) v S ) λ acc a. λ nom n. n reads a E V β (λ E e. e Tense(e) = Perf) (λ acc a. λ nom n. n reads a E ) S η λ acc a. λ nom n. (λ E e. e Tense(e) = Perf) n reads a E S β λ acc a. λ nom n. n reads a Tense(n reads a) = Perf E S The η is a lie! wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

59 Conclusion Associative λ-calculi Justifies shorthands in category theory Chiastic λ-calculi (namely λ χl ) Captures linguistic phenomena Type reduction is CR and SN Term reduction is WCR Current work Is term reduction SN? Can we describe Γ e : τ more directly? What about η? wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

60 fin. wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

61 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Visualizing dimensions Type reduction is strongly normalizing Type reduction is Church Rosser More paradoxical verbal morphology wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

62 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Visualizing dimensions wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

63 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Visualizing dimensions Type reduction is strongly normalizing Type reduction is Church Rosser More paradoxical verbal morphology wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

64 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Type reduction for λ χl is strongly normalizing Definition The length of a type is the number of constructors length(t ) = 1 length(σ A τ) = 1 + length(σ) + length(τ) length( τ A ) = 1 + length(τ) length(σ τ) = 1 + length(σ) + length(τ) Lemma Equivalent types have equal length. Theorem Type reduction diminishes length; i.e., τ, τ. τ τ length(τ) > length(τ ) Back wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

65 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Visualizing dimensions Type reduction is strongly normalizing Type reduction is Church Rosser More paradoxical verbal morphology wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

66 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Type reduction for λ χl is Church Rosser Lemma Type reduction commutes with type equivalence; i.e., τ σ τ * σ Theorem Type reduction is weak Church Rosser. Proof There are no critical pairs. Use the key lemma to resolve potential conflicts between β and itself. Corollary Type reduction is Church Rosser Proof By Newman s Lemma. wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34 Back

67 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Case 2 ρ σ A β(e) (ρ A )(s) r ρ ρ τ ρ σ A e e β(e ) ρ σ A ( σ A )(r) τ A σ τ A σ (τ A )(s) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

68 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Case 3a ρ σ A β(e) ( σ A )(r) r ρ ρ τ ρ σ A e e β(e ) ρ σ A (ρ A )(s) τ A σ τ A σ (τ A )(s) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

69 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Case 3b ρ σ A β(e) ( σ A )(r) r ρ ρ τ ρ σ A e e t τ β(e ) τ A σ τ A σ ( A σ)(t) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

70 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology Visualizing dimensions Type reduction is strongly normalizing Type reduction is Church Rosser More paradoxical verbal morphology wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

71 Visualizing dimensions Type reduction is SN Type reduction is CR More paradoxical verbal morphology More paradoxical verbal morphology -i -te to Interclausal scrambling Adverb between V1 and V2 Argument cluster coordination involving V1 Postposing of VP headed by V1 Clefting of VP headed by V1 Coordination of VP headed by V1 Focus particle between V1 and V2 Reduplication of V2 alone (Kubota 2008) wren ng thornton (Indiana University) Chiastic Lambda-Calculi NLCS, 28 June / 34

A Multi-Modal Combinatory Categorial Grammar. A MMCCG Analysis of Japanese Nonconstituent Clefting

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