Focusing via display Giuseppe Greco - University of Utrecht - - PowerPoint PPT Presentation

Focusing via display Giuseppe Greco - University of Utrecht - (work in progress with W. Fussner, F. Liang, M. Moortgat) – SYSMICS – Chapman University 17 September 2018

Overview 1. The need of structural reasoning 2. The multi-type approach comes in handy 3. Which logic for linguistic analysis? 4. Focusing via display 2 / 33

The need of structural reasoning 3 / 33

Parsing as deduction [Lambek 58]: string of words, [Lambek 61]: bracketed strings (phrases), [Ajdukiewicz 35, Bar-Hillel 53]: AB-grammar ◮ Parts of speech (noun, verb...) � logical formulas - types. ◮ Grammaticality judgement � logical deduction - computation. ⊗ ( np \ s ) ⊗ ((( np \ s ) \ ( np \ s )) / np ) ⊗ ( np / n ) ⊗ np n ⊢ s time flies like an arrow ◮ transitive verb ‘love’: ( np \ s ) / np ◮ kids · (love · games) ◮ conjunction words ‘and/but’: chameleon word ( X \ X ) / X ◮ X = s : (kids like sweets) s but (parents prefer liquor) s ◮ X = np \ s : kids (like sweets) np \ s but (hate vegetables) np \ s ◮ relative pronoun ‘that’: ( n \ n ) / ( s / np ) , i.e. it looks for a noun n to its left and an incomplete sentence to its right ( s / np : it misses a np , the gap at the right) 4 / 33

L: Global Associativity � peripheral gaps √ like ` np ] 1 ( np \ s ) /np [ kids [ /E ] np like · ` np \ s [ \ E ] kids · (like · ) ` s [ A ] (kids · like) · ` s that [ /I ] 1 ( n \ n ) / ( s/np ) kids · like ` s/np games [ /E ] n that · (kids · like) ` n \ n [ \ E ] games · (that · (kids · like)) ` n λx 3 . (( games x 3 ) ^ (( like x 3 ) kids )) 5 / 33

The multi-type approach comes in handy 6 / 33

The language of Lambek calculus A ::= p | A ⊗ A | A / A | A \ A . ⊗ X | X ˇ / X | X ˇ X ::= A | X ˆ \ X 7 / 33

Lambek calculus NL (L = NL + Associativity) ◮ Identity and Cut rules (preorder) X ⊢ A A ⊢ Y Id A ⊢ A Cut X ⊢ Y ◮ Display rules (residuation, adjunction) X ⊢ Z ˇ / Y rp X ˆ ⊗ Y ⊢ Z rp Y ⊢ X ˇ \ Z ◮ Logical rules (arity and tonicity) A ˆ X ⊢ A Y ⊢ B ⊗ B ⊢ Y ⊗ R ⊗ L A ⊗ B ⊢ Y X ˆ ⊗ Y ⊢ A ⊗ B X ⊢ A ˇ X ⊢ A B ⊢ Y \ B \ L \ R A \ B ⊢ X ˇ \ Y X ⊢ A \ B X ⊢ B ˇ X ⊢ A B ⊢ Y / A / L / R B / A ⊢ Y ˇ / X X ⊢ B / A 8 / 33

Proper display calculi [Wansing 98]: proper, [Belnap 82, 89]: display logic, [Mints 72, Dunn 73, 75]: structural connectives Definition A proper DC verifies each of the following conditions: 1. structures can disappear, formulas are forever ; 2. tree-traceable formula-occurrences, via suitably defined congruence relation (same shape, position, non-proliferation); 3. principal = displayed 4. rules are closed under uniform substitution of congruent parameters (Properness!); 5. reduction strategy exists when cut formulas are principal. Theorem (Canonical!) Cut elim. and subformula property hold for any proper DC . 9 / 33

Which logics are properly displayable? [Ciabattoni et al. 15, Greco et al. 16] Complete characterization: 1. the logics of any basic normal (D)LE; 2. axiomatic extensions of these with analytic inductive inequalities : � unified correspondence − ψ + φ ≤ ∧ , ∨ ∧ , ∨ + f , − g − g , + f ∧ , ∨ ∧ , ∨ + g , − f − f , + g + p − p + p + p Fact: cut-elim., subfm. prop., sound-&-completeness, conservativity guaranteed by metatheorem + ALBA-technology. 10 / 33

For many... but not for all. [www.appliedlogictudelft.nl] ◮ The characterization theorem sets hard boundaries to the scope of proper display calculi. ◮ Interesting logics are left out : ◮ DEL, PDL, Logic of Resources and Capabilities ◮ Linear logic ◮ (Lattice logic) ◮ (First order logic) ◮ Inquisitive logic ◮ Semi De Morgan logic ◮ Bi-lattice logic ◮ Rough algebras Can we extend the scope of proper display calculi? Yes: proper display calculi � proper multi-type calculi 11 / 33

Multi-type proper display calculi [Greco et al. 14, ...] Definition A proper DC verifies each of the following conditions: 1. structures can disappear, formulas are forever ; 2. tree-traceable formula-occurrences, via suitably defined congruence relation (same shape, position, non-proliferation) 3. principal = displayed 4. rules are closed under uniform substitution of congruent parameters within each type (Properness!); 5. reduction strategy exists when cut formulas are principal. 6. type-uniformity of derivable sequents; 7. strongly uniform cuts in each/some type(s). Theorem (Canonical!) Cut elim. and subformula property hold for any proper m.DC . 12 / 33

Which logic for linguistic analysis? 13 / 33

L: Global Associativity � overgeneration × hate ` np ] 1 ( np \ s ) /np [ like call of duty parents [ /E ] ( np \ s ) /np np np hate · ` np \ s but kids [ /E ] [ \ E ] np like · call of duty ` np \ s ( s \ s ) /s parents · (hate · ) ` s [ \ E ] [ /E ] kids · (like · call of duty) ` s but · (parents · (hate · )) ` s \ s [ \ E ] (kids · (like · call of duty)) · (but · (parents · (hate · ))) ` s [ A ] (kids · (like · call of duty)) · (but · ((parents · hate) · )) ` s [ A ] (kids · (like · call of duty)) · ((but · (parents · hate)) · ) ` s [ A ] ((kids · (like · call of duty)) · (but · (parents · hate))) · ` s that [ /I ] 1 ( n \ n ) / ( s/np ) (kids · (like · call of duty)) · (but · (parents · hate)) ` s/np games [ /E ] n that · ((kids · (like · call of duty)) · (but · (parents · hate))) ` n \ n [ \ E ] games · (that · ((kids · (like · call of duty)) · (but · (parents · hate)))) ` n [ A ] games · (that · (((kids · (like · call of duty)) · but) · (parents · hate))) ` n [ A ] games · (that · ((((kids · (like · call of duty)) · but) · parents) · hate)) ` n λy 4 . (( games y 4 ) ^ (( but (( hate y 4 ) parents )) (( like call of duty ) kids ))) 14 / 33

NL+SC (licence): Controlled Associativity � peripheral gaps √ [Moortgat 96] ` 2 np ] 2 [ like [ 2 E ] ( np \ s ) /np h i ` np kids [ /E ] np like · h i ` np \ s [ \ E ] kids · (like · h i ) ` s [ MA ] ` } 2 np ] 1 [ (kids · like) · h i ` s [ } E ] 2 (kids · like) · ` s that [ /I ] 1 ( n \ n ) / ( s/ } 2 np ) kids · like ` s/ } 2 np games [ /E ] n that · (kids · like) ` n \ n [ \ E ] games · (that · (kids · like)) ` n λy 3 . (( games y 3 ) ^ (( like y 3 ) kids )) 15 / 33

NL+SC (licence): Controlled Associativity and Exchange � internal gaps √ (Global Assoc. � undergeneration × ) ` 2 np ] 2 [ like [ 2 E ] ( np \ s ) /np h i ` np [ /E ] ` np ] 3 [ like · h i ` np \ s [ \ E ] · (like · h i ) ` s a lot [ \ I ] 3 like · h i ` np \ s ( np \ s ) \ ( np \ s ) kids [ \ E ] np (like · h i ) · a lot ` np \ s [ \ E ] kids · ((like · h i ) · a lot) ` s [ MC ] kids · ((like · a lot) · h i ) ` s [ MA ] ` } 2 np ] 1 [ (kids · (like · a lot)) · h i ` s [ } E ] 2 (kids · (like · a lot)) · ` s that [ /I ] 1 ( n \ n ) / ( s/ } 2 np ) kids · (like · a lot) ` s/ } 2 np games [ /E ] n that · (kids · (like · a lot)) ` n \ n [ \ E ] games · (that · (kids · (like · a lot))) ` n λy 4 . (( games y 4 ) ^ (( a lot ( like y 4 )) kids )) 16 / 33

NL+SC (licence & block): Controlled Associativity � peripheral & internal gaps √ ` 2 np ] 4 [ hate [ 2 E ] ` 2 np ] 6 like [ ( np \ s ) /np h i ` np parents [ 2 E ] [ /E ] ( np \ s ) /np h i ` np np hate · h i ` np \ s kids [ /E ] [ \ E ] np like · h i ` np \ s parents · (hate · h i ) ` s [ \ E ] [ MA ] kids · (like · h i ) ` s [ ` } 2 np ] 3 (parents · hate) · h i ` s [ MA ] [ } E ] 4 [ ` } 2 np ] 5 (kids · like) · h i ` s (parents · hate) · ` s but [ } E ] 6 [ /I ] 3 (kids · like) · ` s (( s/ } 2 np ) \ 2 ( s/np )) / ( s/ } 2 np ) parents · hate ` s/ } 2 np [ /I ] 5 [ /E ] kids · like ` s/ } 2 np but · (parents · hate) ` ( s/ } 2 np ) \ 2 ( s/np ) [ \ E ] (kids · like) · (but · (parents · hate)) ` 2 ( s/np ) [ ` 2 np ] 2 [ 2 E ] [ 2 E ] h (kids · like) · (but · (parents · hate)) i ` s/np h i ` np [ /E ] [ ` } 2 np ] 1 h (kids · like) · (but · (parents · hate)) i · h i ` s [ } E ] 2 h (kids · like) · (but · (parents · hate)) i · ` s that [ /I ] 1 ( n \ n ) / ( s/ } 2 np ) h (kids · like) · (but · (parents · hate)) i ` s/ } 2 np games [ /E ] n that · h (kids · like) · (but · (parents · hate)) i ` n \ n [ \ E ] games · (that · h (kids · like) · (but · (parents · hate)) i ) ` n λy 6 . (( games y 6 ) ^ ((( but λx 3 . (( hate x 3 ) parents )) λx 4 . (( like x 4 ) kids )) y 6 )) 17 / 33

Lambek-Grishin calculus A ::= p | A ⊗ A | A A | A ⊘ A | A ⊕ A | A / A | A \ A ⊘ � I ::= A | I ˆ . ⊗ I | O ˆ I | I ˆ ⊘ O ⊘ X ⊕ O | I ˇ / O | O ˇ O ::= A | O ˇ \ I Full NL = NL + Display and Logical rules for the additional (dual) connectives. ◮ Grishin rules (interactions) X ˆ ⊗ Y ⊢ Z ˇ X ˆ ⊗ Y ⊢ Z ˇ ⊕ W ⊕ W G1 G2 X ⊢ W ˇ Y ⊢ X ˇ Z ˆ Z ˆ / Y \ W ⊘ ⊘ X ˆ ⊗ Y ⊢ Z ˇ X ˆ ⊗ Y ⊢ Z ˇ ⊕ W ⊕ W G3 G4 ⊘ W ⊢ X ˇ ⊘ W ⊢ Z ˇ Y ˆ X ˆ \ Z / Y 18 / 33