Pregroup Calculus as a Logic Functor Annie Foret foret@irisa.fr - PowerPoint PPT Presentation

Pregroup Calculus as a Logic Functor Annie Foret foret@irisa.fr http://www.irisa.fr/prive/foret IRISA – University Rennes1 , FRANCE Wollic 2007 – p.1

PLAN Background Categorial grammars Logic functors Pregroups : properties, tools, applications Pregroup grammars Formal Models, Linguistic examples Pregroup calculus as a logic component a first attempt towards a Logic functor our proposal : FPG Main properties of FPG Lemmas overview Cut elimination, composed calculi Conclusion and remarks Wollic 2007 – p.2

Categorial grammars Σ = alphabet for words of a natural language { John, runs, swims, fast, . . . } Pr = primitive types : ( S, N, SN, SV, · · · ) Types = ex: Tp ::= Pr | Tp \ Tp | Tp/Tp . - with derivation rules on types Logical part AB \ e : A \ B, B ⊢ A and \ e : B, B \ A ⊢ A Wollic 2007 – p.3

Categorial grammars Σ = alphabet for words of a natural language { John, runs, swims, fast, . . . } Pr = primitive types : ( S, N, SN, SV, · · · ) Types = ex: Tp ::= Pr | Tp \ Tp | Tp/Tp . A categorial grammar on Σ - associate types of Tp to words in Σ Lexicon part { John �→ N , runs, swims �→ SN \ S , . . . } - with derivation rules on types Logical part AB \ e : A \ B, B ⊢ A and \ e : B, B \ A ⊢ A Wollic 2007 – p.3

Categorial grammars Σ = alphabet for words of a natural language { John, runs, swims, fast, . . . } Pr = primitive types : ( S, N, SN, SV, · · · ) Types = ex: Tp ::= Pr | Tp \ Tp | Tp/Tp . A categorial grammar on Σ - associate types of Tp to words in Σ Lexicon part { John �→ N , runs, swims �→ SN \ S , . . . } - with derivation rules on types Logical part AB \ e : A \ B, B ⊢ A and \ e : B, B \ A ⊢ A G generates a string c 1 . . . c n ∈ Σ + iff ∃ A 1 , . . . , A n ∈ Tp : G : c i �→ A i (1 ≤ i ≤ n ) and A 1 , . . . , A n ⊢ S L ( G ) = set of strings generated by G (language w.r.t. ⊢ ) Wollic 2007 – p.3

Hierarchy of k -valued categorial grammars {Lambek languages} = {AB categorial languages} = {Context−free languages} . . . . . . . . . . AB rigid 2−valued 3−valued Def: k -valued means at most k types per word (rigid is k=1) Fact: Class of rigid ( k -valued) AB languages learnable "in the limit" (Gold) In contrast to rigid Lambek or Pregroups Wollic 2007 – p.4

On Logic Functors In [Ferré, Ridoux] a logic A is viewed as the association of an abstract syntax AS A , a semantics S A , operations P A (and their implementation ) , including a subsumption (or entailment) relation , ≤ A . and a type T A made of a set of properties, The class of these logics is denoted by L . A logic functor F takes logics L 1 , . . . , L n of L as parameters and returns a logic F ( L 1 , . . . , L n ) in L ; viewed as a tuple ( AS F , S F , P F , T F ) of functions s. t. AS F ( L 1 ,... ,L n ) P F ( L 1 ,... ,L n ) = AS F ( AS L 1 , . . . , AS L n ) = P F ( P L 1 , . . . , P L n ) similarly for S F , T F Wollic 2007 – p.5

On Logic Functors – http://www.irisa.fr/LIS/softwares/ – LogFun ToolBox : implemented in Objective Caml (see) doc/report/) http://www.irisa.fr/LIS/ferre/logfun/ Logical Components, and “Glue” Prop, ... Concrete domains (Atom, Int, Interval,...); Structured Data (Product,...) Provers (decidable fragments) Customized logics Prop(Atom), Prop(Interval(Int)), ... Querying, Navigating in logical contexts http://www.irisa.fr/LIS/ferre/camelis/ Wollic 2007 – p.6

PLAN Background Categorial grammars Logic functors Pregroups : properties, tools, applications Pregroup grammars Formal Models, Linguistic examples Pregroup calculus as a logic component a first attempt towards a Logic functor our proposal : FPG Main properties of FPG Lemmas overview Cut elimination, composed calculi Conclusion and remarks Wollic 2007 – p.7

Pregroup : definitions A pregroup is a structure ( P, ≤ , · , l, r, 1) s. t. ( P, ≤ , · , 1) is a partially ordered monoid in which l, r are unary operations on P that satisfy: a.a r ≤ 1 ≤ a r .a ( PRE ) a l .a ≤ 1 ≤ a.a l and a.b ≤ c ⇔ a ≤ c.b l ⇔ b ≤ a r .c or equivalently: a rl = 1 = a lr Some equations follow from the def. ( a.b ) r = b r .a r , ( a.b ) l = b l .a l , 1 r = 1 = 1 l we also have: a rr � = a � = a ll but not, in general: . . . a ( − 2) = a ll , a ( − 1) = a l , a (0) = a, a (1) = a r , a (2) = a rr . . . iterated adjoints: A monoid is a structure < M, · , 1 > , such that · is associative and has a neutral element 1 A partially ordered monoid is a monoid ( M, · , 1) with a partial order ≤ that satisfies ∀ a, b, c : a ≤ b ⇒ c · a ≤ c · b and a · c ≤ b · c . Wollic 2007 – p.8

Free pregroups the set of atomic types is : P ( Z ) = { p ( i ) | p ∈ P, i ∈ Z } the set of types is Cat ( P, ≤ ) = { p ( i 1 ) · · · p ( i n ) | p k ∈ P, i k ∈ Z for 0 ≤ k ≤ n } n 1 ≤ on Cat ( P, ≤ ) is the smallest reflexive and transitive relation, s.t. for all p, q ∈ Pr , X, Y ∈ Cat ( P, ≤ ) and n ∈ Z : Xp ( n ) p ( n +1) Y ≤ XY (contraction), XY ≤ Xp ( n +1) p ( n ) Y (expansion), Xp ( n ) Y ≤ Xq ( n ) Y , (induction) if p ≤ q with n even or q ≤ p with n odd the free pregroup generated by ( P, ≤ ) is defined on classes [ ... ] modulo ∼ s.t. X ∼ Y iff X ≤ Y and Y ≤ X Wollic 2007 – p.9

Deductions in Free Pregroups Deduction system (Buszkowski), S Adj For X, Y ∈ Cat ( P, ≤ ) , we have: X ≤ Y iff it is deducible in: Xq ( k ) Y ≤ Z XY ≤ Z X ≤ X ( Id ) ( A L ) ( IND L ) Xp ( n ) p ( n +1) Y ≤ Z Xp ( k ) Y ≤ Z X ≤ Y p ( k ) Z X ≤ Y Z X ≤ Y Y ≤ Z ( A R ) ( Cut ) ( IND R ) X ≤ Y p ( n +1) p ( n ) Z X ≤ Y q ( k ) Z X ≤ Z with q ≤ p if k is even or p ≤ q if k is odd Cut Elimination Every derivable inequality has a Cut-free derivation symbol corrigendum: in the proceedings permute symbols p and q for IND R of S Adj . Wollic 2007 – p.10

Free Pregroup Interpretation FP = free pregroup on ( Pr, =) Interpretation [ · ] ] from formulas in L or NL , to FP if A is a primitive type of Pr [ [ A ] ] = A ] r [ [ [ C 1 \ C 2 ] ] = [ [ C 1 ] [ C 2 ] ] ] l [ [ C 1 / C 2 ] = [ [ C 1 ] ][ [ C 2 ] [ [ C 1 • C 2 ] = [ [ C 1 ] ][ [ C 2 ] ] The notation extends to sequents by: [ [ A 1 , . . . , A n ] ] = [ [ A 1 ] ] · · · [ [ A n ] ] Property FP is a model for L (hence for NL ): if Γ ⊢ L C then [ [Γ] ] ≤ FP [ [ C ] ] The converse does not hold: ] = a.b.c l ( a.b ) / c �⊢ a. ( b / c ) [ [( a.b ) / c ] ] = [ [ a. ( b / c )] pp ll p ll p l p l ≤ p ( p / (( p / p ) / p )) / p �⊢ p Wollic 2007 – p.11

A linguistic example : PG sentence: The film explains this situation ( n s c l ) ( π 3 r s 1 o l ) ( n s c l ) types: c c using primitive types and order postulates as follows: c = count noun (film, situation) n s = singular noun phrase (John) n s ≤ π 3 π k = k th personal subject pronoun ( π 3 =he/she/it) o = direct object n s ≤ o s 1 = statement in present tense s 1 ≤ s ≤ s s = declarative sentence (no tense) s =indirect sentence Wollic 2007 – p.12

A linguistic example : Lambek-like this situation n s / c c The film explains n s n s / c c ( π 3 \ s 1 ) / o n s π 3 \ s 1 s 1 where types n s , π 3 , o . . . are to be replaced with complex types such that: n s ⊢ π 3 , n s ⊢ o , and s 1 ⊢ s ⊢ s Wollic 2007 – p.13

PLAN Background Categorial grammars Logic functors Pregroups : properties, tools, applications Pregroup grammars Formal Models, Linguistic examples Pregroup calculus as a logic component a first attempt towards a Logic functor our proposal : FPG Main properties of FPG Lemmas overview Cut elimination, composed calculi Conclusion and remarks Wollic 2007 – p.14

Functor FPG ( A ) , a first proposal: S Adj [ A ] - where p, q are formulas in the logic A (parameter) , n, k ∈ Z and X, Y, Z ∈ Cat [ A ] - Xp ( k ) Y ≤ Z XY ≤ Z X ≤ X ( Id ) ( A L ) ( IND L + ) Xp ( n ) p ( n +1) Y ≤ Z Xq ( k ) Y ≤ Z X ≤ Y q ( k ) Z X ≤ Y Z X ≤ Y Y ≤ Z ( Cut ) ( A R ) ( IND R + ) X ≤ Y p ( n +1) p ( n ) Z X ≤ Y p ( k ) Z X ≤ Z with q ≤ A p if k is even ( A has ≤ A as subsumption) or p ≤ A q if k is odd for rules ( IND L + ) , ( IND R + ) This is direct adaptation of S Adj . However some drawbacks of IND L + and IND R + : IND L + , IND R + do not have the subformula property for the given q in the conclusion, { p ∈ AS A | q ≤ A p } is potentially infinite (in constrast to PG-grammars based on finite posets). symbol corrigendum: in the proceedings permute p and q for IND + R of S Adj [ A ] . Wollic 2007 – p.15

′ Functor FPG ( A ) , snd proposal: S [ A ] , S [ A ] - where a, b are formulas of A , n, k ∈ Z and X, Y, Z ∈ Cat [ A ] - a ≤ A b, if m is even XY ≤ Z a ≤ A b, if m is even ( Sub ) ( A L + ) a ( m ) ≤ b ( m ) Xa ( m ) b ( m +1) Y ≤ Z b ≤ A a, if m is odd XY ≤ Z b ≤ A a, if m is odd ( Sub ) ( A L + ) a ( m ) ≤ b ( m ) Xa ( m ) b ( m +1) Y ≤ Z Xa ( m +1) ≤ Y X ≤ Y Y ≤ Z X ≤ X ( Id ) ( Cut ) ( I R ) X ≤ Y a ( m ) X ≤ Z ′ S [ A ] denotes the same system as S [ A ] without the cut rule. Pregroup grammars on A and their language are defined as before, but using S [ A ] instead. Wollic 2007 – p.16