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

• ur 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 c1 . . . cn ∈ Σ+ iff ∃A1, . . . , An ∈ Tp : G : ci → Ai (1 ≤ i ≤ n)

and A1, . . . , An⊢ S

L(G) = set of strings generated by G (language w.r.t. ⊢)

Wollic 2007 – p.3

Hierarchy of k-valued categorial grammars

. . . . . . . . . .

AB rigid 2−valued 3−valued = {AB categorial languages} = {Context−free languages} {Lambek languages}

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 ASA, a semantics SA,

• perations PA (and their implementation) ,

including a subsumption (or entailment) relation, ≤A. and a type TA made of a set of properties, The class of these logics is denoted by L. A logic functor F takes logics L1, . . . , Ln of L as parameters and returns a logic F(L1, . . . , Ln) in L;

viewed as a tuple (ASF , SF, PF , TF) of functions s. t. ASF (L1,... ,Ln) PF (L1,... ,Ln) = ASF (ASL1, . . . , ASLn) = PF (PL1, . . . , PLn) similarly for SF , TF

Wollic 2007 – p.5

On Logic Functors

– http://www.irisa.fr/LIS/softwares/ –

LogFun ToolBox : implemented in Objective Caml

http://www.irisa.fr/LIS/ferre/logfun/

(see) doc/report/) 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

• ur 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

• rdered monoid

in which l, r are unary operations on P that satisfy: (PRE) al.a ≤ 1 ≤ a.al and a.ar ≤ 1 ≤ ar.a

• r equivalently:

a.b ≤ c ⇔ a ≤ c.bl ⇔ b ≤ ar.c Some equations follow from the def. arl = 1 = alr we also have: (a.b)r = br.ar , (a.b)l = bl.al , 1r = 1 = 1l but not, in general: arr = a = all iterated adjoints: . . . a(−2) =all, a(−1) =al, a(0) =a, a(1) =ar, a(2) =arr . . .

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(i1)

1

· · · p(in)

n

| pk ∈ P, ik ∈ Z for 0≤k≤n} ≤ 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), SAdj For X, Y ∈ Cat(P,≤), we have: X ≤ Y iff it is deducible in:

X ≤ X (Id) XY ≤Z (AL) Xp(n)p(n+1)Y ≤Z Xq(k)Y ≤Z (INDL) Xp(k)Y ≤Z X ≤Y Y ≤Z (Cut) X ≤ Z X ≤Y Z (AR) X ≤Y p(n+1)p(n)Z X ≤Y p(k)Z (INDR) X ≤Y q(k)Z with q ≤ p if k is even

• r 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 INDR of SAdj .

Wollic 2007 – p.10

Free Pregroup Interpretation

FP = free pregroup on (Pr, =)

Interpretation [

·] ] from formulas in L or NL, to FP [ [A] ] = A

if A is a primitive type of Pr

[ [C1 \ C2] ] = [ [C1] ]r[ [C2] ] [ [C1 / C2] = [ [C1] ][ [C2] ]l [ [C1 • C2] = [ [C1] ][ [C2] ]

The notation extends to sequents by:

[ [A1, . . . , An] ] = [ [A1] ] · · · [ [An] ]

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 ⊢ a.(b / c) [ [(a.b) / c] ] = [ [a.(b / c)] ] = a.b.cl (p / ((p / p) / p)) / p ⊢ p ppllpllplpl ≤ p

Wollic 2007 – p.11

A linguistic example : PG

sentence: The film explains this situation types: (ns cl )

c

(π3r s1 ol) (ns cl)

c

using primitive types and order postulates as follows:

c = count noun

(film, situation)

ns ≤ π3 ns = singular noun phrase

(John)

πk = kth personal subject pronoun

(π3=he/she/it)

ns ≤ o

• = direct object

s1 ≤ s ≤ s s1 = statement in present tense s = declarative sentence

(no tense)

s=indirect sentence

Wollic 2007 – p.12

A linguistic example : Lambek-like

s1 ns

The

ns / c

film

c π3 \ s1

explains

(π3 \ s1) / o ns

this

ns / c

situation

c

where types ns, π3, o . . . are to be replaced with complex types such that: ns ⊢ π3, ns ⊢ o, and s1 ⊢ 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

• ur 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: SAdj[A]

• where p, q are formulas in the logic A (parameter) , n, k ∈ Z and X, Y, Z ∈ Cat[A] -

X ≤ X (Id) XY ≤Z (AL) Xp(n)p(n+1)Y ≤Z Xp(k)Y ≤Z (INDL+) Xq(k)Y ≤Z X ≤Y Y ≤Z (Cut) X ≤ Z X ≤Y Z (AR) X ≤Y p(n+1)p(n)Z X ≤Y q(k)Z (INDR+) X ≤Y p(k)Z (A has ≤A as subsumption) with q ≤A p if k is even

• r p ≤A q if k is odd

for rules (INDL+), (INDR+) This is direct adaptation of SAdj. However some drawbacks of INDL+ and INDR+: INDL+ , INDR+ do not have the subformula property for the given q in the conclusion, {p ∈ ASA | 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 SAdj[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 (Sub) a(m) ≤ b(m) XY ≤Z a ≤A b, if m is even (AL+) Xa(m)b(m+1)Y ≤Z b ≤A a, if m is odd (Sub) a(m) ≤ b(m) XY ≤Z b ≤A a, if m is odd (AL+) Xa(m)b(m+1)Y ≤Z X ≤ X (Id) Xa(m+1) ≤Y (IR) X ≤Y a(m) X ≤Y Y ≤Z (Cut) 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

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 (Sub) a(m) ≤ b(m) XY ≤Z a ≤A b, if m is even (AL+) Xa(m)b(m+1)Y ≤Z b ≤A a, if m is odd (Sub) a(m) ≤ b(m) XY ≤Z b ≤A a, if m is odd (AL+) Xa(m)b(m+1)Y ≤Z X ≤ X (Id) Xa(m+1) ≤Y (IR) X ≤Y a(m) X ≤Y Y ≤Z (Cut) X ≤ Z S[A] denotes the same system as S

′

[A] without the cut rule.

condensed presentation for rules (Sub) and (AL+): where a(m) ≤A b(m) stands for a≤A b if m even, b≤A a if m is odd a(m) ≤A b(m) (Sub) a(m) ≤ b(m) XY ≤Z a(m) ≤A b(m) (AL+) Xa(m)b(m+1)Y ≤Z

Wollic 2007 – p.16

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

• ur proposal : FPG

Main properties of FPG Lemmas overview Cut elimination, composed calculi Conclusion and remarks

Wollic 2007 – p.17

Functor FPG(A) , lemmas on S[A], S

′

[A]

a ≤A b, if m is even (Sub) a(m) ≤ b(m) XY ≤Z a ≤A b, if m is even (AL+) Xa(m)b(m+1)Y ≤Z . . . if m is odd X ≤ X (Id) Xa(m+1) ≤Y (IR) X ≤Y a(m) X ≤Y Y ≤Z (Cut) X ≤ Z

• 1. rule IR is reversible in both S[A] (without cut) and S

′

[A]

using ≤A reflexive

Wollic 2007 – p.18

Functor FPG(A) , lemmas on S[A], S

′

[A]

a ≤A b, if m is even (Sub) a(m) ≤ b(m) XY ≤Z a ≤A b, if m is even (AL+) Xa(m)b(m+1)Y ≤Z . . . if m is odd X ≤ X (Id) Xa(m+1) ≤Y (IR) X ≤Y a(m) X ≤Y Y ≤Z (Cut) X ≤ Z

• 1. rule IR is reversible in both S[A] (without cut) and S

′

[A]

using ≤A reflexive . INDL and INDR− (weak form of (INDR)) hold in S[A] and S

′

[A]

Xb(k)Y ≤Z (INDL+) Xa(k)Y ≤Z a≤A b if m even b≤A a if m is odd X ≤Y a(k) (INDR−) X ≤Y b(k)

Wollic 2007 – p.18

Functor FPG(A) , lemmas on S[A], S

′

[A]

a ≤A b, if m is even (Sub) a(m) ≤ b(m) XY ≤Z a ≤A b, if m is even (AL+) Xa(m)b(m+1)Y ≤Z . . . if m is odd X ≤ X (Id) Xa(m+1) ≤Y (IR) X ≤Y a(m) X ≤Y Y ≤Z (Cut) X ≤ Z

• 1. rule IR is reversible in both S[A] (without cut) and S

′

[A]

using ≤A reflexive . INDL and INDR− (weak form of (INDR)) hold in S[A] and S

′

[A]

Xb(k)Y ≤Z (INDL+) Xa(k)Y ≤Z a≤A b if m even b≤A a if m is odd X ≤Y a(k) (INDR−) X ≤Y b(k)

• 2. if Xa(m+1)b(m)Y ≤Z and a(m) ≤A b(m) then XY ≤Z

analogue of 1 ≤ ara

Wollic 2007 – p.18

Cut elimination

– from Lemma 1, 2, ≤A –

Theorem X ≤ Y in S[A] iff X ≤ Y in S

′

[A]

(∀X, Y ∈ Cat[A])

induction on the number of Cut and the length of a derivation P in S

′

[A], ending in Cut:

γl        . . . Rl X ≤ Y γr        . . . Rr Y ≤ Z Cut X ≤ Z induction on Y (lemmas 1,2) when Y is simple, for Rl (left) and Rr (right): (no AL+ as Rr, no IR as Rl, for Y simple) Rl Rr method Sub IR lemma 1 [INDL] AL+ Sub lemma 1 [INDR−] AL+ IR permute Rl with cut Sub Sub transitivity of ≤A

Wollic 2007 – p.19

Functor FPG(A) , lemmas on S[A], S

′

[A]

Lemma Rule [INDR], [AR] ([AR+]) are admissible in S

′

[A]:

X ≤Y a(k)Z and a(k) ≤A b(k) (INDR+) X ≤Y b(k)Z (a, b ∈ A SA) X ≤Y Z (AR) X ≤Y a(n+1)a(n)Z X ≤Y Z and a(n) ≤A b(n) (AR+) X ≤Y a(n+1)b(n)Z

Theorem

X ≤ Y is provable in S[A] iff it is provable in SAdj[A].

Wollic 2007 – p.20

Functor FPG(A) , lemmas on S[A], S

′

[A]

Lemma Rule [INDR], [AR] ([AR+]) are admissible in S

′

[A]:

X ≤Y a(k)Z and a(k) ≤A b(k) (INDR+) X ≤Y b(k)Z (a, b ∈ A SA) X ≤Y Z (AR) X ≤Y a(n+1)a(n)Z X ≤Y Z and a(n) ≤A b(n) (AR+) X ≤Y a(n+1)b(n)Z

Theorem

X ≤ Y is provable in S[A] iff it is provable in SAdj[A].

equivalence with Pregroups (for ≤A as ≤) as a particular case

Wollic 2007 – p.20

Other Properties of composed calculi

• Proposition. Let A = A

SA, ≤A where ≤A is a preorder.

If ≤A is a decidable calculus, then S[A] and S

′

[A] (applied to A) are decidable.

This is clear, using the cut elimination theorem, and the subformula property for S[A] (with the special case of IR, having only one possible antecedent).

Wollic 2007 – p.21

Other Properties of composed calculi

• Proposition. Let A = A

SA, ≤A where ≤A is a preorder.

If ≤A is a decidable calculus, then S[A] and S

′

[A] (applied to A) are decidable.

This is clear, using the cut elimination theorem, and the subformula property for S[A] (with the special case of IR, having only one possible antecedent).

• Proposition. The language generated by a PG-grammar

G on a Sub-Logic A = A SA, ≤A where ≤A is a

preorder (G based on the deduction system S[A] or equivalently S

′

[A]) is a context-free language.

This can be shown by associating to G a free PG-grammar GP G,

• btained by replacing, in the assignment, all subformulas F that

belong to A by a new constant cF , with cF ≤ cF ′ whenever F ≤A F ′ : GP G generates the same language as G.

Wollic 2007 – p.21

Conclusion and remarks

We have reformulated and proposed to extend the pregroup calculus, for its composition with other logics and calculi. Formal results The cut elimination property and the decidability property. Equivalence with PG, when A is reduced to Pr. Practical issues of "parameterized pregroups" Ready as a decision procedure and a parsing algorithm. Customized calculi , ex : structure to the basic types. Other perspectives PG as argument of another logic functor ;

• ther Lambek calculi ;

enrich types in a clear way, both formally and practically.

Wollic 2007 – p.22

A schema

LAB(G) LL(G) LNL∅(G) LPG(G)

. . .

LPG[A](G[A]) ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ LL∅(G) LNL(G)

✲ ✲ ❄ ❄ ✲ ✲ ✲

Wollic 2007 – p.23