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1 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Plan

1

Introduction

2

On categorial grammars and learnability

3

Logical Information Systems (LIS)

4

Categorial grammars and/as LIS

5

Annex

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

On Categorial Grammatical Inference and Logical Information Systems

Annie Foret IRISA & Univ. Rennes, France

Email: foret@irisa.fr LACompLing 2018, Stockholm, August 28–31 2018

work in particular with: SemLIS team at Univ. Rennes (S. Ferr´ e) Data and Knowledge Management department and Univ. Nantes (D. B´ echet)

2

3 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

modelling (natural languages, sentences structures) [lexicon, corpora] via formal grammars (finite description) [categorial, dependencies] inference ? Logic Computing Language from theoretical . . . to practical issues parsing (structures) . . . as proof (trees) DATA (nature: linguistic | general | mixed) LIS USER (mode: data specialist | data exploration | action) HELP (system: reliable & informative & easy to use)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

4 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Abstract

We shall consider several classes of categorial grammars and discuss their learnability. We consider learning as a symbolic issue in an unsupervised setting, from raw or from structured data, for some variants of Lambek grammars and of categorial dependency grammars. In that perspective, we discuss for these frameworks different type constructors and structures, some limitations (negative results) but also some algorithms (positive results) under some hypothesis. On the experimental side, we also consider the Logical Information Systems approach, that allows for navigation, querying, updating, and analysis of heterogeneous data collections where data are given (logical) descriptors. Categorial grammars can be seen as a particular case of Logical Information System.

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

5 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Plan

1

Introduction

2

On categorial grammars and learnability background (un)-learnability from strings learning from structures

• ther type constructions

3

Logical Information Systems (LIS)

4

Categorial grammars and/as LIS

5

Annex

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

6 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Non-Commutative Type Calculi and grammar languages

Categorial grammars : {wordi → {typei,1, typei,2, ...}} ⊲

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

6 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Non-Commutative Type Calculi and grammar languages

Categorial grammars : {wordi → {typei,1, typei,2, ...}} ⊲ John N likes (N \ S) / N Mary N ∈ ⊢ L(G) S in AB, NL, L A ⊢ A

L a m b e k

N L , L

Γ ⊢ A ∆ ⊢ C / A /e (∆, Γ) ⊢ C (Γ, A) ⊢ B /i Γ ⊢ B / A Γ ⊢ A ∆ ⊢ A \ C \e (Γ, ∆) ⊢ C (A, Γ) ⊢ B \i Γ ⊢ A \ B

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

6 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Non-Commutative Type Calculi and grammar languages

Categorial grammars : {wordi → {typei,1, typei,2, ...}} ⊲ John N likes (N \ S) / N Mary N ∈ ⊢ L(G) S in AB, NL, L N(0) N(1) S(0) N(−1) N(0) in Pregroup ⊲ A ⊢ A

L a m b e k

N L , L

Γ ⊢ A ∆ ⊢ C / A /e (∆, Γ) ⊢ C (Γ, A) ⊢ B /i Γ ⊢ B / A Γ ⊢ A ∆ ⊢ A \ C \e (Γ, ∆) ⊢ C (A, Γ) ⊢ B \i Γ ⊢ A \ B

i n P G Γ, ∆ ⊢ C Γ, p(n), q(n+1), ∆ ⊢ C for (p ≤ q, n even) or (q ≤ p, n odd)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

6 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Non-Commutative Type Calculi and grammar languages

Categorial grammars : {wordi → {typei,1, typei,2, ...}} ⊲ John N likes (N \ S) / N Mary N ∈ ⊢ L(G) S in AB, NL, L A ⊢ A

L a m b e k

N L , L

Γ ⊢ A ∆ ⊢ C / A /e (∆, Γ) ⊢ C (Γ, A) ⊢ B /i Γ ⊢ B / A Γ ⊢ A ∆ ⊢ A \ C \e (Γ, ∆) ⊢ C (A, Γ) ⊢ B \i Γ ⊢ A \ B

i n C D G

ran : N \ S / A

⊲

anbncn ⊲ mix ⊲ rules ⊲ Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

7 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Identification in the limit (Gold)

Algorithm : Input : a finite set of sentences (positive examples). Output : a grammar in the class that generates the sentences ; the algorithm is required to converge Formally :

G : class of grammars V : alphabet φ : function from finite subsets of V ∗ to G such that ∀G ∈ G, ∀eii∈N with L(G) = eii∈N : ∃G ′ ∈ G with L(G ′) = L(G) ∃n0 ∈ N : ∀n > n0 φ({e1, . . . , en}) = G ′ ∈ G where L(G) denotes the language1 associated to G

1of strings or more generally of structures Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

8 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

”Inductive Inference from positive data is powerful” [T. Shinohara 1989]

(FT → IP) (¬FE) (FT → FE) (FE → IP) Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

9 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Inductive Inference of monotonic formal systems from positive data [T. Shinohara, ALT 1990]

Given (U, E, M) U of objects, (a universe) E of expressions, M a semantic mapping from finite subsets of E (formal systems) to subsets of U (concepts) such as: U:strings over Σ E:grammar rules M:language (monotonic) holds for any n: class of languages of context-sensitive grammars with at most k rules (learnable) Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

10 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Finite Elasticity

Finite elasticity is a nice property : it can be extended from a class to every class obtained by a finite-valued relation. Theorem [Kanazawa 1998] Let M be a class of languages over G that has finite elasticity, and let R ⊆ Σ∗ × G∗ be a finite-valued relation. Then the class of languages {R(−1)[M] | M ∈ M} has finite elasticity. where R(−1)[M] = {s ∈ Σ∗|∃u ∈ M ∧ (s, u) ∈ R} A relation R ⊆ Σ∗ × G∗ is finite-valued iff for every s ∈ Σ∗ , there are at most finitely many u ∈ G∗ such that (s, u) ∈ R

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

11 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Limit points – tool − → not Learnable

• Definition. A class CL of langages has a limit point

iff ∃ Lnn∈N infinite sequence of langages in CL such that : L0 L1 . . . ... Ln . . . L∗ =

n∈N Ln ∈ CL

Property. The languages of grammars of G have a limit point = ⇒ the class G is not learnable no learning algorithm! n0 n bc abc aabc · · · · · · · · · Language? a(≤n0)bc · · · ⊆ a(≤n)bc · · · ⊆ a∗bc

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

11 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Limit points – tool − → not Learnable

• Definition. A class CL of langages has a limit point

iff ∃ Lnn∈N infinite sequence of langages in CL such that : L0 L1 . . . ... Ln . . . L∗ =

n∈N Ln ∈ CL

Property. The languages of grammars of G have a limit point = ⇒ the class G is not learnable no learning algorithm! n0 n bc abc aabc · · · · · · · · · Language? a(≤n0)bc · · · ⊆ a(≤n)bc · · · ⊆ a∗bc

In contrast to rigid AB-grammars (learnable from strings) rigid L-grammars and NL-grammars admit string limit points [2002,2003]

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

12 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Finite elasticity – tool − → Learnable

[Wright (1989)+ Motoki, Shinohara]

• Definition. A class CL of languages has infinite elasticity

iff ∃eii∈N sentences ∃Lii∈N languages in CL ∀i ∈ N : {e1, . . . , en} ⊆ Ln+1 and ei ∈ Li. has finite (not infinite) elasticity = ⇒ is learnable

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

12 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Finite elasticity – tool − → Learnable

[Wright (1989)+ Motoki, Shinohara]

• Definition. A class CL of languages has infinite elasticity

iff ∃eii∈N sentences ∃Lii∈N languages in CL ∀i ∈ N : {e1, . . . , en} ⊆ Ln+1 and ei ∈ Li. has finite (not infinite) elasticity = ⇒ is learnable

• Definition. A class CL of langages has a limit point

iff ∃ Lnn∈N infinite sequence of langages in CL such that :

• L0 L1 . . . ... Ln . . .

L∗ =

n∈N Ln ∈ CL

has a limit point ⇒ is unlearnable ⇒ has infinite elasticity

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

12 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Finite elasticity – tool − → Learnable

[Wright (1989)+ Motoki, Shinohara]

• Definition. A class CL of languages has infinite elasticity

iff ∃eii∈N sentences ∃Lii∈N languages in CL ∀i ∈ N : {e1, . . . , en} ⊆ Ln+1 and ei ∈ Li. has finite (not infinite) elasticity = ⇒ is learnable

• Definition. A class CL of langages has a limit point

iff ∃ Lnn∈N infinite sequence of langages in CL such that :

• L0 L1 . . . ... Ln . . .

L∗ =

n∈N Ln ∈ CL

has a limit point ⇒ is unlearnable ⇒ has infinite elasticity

In contrast to rigid AB-languages (finite elasticity = ⇒ learnable) rigid L-languages and NL-grammars do not have string finite elasticity

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

13 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A chain of types in L and in pregroups

A preliminary question Type-raising in L X Y =def Y / (X \ Y ) with X ⊢ X Y Type-raising in pregroups [X Y ] = [Y ] [Y ]l [X] Iterating with the same exponent (X Y )Y ⊢ X Y and X Y ⊢ (X Y )Y [(X Y )Y ] = [X Y ] closure

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

13 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A chain of types in L and in pregroups

A preliminary question Type-raising in L X Y =def Y / (X \ Y ) with X ⊢ X Y Type-raising in pregroups [X Y ] = [Y ] [Y ]l [X] Iterating with the same exponent (X Y )Y ⊢ X Y and X Y ⊢ (X Y )Y [(X Y )Y ] = [X Y ] closure Iterating and alternating exponents A0 Am =def ((Am−1)y)z (x, y, z primitive) A0 ⊢ A1 . . . Am−1 ⊢ Am (strict derivations)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

13 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A chain of types in L and in pregroups

A preliminary question Type-raising in L X Y =def Y / (X \ Y ) with X ⊢ X Y Type-raising in pregroups [X Y ] = [Y ] [Y ]l [X] Iterating with the same exponent (X Y )Y ⊢ X Y and X Y ⊢ (X Y )Y [(X Y )Y ] = [X Y ] closure Iterating and alternating exponents A0 Am =def ((Am−1)y)z (x, y, z primitive) A0 ⊢ A1 . . . Am−1 ⊢ Am (strict derivations) [A0] ≤ [A1] . . . [Am−1[≤ [Am] (strict) A0 = x ≤ zzlyylx . . . zzlyyl

m−1

x ≤ zzlyyl

m

x (strict)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

14 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Method overview for limit points

Grammars of the form : Gn = {a → A; b → B; c → Dn} Chains Dn ⊢ Dn−1 ⊢ . . . ... ensures: L(Gn) ⊇ L(Gn−1) ⊇ . . . ... Inclusion

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

14 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Method overview for limit points

Grammars of the form : Gn = {a → A; b → B; c → Dn} Chains Dn ⊢ Dn−1 ⊢ . . . ... ensures: L(Gn) ⊇ L(Gn−1) ⊇ . . . ... Inclusion Schema in L∅ and NL∅: D / A

Dn

⊢ D

• Dn−1

with ⊢ A iterated (tautology ex : A = p / p) and alternated (using B = q / q) Strictness

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

14 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Method overview for limit points

Grammars of the form : Gn = {a → A; b → B; c → Dn} Chains Dn ⊢ Dn−1 ⊢ . . . ... ensures: L(Gn) ⊇ L(Gn−1) ⊇ . . . ... Inclusion Schema in L∅ and NL∅: D / A

Dn

⊢ D

• Dn−1

with ⊢ A iterated (tautology ex : A = p / p) and alternated (using B = q / q) Strictness

• therwise . . . = L(Gn) = L(Gn−1) = . . .

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

15 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

System Grammars and types Languages Ref Subclasses L∅ G1,n = {a → p / p ; b → q / q ; c → D1,n} c(b∗a∗)n [FN02a] / only G1,∗ = {a → p / p ; b → p / p ; c → S / (p / p)} c{a, b}∗

• rder 2

where  D1,0 = S D1,n = (D1,n−1 / (p / p)) / (q / q) L

· ,/

G2,n = {a → p / p ; b → q / q ; c → D2,n} c(b∗a∗)n [FN02a] no \ G2,∗ = {a → A ; b → A ; c → (S / A)·A} c{a, b}∗ where  D2,0 = S, A = p / p, B = q / q D2,n = (((D2,n−1 / A)·A) / B)·B NL∅ G′

1,n = {a → p / p ; c → D′ 1,n}

{cak/0 ≤ k ≤ n} [FN02c]2 / only G′

1,∗ = {a → (p / p) / (p / p) ; c → S / (p / p)}

c{a}∗

• rder 2

where  D′

1,0 = S

D′

1,n = D′ 1,n−1 / (p / p)

L/,\ G′

3,n = {a → p\p ; b → q\q ; c → D′ 3,n}

c(b∗a∗)n − {c} [FN02c] no product G′

2,n = {a → p\p ; b → q\q ; c → D′ 2,n}

(by ⊢⊣) G′

3,∗ = {a → p\p ; b → p\p ; c → S / (p\p)}

c{a, b}+ where 8 < : D′

2,0 = S

D′

2,n = (S / p)·((p / q)·(q / p))n−1·(p / q)·q

D′

3,n = S / (D′ 2,n\S)

pregroups GP1,n = {a → ppl ; b → qql ; c → Cn} . . . [For02] “order” 2 from L∅

• r L

· ,/

where  C0 = S Cn = (Cn−1)pllplqllql GP1,∗ = {a → ppl ; b → ppl ; c → Spllpl} . . . NL (also NL

∅

L, L∅) Gn = 8 < : a → A / B = (q / (p\q)) / p b → Dn c → En\S 9 = ; {akbc/0 ≤ k ≤ n} [BF03b] no product

• rder 5

G∗ = {a → p / p ; b → p ; c → p\S} a∗bc where 8 > < > : A = D0 = E0 = q / (p\q) B = p Dn+1 = (A / B)\Dn En+1 = (A / A)\En pregroups from NL GPn = 8 > > < > > : a → qqlppl = [A / B] b → pprqqr | {z }

n

qqlp = [Dn] c → prqqrS = [En\S] 9 > > = > > ; {akbc/0 ≤ k ≤ n} [BF03b] “order” 1 GP∗ = {a → ppl ; b → p ; c → prS} a∗bc pregroups GP ′

n =

 a → (pl)nql ; b → qpql ; c → qrl d → rplrl ; e → rpns ff {abkcdke/0 ≤ k ≤ n} taln “order” 1/2 GP ′

∗ =

 a → ql ; b → qplql ; c → qrl d → rprl ; e → rs ff {abkcdke/k ≥ 0}

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

15 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

L∅ G1,n = {a → p / p ; b → q / q ; c → D1,n} G1,∗ = {a → p / p ; b → p / p ; c → S / (p / p)} where  D1,0 = S D1,n = (D1,n−1 / (p / p)) / (q / q) L

· ,/

G2,n = {a → p / p ; b → q / q ; c → D2,n} G2,∗ = {a → A ; b → A ; c → (S / A)·A} where  D2,0 = S, A = p / p, B = q / q D2,n = (((D2,n−1 / A)·A) / B)·B NL∅ G′

1,n = {a → p / p ; c → D′ 1,n}

G′

1,∗ = {a → (p / p) / (p / p) ; c → S / (p / p)}

where  D′

1,0 = S

D′

1,n = D′ 1,n−1 / (p / p)

L/,\ G′

3,n = {a → p\p ; b → q\q ; c → D′ 3,n}

G′

2,n = {a → p\p ; b → q\q ; c → D′ 2,n}

G′

3,∗ = {a → p\p ; b → p\p ; c → S / (p\p)}

where 8 < : D′

2,0 = S

D′

2,n = (S / p)·((p / q)·(q / p))n−1·(p / q)·q

D′

3,n = S / (D′ 2,n\S)

pregroups GP1,n = {a → ppl ; b → qql ; c → Cn} from L∅

• r L

· ,/

where  C0 = S Cn = (Cn−1)pllplqllql GP1,∗ = {a → ppl ; b → ppl ; c → Spllpl}

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

CG Acquisition and hierarchies

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

Fact: a strict hierarchy for AB languages

⊕⊲

LAB(G) . . . . . . . . . LCDG?,∗,...(G...) LL(G) LNL∅(G) LPG([G]) . . . LPG?,∗([G]...) ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ LL∅(G) LNL(G) ✲ ✲ ❄ ❄ ✲ ✲ ✲

⊕⊲

GAB,k : k-valued AB (Buszkowski,Penn 1990, Kanazawa 1998)

Learnable from FA-structures Learnable from strings

GL,k : k-valued Associative Lambek

Learnable from full derivation (Bonato,Retore 2001) . . . . . . . . . . . Not learnable from strings

(Foret,Le Nir 2002)

GNL,k : k-valued Non-Associative Lambek

Learnable from generalized FA-structures (Bechet,Foret 2003) Not Learnable from strings (bracketed) (” ”,” ” ACL03,CG04) + “arity”-bounded: learnable from strings (Bechet,Foret 03)

GPG,k : k-valued Pregroups

Not learnable from strings

(2002,ACL03,ICGI04)

+ algorithm from ”features”

(B´ echet,Foret,Tellier: SL 07)

+ bounded width, learnable from strings

(”: ICGI04, SL07)

GCDG,. : categorial dependency (B´

echet, Foret, Dikovsky -2010)

k-valued Not learnable from strings or FA-structures + algorithms from dependency structures

16

Learning algorithm example ...

RG on FA-structures AB

details ⊲

from D = { , }

(i) General form GF(D) (ii) Unification σ (iii) Rigid grammar RG(D) a X1 / X2, X3 / X4 X1 = X3, X2 = X4 X1 / X2 fast X5\(X3\S) (X1\S)\(X1\S) fish X4 X2 man X2 X2 swims X1\S, X5 X5 = X1\S X1\S

17

Learning algorithm example ...

RG on FA-structures AB

details ⊲

from D = { , }

(i) General form GF(D) (ii) Unification σ (iii) Rigid grammar RG(D) a X1 / X2, X3 / X4 X1 = X3, X2 = X4 X1 / X2 fast X5\(X3\S) (X1\S)\(X1\S) fish X4 X2 man X2 X2 swims X1\S, X5 X5 = X1\S X1\S possibly incremental, kernel grammar, adaptations

17

Learning algorithm example ...

RG on FA-structures AB

details ⊲

from D = { , }

(i) General form GF(D) (ii) Unification σ (iii) Rigid grammar RG(D) a X1 / X2, X3 / X4 X1 = X3, X2 = X4 X1 / X2 fast X5\(X3\S) (X1\S)\(X1\S) fish X4 X2 man X2 X2 swims X1\S, X5 X5 = X1\S X1\S possibly incremental, kernel grammar, adaptations Di ⊆ Dj . . . ⊆ F L(G) G is AB rigid → RG(Di) ⊑ RG(Dj) . . . where Gi ⊑ G is : → F L((...i)) ⊆ F L((...j)) . . . ∃σ : σ(Gi) ⊆ G

17

18 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A summary for NL learnability

non-associative Lambek

Adaptation of RG to NL : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A / x′

i

A xi . . . add the constraint xi ⊢ x′

i

Classes of NL grammars from (structured) examples

[TCS (Bechet, Foret)]

Restriction \ Structure strings Well bracketed strings Generalized F A all no no no . . . . . . . . . . . . . . . . . k-valued no

[ACL 03]

no

[CG 04]

yes

[TCS 06]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . k-valued and t-arity bounded yes

[CAP 03]

yes

corollary of [CAP 03]

yes

corollary of [CAP 03]

k-valued and FA-arity bounded yes [TCS 06] yes [TCS 06] yes [TCS 06]

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

19 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG iterated types, dependencies and parses

example

With the lexicon

(empty potentials)

John → N ran → [N\S/A∗] fast, yesterday → A Derivation

John

N

ran

[N \ S / A∗]

fast

A Lr [N \ S / A∗]

yesterday

A Ir [N \ S / A∗] Ωr [N \ S] Ll S

Dependency structures

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

19 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG iterated types, dependencies and parses

example

With the lexicon

(empty potentials)

John → N ran → [N\S/A∗] fast, yesterday → A Derivation

John

N

ran

[N \ S / A∗]

fast

A Lr [N \ S / A∗]

yesterday

A Ir [N \ S / A∗] Ωr [N \ S] Ll S

Dependency structures

Ll HP1[H\β]P2 ⊢ [β]P1P2 Lr [β / H]P2HP1 ⊢ [β]P2P1 Il C P1[C ∗\β]P2 ⊢ [C ∗\β]P1P2 Ir [β / C ∗]P2C P1 ⊢ [β / C ∗]P2P1 Ωl [C ∗\β]P ⊢ [β]P Ωr [β / C ∗]P ⊢ [β]P

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20 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Option ? and Iteration *

In categorial dependency grammars (CDG) [A. Dikovsky 04] these constructions are defined using reduction rules

• Ol. a[a?\β] ⊢ β

Ωl

?. [a?\β] ⊢ β

• Or. [β / a?]a ⊢ β

Ωr

?. [β / a?] ⊢ β A

• D

first

• ρ

student

• ρ?\D \ N

Ol D \ N

\e

N

ran

• N\S/ a∗

fast

• a

I r N\S/ a∗

yesterday

• a

I r N\S/ a∗ Ωr N\S

\e

S

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21 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG (un)-learnability

String limit point

[BDFM (FG 2004)]

Let S, A, B be local dependency names. Grammars G ′

n, G ′ ∗ are :

G ′

0 = {a → A, b → B, c → C ′ 0}

G ′

n = {a → A, b → B, c → [C ′ n]}

G ′

∗ = {a → A, b → A, c → [S / A∗]}

C ′

0 = S

C ′

n+1 = C ′ n / A∗ / B∗

L(G ′

n) = {c(b∗a∗)k | k ≤ n} and L(G ′ ∗) = c{b, a}∗

The constructions show the non-learnability from strings for the classes of (rigid) grammars allowing iterative subtypes (A∗).

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21 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG (un)-learnability

String limit point

[BDFM (FG 2004)]

Let S, A, B be local dependency names. Grammars G ′

n, G ′ ∗ are :

G ′

0 = {a → A, b → B, c → C ′ 0}

G ′

n = {a → A, b → B, c → [C ′ n]}

G ′

∗ = {a → A, b → A, c → [S / A∗]}

C ′

0 = S

C ′

n+1 = C ′ n / A∗ / B∗

L(G ′

n) = {c(b∗a∗)k | k ≤ n} and L(G ′ ∗) = c{b, a}∗

The constructions show the non-learnability from strings for the classes of (rigid) grammars allowing iterative subtypes (A∗). also a limit point for (rigid) FA-structure languages [BDF (FG 2010)] The number of iterative subtypes (A∗) is not bound !

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21 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG (un)-learnability

String limit point

[BDFM (FG 2004)]

Let S, A, B be local dependency names. Grammars G ′

n, G ′ ∗ are :

G ′

0 = {a → A, b → B, c → C ′ 0}

G ′

n = {a → A, b → B, c → [C ′ n]}

G ′

∗ = {a → A, b → A, c → [S / A∗]}

C ′

0 = S

C ′

n+1 = C ′ n / A∗ / B∗

L(G ′

n) = {c(b∗a∗)k | k ≤ n} and L(G ′ ∗) = c{b, a}∗

The constructions show the non-learnability from strings for the classes of (rigid) grammars allowing iterative subtypes (A∗). also a limit point for (rigid) FA-structure languages [BDF (FG 2010)] The number of iterative subtypes (A∗) is not bound ! Learning from dependency structures

[BDF (FG2010→))]

based on a repetition-undiscernability parameter (Meluk)

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22 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Iteration (*) – Option (?) – Repetition (+)

Name Rule Ol C P1[C ?\β]P2 ⊢ [β]P1P2

the D cities D? \ N ⊢ N

Ωl [C ?\β]P ⊢ [β]P

city A? \ N ⊢ N

Rl C P1[C +\β]P2 ⊢ [C ∗\β]P1P2

nice A city A+ \ N ⊢ A∗ \ N

. . . For rigid and k-valued classes

Class Learnable from strings Finite elasticity

• n strings

Finite elasticity

• n structures

Finite-valued Relation A∗ no → no yes → no A? no → no yes → no A+ yes ← yes ← yes yes

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23 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Interpretation of Mel’cuk’s principle

page-livre

Principle of repeatable optional dependencies [Mel’cuk 1988] Any dependency must be either non-repeatable, or repeatable (...)

Principle not precised about the order of subordinates. Several readings : consecutive repetitions [BDF2010] (sequential iterations are frequent, but reading not linguistically founded) ; dispersed iterations : a repeatable dependency can be placed anywhere on the left (or right) of the head ;

[CAP and LACL 2011]

with iterated choices : disjunctive choices of repeatable dependencies can occur at the same argument position.

[LACL 2011]

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23 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Interpretation of Mel’cuk’s principle

page-livre

Principle of repeatable optional dependencies [Mel’cuk 1988] Any dependency must be either non-repeatable, or repeatable (...)

Principle not precised about the order of subordinates. Several readings : consecutive repetitions [BDF2010] (sequential iterations are frequent, but reading not linguistically founded) ; dispersed iterations : a repeatable dependency can be placed anywhere on the left (or right) of the head ;

[CAP and LACL 2011]

• IDl. C P1[{α1, C ∗, α2}\β/{γ}]P2 ⊢ [{α1, C ∗, α2}\β/{γ}]P1P2

...

dispersed iteration

with iterated choices : disjunctive choices of repeatable dependencies can occur at the same argument position.

[LACL 2011]

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23 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Interpretation of Mel’cuk’s principle

page-livre

Principle of repeatable optional dependencies [Mel’cuk 1988] Any dependency must be either non-repeatable, or repeatable (...)

Principle not precised about the order of subordinates. Several readings : consecutive repetitions [BDF2010] (sequential iterations are frequent, but reading not linguistically founded) ; dispersed iterations : a repeatable dependency can be placed anywhere on the left (or right) of the head ;

[CAP and LACL 2011]

with iterated choices : disjunctive choices of repeatable dependencies can occur at the same argument position.

[LACL 2011]

We have explored the learnability of CDG according to the way these dependencies are expressed.

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24 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Learning Algorithm

Learning from dependency structures

[BDF (FG2010→))]

based on a repetition-undiscernability parameter (Meluk) (no k-valued bound) Computes words’ types from their VICINITIES in DS Vicinity V (w, D) of word w in DS D:

V (partition, D) = [det\a−obj/modif /attr/attr/modif ], V (de, D) = [attr/prepos−gp]

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25 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Algorithm TGE(K)

disp (type-generalize-expand): ex ⊲

Input: σ[i] (σ being a training sequence). Output: GCD TGE(K)

disp(σ[i]).

let GH = (WH, CH, S, λH) where WH := ∅; CH := {S}; λH := ∅; k := 0 (loop) for i ≥ 0 //Infinite loop on σ let σ[i + 1] = σ[i] · D; let (x, E) = D; // D is the new example (loop) for every w ∈ x // w is a word in D WH := WH ∪ {w}; let V (w, D) = [lm\ · · · \l1\h/r1/ · · · /rn]P // intermediary type for w in D let LT := {l1} ∪ · · · ∪ {lm} let LF := {d : d ∈ LT, card ({i : 1 ≤ i ≤ m, li = d}) ≥ K } // find on the left let RT := {r1} ∪ · · · ∪ {rn} let RF := {d : d ∈ RT, card ({i : 1 ≤ i ≤ n, ri = d}) ≥ K } // find on the right let tw := [ {lf ∗

1 , . . . , lf ∗ p }\ l′ m′\ · · · \l′ 1\h/r′ 1/ · · · /r′ n′ /{rf ∗ 1 , . . . , rf ∗ q }]P

// generalised type where {lf1, . . . , lfp} = LF, {rf1, . . . , rfq} = RF, where l′

m′, . . . , l′ 1 is the sublist of lm, . . . , l1 without elements in LF

where r′

1, . . . , r′ n′ is the sublist of r1, . . . , rn without elements in RF.

λH(w) := λH(w) ∪ {tw}; // expansion end end

Figure: Inference algorithm TGE(K)

disp

prop ⊲

The class of dispersed K-star revealing CDG is (incrementally) learnable from DS .

grammars equiv. to their generalisation CK

disp(G) not distinguishing dependencies that repeat

K times from ∗ Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

CG Acquisition and hierarchies

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

Fact: a strict hierarchy for AB languages

⊕⊲

LAB(G) . . . . . . . . . LCDG?,∗,...(G...) LL(G) LNL∅(G) LPG([G]) . . . LPG?,∗([G]...) ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ LL∅(G) LNL(G) ✲ ✲ ❄ ❄ ✲ ✲ ✲

⊕⊲

GAB,k : k-valued AB (Buszkowski,Penn 1990, Kanazawa 1998)

Learnable from FA-structures Learnable from strings

GL,k : k-valued Associative Lambek

Learnable from full derivation (Bonato,Retore 2001) . . . . . . . . . . . Not learnable from strings

(Foret,Le Nir 2002)

GNL,k : k-valued Non-Associative Lambek

Learnable from generalized FA-structures (Bechet,Foret 2003) Not Learnable from strings (bracketed) (” ”,” ” ACL03,CG04) + “arity”-bounded: learnable from strings (Bechet,Foret 03)

GPG,k : k-valued Pregroups

Not learnable from strings

(2002,ACL03,ICGI04)

+ algorithm from ”features”

(B´ echet,Foret,Tellier: SL 07)

+ bounded width, learnable from strings

(”: ICGI04, SL07)

GCDG,. : categorial dependency (B´

echet, Foret, Dikovsky -2010)

k-valued Not learnable from strings or FA-structures + algorithms from dependency structures

26

27 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Calculi and grammar languages

LAB(G) . . . . . . . . . LCDG?,∗,...(G...) LL(G) LNL∅(G) LPG([G]) . . . LPG?,∗([G]...) ⊆ ⊆ ⊆ ⊆ ⊆ ⊆ LL∅(G) LNL(G) ✲ ✲ ❄ ❄ ✲ ✲ ✲

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28 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Some remarks

Examples =positive raw or structured structure-arity Lexicon ambiguity degree arity

• rder degree

Formalism constructors+rules flexible

On data and subclasses :

• n strings, some bounds :

a lexicon ambiguity degree (k-valued) the arity (L), or pseudo-arity (CDG)

• n structures :

(bracketing may be not enough) full/partial derivations annotated structures Other learning paradigms (considering presentations, informant, negative examples, ...)

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29 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Plan

1

Introduction

2

On categorial grammars and learnability

3

Logical Information Systems (LIS)

4

Categorial grammars and/as LIS

5

Annex

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30 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

LCA = Logical Concept Analysis [Ferr´ e and Ridoux, 2004] (an extension of FCA) LCA methods provide guidance and serendipity (cf experiment) using a logical information context management system (Camelis) allows to explore the information content : linguistic info., in a flexible way, without a priori knowledge, with data quality feedbacks can ease other treatments, after appropriate selection in an extensible way, it can be enriched : by adding diverse objects without confusion by adding properties to expand the browsing possibilities by adding triggered actions on objects, according to contextual arguments in a safe way.

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31 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Logical Information Systems Approach

LCA = Logical Concept Analysis [Ferr´ e and Ridoux, 2004] (an extension of FCA) LIS Tools : Camelis

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32 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Logical Information Systems Approach

and tools

LCA = Logical Concept Analysis (an extension of FCA) LIS Tools : Camelis

from [ http://www.irisa.fr/LIS/softwares]

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33 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Introduction LIS

A logic is a pre-order LT = (L, ⊑T), where: L is a set of formulas, T is a customizable parameter of the logic, and ⊑T is a subsumption relation that depends on T.

f ⊑T g reads“f is more specific than g”or“f is subsumed by g” , also denotes the partial ordering induced from the pre-order (refl., trans.)

Logic For any object o, a formula d(o) denotes the description of o Information Context → A query is a logical formula. A logic functor is a function F that:

• takes logics L1, ..., Ln as arguments (n ≥ 0),
• returns a composed logic LT = F(L1, ..., Ln).

→ Customized logics for querying, navigating data. System Tool

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34 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Context = collection of objects + their logical description (vs set of attributes)

• Def. A logical context is a tuple K = (O, LT, X, d), where:

O is a finite set of objects, ← LT = (L, ⊑T) is a Logic, X ⊆ L is finite, called the navigation vocabulary, ← d ∈ (O → LT) is a mapping from objects to formulas. ← For any object o, the formula d(o) denotes the description of o. Answer (extent) of a query formula q = the set of objects whose description is subsumed by this formula

• Def. Extent

Let K be a logical context, and q ∈ L be a query formula. The extent of q in K is: K.ext(q) = {o ∈ O | d(o) ⊑T q}.

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35 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

LIS – Logic Components

A logic functor is a function F that:

• takes logics L1, ..., Ln as arguments (n ≥ 0),
• returns a composed logic LT = F(L1, ..., Ln).

As for any logic, one has LT = (L, ⊑T), but L (resp. ⊑T) is function of the sets of formulas (resp. subsumption) of arguments Li

LogFun ToolBox: in Objective Caml

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/

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36 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

LIS – Navigating in a Logical Context

The concept lattice → local view + moves

intent extent increments current query further queries queries properties

• bjects

navigation loop navigation links concept

Such a local view already gives a rich information about the context enables to move in every direction in the lattice

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37 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

LIS – Characteristics and Advantages

Properties Navigation Completeness No dead-end Expected qualities: extensible descriptions (by users) rich datatypes entailment relations between values expressive querying: patterns, AND, OR, NOT flexible navigation (guiding users) and interactions with other tools

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38 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Exploring variants and false-friends.

A state from selections in the index tree ; links of a same color characterize the same set of objects (concept)

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39 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Refinements - adaptations - linkage

This navigation mode allows to detect abnormalities, in particular pseudo-empty properties appear on other facets (through a link like Definition is ” ” ) these cases often correspond to existing but empty XML elements (but are not XML errors). low cardinalities in the navigation tree may suggest to explore the link, by selecting it and opening other facets simultaneously ; we can analyse this way ” the words without translation, following the not Equivalent? link. in case of redundancies, noticeable through browsing: exploring the Antonymes facet, we can see XML structuring redundancies (this information being carried by two source elements).

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

40 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex ✐♥tr♦❞✉❝t✐♦♥ ❖♥ ▲■❙ ❛♥❞ ▲❈❆ ❚❡r♠✐♥♦❧♦❣✐❝❛❧ r❡s♦✉r❝❡ ❚r❛♥s❞✉❝❡r ❯s✐♥❣ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥t❡①t ❢❛❝❡ts ❘❡✜♥❡♠❡♥ts ❛♥❞ ✉s❡r ♣r❡❢❡r❡♥❝❡s ❈♦♥❝❧✉s✐♦♥ ❆♥♥❡①

❘❡✜♥❡♠❡♥ts ✶ ✕ tr❛♥s❧❛t✐♥❣ ✐♥❞❡①

❆❞❛♣t✐♥❣ ❢❛❝❡ts ✉s✐♥❣ r✉❧❡s ❉♦♠❛✐♥s ❛♥❞ ❙✉❜❉♦♠❛✐♥s ❞❛t❛ ❤❛✈❡ ❜❡❡♥ tr❛♥s❧❛t❡❞✳ ❚❤❡ r❡s✉❧t✐♥❣ t✇♦ ❝♦♥t❡①t ✜❧❡s ❝♦♥t❛✐♥ ✉♣❞❛t❡ r✉❧❡s ♦❢ t❤❡ ❢♦r♠✿

r✉❧❡❴❡①tr ❉♦♠❛✐♥❡ ✐s ✧❆❝♦✉st✐q✉❡✧ ✕❃ ❉♦♠❛✐♥ ✐s ✧❆❝♦✉st✐❝✧

✇❤❡♥ ❧♦❛❞❡❞ ✐♥ t❤❡ ❝♦♥t❡①t✱ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ❛r❡ ❛❞❞❡❞ t♦ ❛❧❧ ♦❜❥❡❝ts ✈❡r✐❢②✐♥❣ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡✳

❆♥♥✐❡ ❋♦r❡t ▲■❙ ❛♥❞ ❚❡r♠✐♥♦❧♦❣② Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

41 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex ✐♥tr♦❞✉❝t✐♦♥ ❖♥ ▲■❙ ❛♥❞ ▲❈❆ ❚❡r♠✐♥♦❧♦❣✐❝❛❧ r❡s♦✉r❝❡ ❚r❛♥s❞✉❝❡r ❯s✐♥❣ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥t❡①t ❢❛❝❡ts ❘❡✜♥❡♠❡♥ts ❛♥❞ ✉s❡r ♣r❡❢❡r❡♥❝❡s ❈♦♥❝❧✉s✐♦♥ ❆♥♥❡①

❘❡✜♥❡♠❡♥ts ✷ ✕ ❤❛r♠♦♥✐③❛t✐♦♥

■♠♣r♦✈✐♥❣ ❣r❛♠♠❛t✐❝❛❧ ❝❛t❡❣♦r✐❡s ✉s✐♥❣ r✉❧❡s ❛♥❞ ❛①✐♦♠s ■♥ t❤❡ ♦r✐❣✐♥❛❧ ❝♦♥t❡①t✱ ❣r❛♠♠❛t✐❝❛❧ ❝❛t❡❣♦r✐❡s ❛r❡ ❧✐st❡❞ ✇✐t❤ ✈❛r✐♦✉s ✈❛❧✉❡s ✲ ❛ ❞✐s❥✉♥❝t✐♦♥✱ ❛s ✐♥✿

❝❛t❡❣♦r✐❡ ✐s ✧❛❞❥✳ ♦✉ ♥✳♠✳✧ ❛♥❞ ❊q✉✐✈❛❧❡♥t ❴❡♥ ✐s ✧❝r♦ss♠❡❞✐❛ ✭♥✳ ♦✉ ❛❞❥✳✮✧

✇❤✐❝❤ s❡❧❡❝ts t❤❡ t❡r♠ tr❛♥s♠é❞✐❛❀ ❜✉t ✇❡ ❛❧s♦ ♦❜s❡r✈❡ ✐ts ♣❡r♠✉t❛t✐♦♥ ❝❛t❡❣♦r✐❡ ✐s ✧♥✳♠✳

♦✉ ❛❞❥✳✧❀

✲ ❛ ♠♦r❡ ♦r ❧❡ss ✜♥❡ ❣r❛♥✉❧❛r✐t②✱ ❛s ✐♥✿ ❝❛t❡❣♦r✐❡ ✐s ✧♥✳♠✳✐♥✈✳✧ ❲❡ ❛❞❞ r✉❧❡s ❛♥❞ ❛①✐♦♠s ✐♥ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥t❡①t s✉❝❤ ❛s✿

r✉❧❡❴❡①tr ❝❛t❡❣♦r✐❡ ✐s ✧♥✳♠✳✐♥✈✳✧ ✕❃ ❈❛t❡❣♦r✐❡❴♥❴♠❴✐♥✈ ✳✳✳ ❈❛t❡❣♦r✐❡❴♥❴♠❴✐♥✈ ❛①✐♦♠✱ ❈❛t❡❣♦r✐❡❴♥❴♠ ❈❛t❡❣♦r✐❡❴♥❴♠ ❛①✐♦♠✱ ❈❛t❡❣♦r✐❡❴♥

✳✳✳ t❤❛t ❞❡✜♥❡ ❛ ❤✐❡r❛r❝❤② ♦❢ ❝❛t❡❣♦r✐❡s✱ ❛♥❞ ❛ ❜❡tt❡r ✐♥❞❡① ■t ❝♦✉❧❞ ❛❧s♦ ✐♥✈♦❧✈❡ ❧✐♥❣✉✐st✐❝ ❛♥❛❧②s✐s ♦r ♦t❤❡r ❧❡①✐❝♦♥s✳

❆♥♥✐❡ ❋♦r❡t ▲■❙ ❛♥❞ ❚❡r♠✐♥♦❧♦❣② Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

42 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

From a small breton lexicon (and Apertium data)

by theme : “at home” ,“at school” name with det, gender : al logodenn* → Lexicon (simple xml) Expressions (has image) written by theme : with gender no image no translation enriched with : mutation marks comments ↓ LIS inform. context

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

43 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Monolingual context → after a new selection in the index on the left (dynamic, re-

• rded)

→ coherence between win- dows → indicators :

• size
• same color when same subset
• new links for exploring

(local view of the LCA lattice concept)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

44 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Plan

1

Introduction

2

On categorial grammars and learnability

3

Logical Information Systems (LIS)

4

Categorial grammars and/as LIS Pregroups Experiments on dependency treebanks

5

Annex

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

Categorial Grammars as LIS

Categorial grammars (such as pregroups) can be considered as LIS both theoretically, and practically. Enabled by:

their lexicalized nature the logical nature of this linguistic framework

Logical Concept Analysis can provide:

better representations better propagation of properties

The combination, through the toolbox,

helps the creation, control and query of linguistic grammars illustrates the strength and elegance of logic functors [WOLLIC 2007, Information and Computation 2010]

45

46 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A toy grammar, with other information

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

47 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

using PG axioms

• Fig. 3. A toy grammar as a LIS context:

pgtype [o] vs pgtype [n] (n ≤ o in the pregroup grammar)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

48 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

49 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

50 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex S´ ebastien Ferr´ e and Olivier Ridoux. 2004. Introduction to logical information systems.

• Inf. Process. Manage., 40(3):383–419.

Bernhard Ganter and Rudolf Wille. 1999. Formal concept analysis - mathematical foundations. Springer.

fundational

Peggy Cellier, S´ ebastien Ferr´ e, Annie Foret and Olivier Ridoux. 2016 Exploration des Donn´ ees du D´ efi EGC 2016 ` a l’aide d’un Syst` eme d’Information Logique. In Defi EGC 2016. Annie Foret. 2015 A logical information system proposal for browsing terminological resources. In Terminology and Artificial Intelligence (TIA) 2015. Annie Foret, Val´ erie Bellynck and Christian Boitet 2015 Akenou, un projet de plate-forme valorisant des ressources et outils informatiques et linguistiques pour le breton In Traitement Automatique des langues r´ egionales (TALARE), TALN workshop, 2015. Annie Foret and S´ ebastien Ferr´ e. 2010. On categorial grammars as logical information systems. In L´ eonard Kwuida and Baris Sertkaya, editors, Formal Concept Analysis, 8th International Conference, ICFCA 2010, Agadir, Morocco, March 15-18, 2010. Proceedings, volume 5986 of Lecture Notes in Computer Science, pages 225–240. Springer. Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

51 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

From dependency treebanks to vicinities, then CDG

[LACL 2016, ICGI 2016]

Figure: Simplified vicinities computed on corpus Sequoia

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

52 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Summarizing

Learning as a symbolic issue in an unsupervised setting, – from raw or from structured data, – for some variants of Lambek grammars and of categorial dependency grammars. – different type constructors and structures, Some limitations (negative results) but also some algorithms (positive results) under some hypothesis, now applied to treebanks. The Logical Information Systems approach, allows for navigation, querying (classify), updating, and analysis of heterogeneous data collections (linguistic data) where data are given (logical) descriptors. (Inferred) Categorial grammars as a case of Logical Information System also formally as logic functors to extend them by composition with other logics and calculi.

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

53 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Plan

1

Introduction

2

On categorial grammars and learnability

3

Logical Information Systems (LIS)

4

Categorial grammars and/as LIS

5

Annex

Pregroup calculus On CDG

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

54 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

A pregroup is a structure (P, ≤, ·, l, r, 1) s. t.

• (P, ≤, ·, 1) is a partially ordered monoid (· associative, 1 neutral)

A partially ordered monoid is a monoid (M, ·, 1) with a partial

• rder ≤ that satisfies ∀a, b, c: a ≤ b ⇒ c · a ≤ c · b ∧ a · c ≤ b · c.
• 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 . . .

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

55 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

Pregroup type calculus

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 Xp(k)Y≤Z (INDL) Xq(k)Y ≤Z X ≤Y Y ≤Z (Cut) X ≤ Z X ≤YZ (AR) X ≤Yp(n+1)p(n)Z X ≤Yq(k)Z (INDR) X ≤Yp(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

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

56 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG example: mix [LACL2005]

Figure: A CDG grammar for mix, in the ”CDG Parser” format, with a parse example In the above grammar2 , some types have empty heads ; other grammars avoiding empty heads can be provided, but the above one is simpler.

2the CDG Parser is a part of the CDG Lab in LINA, Nantes Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS

57 Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex

CDG example: anbncn

Figure: A CDG grammar for anbncn, in the ”CDG Parser” format, with a parse example (the CDG Parser is a part of the CDG Lab in LINA, Nantes)

Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS