Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex Plan Introduction 1 On categorial grammars and learnability 2 Logical Information Systems (LIS) 3 Categorial grammars and/as LIS 4 Annex 5 1 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
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 ? Language parsing (structures) from theoretical . . . . . to . as practical issues proof (trees) Logic Computing DATA (nature: linguistic | general | mixed) LIS USER (mode: data specialist | data exploration | action) HELP (system: reliable & informative & easy to use) 3 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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. 4 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex Plan Introduction 1 On categorial grammars and learnability 2 background (un)-learnability from strings learning from structures other type constructions Logical Information Systems (LIS) 3 Categorial grammars and/as LIS 4 Annex 5 5 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 : { word i �→ { type i , 1 , type i , 2 , ... }} ⊲ 6 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 : { word i �→ { type i , 1 , type i , 2 , ... }} ⊲ ∈ L ( G ) John likes Mary N ( N \ S ) / N N ⊢ S in AB, NL, L L a m N b e L k A ⊢ A , L Γ ⊢ A ∆ ⊢ C / A (Γ , A ) ⊢ B / e / i (∆ , Γ) ⊢ C Γ ⊢ B / A Γ ⊢ A ∆ ⊢ A \ C ( A , Γ) ⊢ B \ e \ i (Γ , ∆) ⊢ C Γ ⊢ A \ B 6 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 : { word i �→ { type i , 1 , type i , 2 , ... }} ⊲ ∈ L ( G ) John likes Mary N ( N \ S ) / N N ⊢ S in AB, NL, L N (1) S (0) N ( − 1) N (0) N (0) in Pregroup ⊲ L a m N b e L k A ⊢ A , L G Γ , ∆ ⊢ C P n i Γ ⊢ A ∆ ⊢ C / A (Γ , A ) ⊢ B Γ , p ( n ) , q ( n +1) , ∆ ⊢ C / e / i for (p ≤ q, n even) or (q ≤ p, n odd) (∆ , Γ) ⊢ C Γ ⊢ B / A Γ ⊢ A ∆ ⊢ A \ C ( A , Γ) ⊢ B \ e \ i (Γ , ∆) ⊢ C Γ ⊢ A \ B 6 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 : { word i �→ { type i , 1 , type i , 2 , ... }} ⊲ ∈ L ( G ) John likes Mary N ( N \ S ) / N N ⊢ S in AB, NL, L L a m N b e L k G A ⊢ A , L D ran : N \ S / A C n i Γ ⊢ A ∆ ⊢ C / A (Γ , A ) ⊢ B / e / i (∆ , Γ) ⊢ C Γ ⊢ B / A ⊲ Γ ⊢ A ∆ ⊢ A \ C ( A , Γ) ⊢ B a n b n c n ⊲ mix ⊲ rules ⊲ \ e \ i (Γ , ∆) ⊢ C Γ ⊢ A \ B 6 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 , ∀� e i � i ∈ N with L ( G ) = � e i � i ∈ N : ∃ G ′ ∈ G with L ( G ′ ) = L ( G ) φ ( { e 1 , . . . , e n } ) = G ′ ∈ G ∃ n 0 ∈ N : ∀ n > n 0 where L ( G ) denotes the language 1 associated to G 1 of strings or more generally of structures 7 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 ) 8 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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) 9 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 10 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 ∃ � L n � n ∈ N infinite sequence of langages in CL � L 0 � L 1 . . . � ... � L n � . . . such that : L ∗ = � n ∈ N L n ∈ CL Property. The languages of grammars of G have a limit point = ⇒ the class G is not learnable no learning algorithm! n 0 n bc abc aabc · · · · · · · · · a ( ≤ n 0 ) bc a ( ≤ n ) bc · · · ⊆ a ∗ bc Language ? · · · ⊆ 11 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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 ∃ � L n � n ∈ N infinite sequence of langages in CL � L 0 � L 1 . . . � ... � L n � . . . such that : L ∗ = � n ∈ N L n ∈ CL Property. The languages of grammars of G have a limit point = ⇒ the class G is not learnable no learning algorithm! n 0 n bc abc aabc · · · · · · · · · a ( ≤ n 0 ) bc a ( ≤ n ) bc · · · ⊆ a ∗ bc Language ? · · · ⊆ In contrast to rigid AB-grammars (learnable from strings) rigid L-grammars and NL-grammars admit string limit points [2002,2003] 11 Annie Foret IRISA & Univ. Rennes, France Categorial Inference and LIS
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