On Categorial Grammatical Inference And Logical .
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexPlan1Introduction2On categorial grammars and learnability3Logical Information Systems (LIS)4Categorial grammars and/as LIS5Annex1Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
On Categorial Grammatical InferenceandLogical Information SystemsAnnie ForetIRISA & Univ. Rennes, FranceEmail: foret@irisa.frLACompLing 2018, Stockholm, August 28–31 2018work in particular with:SemLIS team at Univ. Rennes (S. Ferré)Data and Knowledge Management departmentand Univ. Nantes (D. Béchet)2
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annexmodelling (natural languages, sentences structures),modelling (natural languages, sentences structures)[lexicon, corpora]via formal grammars (finite description, idealizations),via formal grammars (finite description)[categorial, dependencies]several frameworks and traditions, no winning oneinference ?Languageparsing (structures). asproof (trees)LISLogicfrom theoretical. topractical issuesComputingDATA (nature: linguistic general mixed)USER (mode: data specialist data exploration action)HELP (system: reliable & informative & easy to use)2/43Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexAbstractWe shall consider several classes of categorial grammars anddiscuss their learnability.We consider learning as a symbolic issue in an unsupervisedsetting, from raw or from structured data, for some variants ofLambek grammars and of categorial dependency grammars. Inthat perspective, we discuss for these frameworks different typeconstructors and structures, some limitations (negative results) butalso some algorithms (positive results) under some hypothesis.On the experimental side, we also consider the Logical InformationSystems approach, that allows for navigation, querying, updating,and analysis of heterogeneous data collections where data aregiven (logical) descriptors.Categorial grammars can be seen as a particular case of LogicalInformation System.4Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexPlan1Introduction2On categorial grammars and learnabilitybackground(un)-learnability from stringslearning from structuresother type constructions3Logical Information Systems (LIS)4Categorial grammars and/as LIS5Annex5Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexNon-Commutative Type Calculi and grammar languagesCategorial grammars : {wordi 7 {typei,1 , typei,2 , .}}.6Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexNon-Commutative Type Calculi and grammar languagesCategorial grammars : {wordi 7 {typei,1 , typei,2 , .}}JohnNlikes(N \ S) / NMaryN L(G )S.in AB, NL, LmLaΓ A A\C(Γ, ) C/e\e(Γ, A) BΓ B /A(A, Γ) BΓ A\BAnnie Foret IRISA & Univ. Rennes, Francek( , Γ) CLL,Γ A C/AbeNA A/i\iCategorial Inference and LIS6
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexNon-Commutative Type Calculi and grammar languagesCategorial grammars : {wordi 7 {typei,1 , typei,2 , .}}JohnNN (0)likes(N \ S) / NN (1)S (0)MaryNN ( 1)N (0) L(G )S.in AB, NL, Lin Pregroup .mLaΓ A A\C(Γ, ) C/e\e(Γ, A) BΓ B /A(A, Γ) BΓ A\BAnnie Foret IRISA & Univ. Rennes, FranceinPGk( , Γ) CLL,Γ A C/AbeNA A/iΓ, CΓ, p (n) , q (n 1) , Cfor (p q, n even) or (q p, n odd)\iCategorial Inference and LIS6
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexNon-Commutative Type Calculi and grammar languagesCategorial grammars : {wordi 7 {typei,1 , typei,2 , .}}JohnNlikes(N \ S) / NMaryN L(G )S.in AB, NL, LmLaΓ A A\C(Γ, ) CGk( , Γ) CCDran : N \ S / AinLL,Γ A C/AbeNA A/e\e(Γ, A) BΓ B /A(A, Γ) BΓ A\BAnnie Foret IRISA & Univ. Rennes, France/i.\ian b n c n.mix.Categorial Inference and LISrules.6
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexIdentification in the limit (Gold)Algorithm :Input : a finite set of sentences (positive examples).Output : a grammar in the class that generates thesentences ; the algorithm is required to convergeFormally :G : class of grammarsV : alphabetφ : function from finite subsets of V to G such that G G, hei ii N with L(G ) hei ii N : G 0 G with L(G 0 ) L(G ) n0 N : n n0 φ({e1 , . . . , en }) G 0 Gwhere L(G ) denotes the language1 associated to G1of strings or more generally of structuresAnnie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS7
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)8Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexInductive 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 mappingfrom finite subsets of E(formal systems)to subsets of U (concepts)such as:U:strings over ΣE :grammar rulesM:language(monotonic)holds for any n:class of languages ofcontext-sensitive grammarswith at most k rules(learnable)9Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexFinite ElasticityFinite elasticity is a nice property :it can be extended from a class to every class obtained by afinite-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 finiteelasticity.where R ( 1) [M] {s Σ u M (s, u) R}A relation R Σ G is finite-valued iff for every s Σ , thereare at most finitely many u G such that (s, u) R10Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexLimit points – tool not LearnableDefinition. A class CL of langages has a limit pointiff hLn in N infinite sequence of langages in CLL0 ( LS1 . . . ( . ( Ln ( . . .such that :L n N Ln CLProperty.The languages of grammars of G have a limit point the class G is not learnableno learning algorithm!n0nbc abc aabc·········( n)( n)0Language?abc · · · abc · · · a bc11Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexLimit points – tool not LearnableDefinition. A class CL of langages has a limit pointiff hLn in N infinite sequence of langages in CLL0 ( LS1 . . . ( . ( Ln ( . . .such that :L n N Ln CLProperty.The languages of grammars of G have a limit point the class G is not learnableno learning algorithm!n0nbc abc aabc·········( n)( n)0Language?abc · · · abc · · · a bcIn contrast to rigid AB-grammars (learnable from strings)rigid L-grammars and NL-grammars admit string limit points [2002,2003]11Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex LearnableFinite elasticity – tool[Wright (1989) Motoki, Shinohara]Definition. A class CL of languages has infinite elasticityiff hei ii N sentences hLi ii N languages in CL i N : {e1 , . . . , en } Ln 1 and ei 6 Li .has finite (not infinite) elasticity is learnable12Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex LearnableFinite elasticity – tool[Wright (1989) Motoki, Shinohara]Definition. A class CL of languages has infinite elasticityiff hei ii N sentences hLi ii N languages in CL i N : {e1 , . . . , en } Ln 1 and ei 6 Li .has finite (not infinite) elasticity is learnableDefinition. A class CL of langages has a limit pointiff hLn in N infinite sequence of langages in CLL0 ( LS1 . . . ( . ( Ln ( . . .such that :L n N Ln CLhas a limit point is unlearnable has infinite elasticity12Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annex LearnableFinite elasticity – tool[Wright (1989) Motoki, Shinohara]Definition. A class CL of languages has infinite elasticityiff hei ii N sentences hLi ii N languages in CL i N : {e1 , . . . , en } Ln 1 and ei 6 Li .has finite (not infinite) elasticity is learnableDefinition. A class CL of langages has a limit pointiff hLn in N infinite sequence of langages in CLL0 ( LS1 . . . ( . ( Ln ( . . .such that :L n N Ln CLhas a limit point is unlearnable has infinite elasticityIn contrast to rigid AB-languages (finite elasticity learnable)rigid L-languages and NL-grammars do not have string finite elasticity12Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexA chain of types in L and in pregroupsType-raising in LX Y def Y / (X \ Y ) with X X YA preliminary questionType-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 ]closure13Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexA chain of types in L and in pregroupsType-raising in LX Y def Y / (X \ Y ) with X X YA preliminary questionType-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 ]closureIterating and alternating exponentsA0 Am def ((Am 1 )y )z (x, y , z primitive)A0 A1 . . . Am 1 Am(strict derivations)13Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexA chain of types in L and in pregroupsType-raising in LX Y def Y / (X \ Y ) with X X YA preliminary questionType-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 ]closureIterating and alternating exponentsA0 Am def ((Am 1 )y )z (x, y , z primitive)A0 A1 . . . Am 1 Am(strict derivations)[A0 ] [A1 ] . . . [Am 1 [ [Am ](strict)lllllA0 x zz yy x . . . zz yy x zz yy l x(strict) {z } {z }m 1m13Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexMethod overview for limit pointsGrammars of the form :Gn {a 7 A; b 7 B; c 7 Dn }InclusionChainsensures:Dn Dn 1 . . . .L(Gn ) L(Gn 1 ) . . . .14Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexMethod overview for limit pointsGrammars of the form :Gn {a 7 A; b 7 B; c 7 Dn }InclusionChainsensures:Dn Dn 1 . . . .L(Gn ) L(Gn 1 ) . . . .StrictnessSchema in L and NL : Aiteratedand alternatedDD / A {z} {z }DnwithDn 1(tautology ex : A p / p)(using B q / q)14Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexMethod overview for limit pointsGrammars of the form :Gn {a 7 A; b 7 B; c 7 Dn }InclusionChainsensures:Dn Dn 1 . . . .L(Gn ) L(Gn 1 ) . . . .StrictnessSchema in L and NL : Aiteratedand alternatedDD / A {z} {z }DnwithDn 1(tautology ex : A p / p)(using B q / q)otherwise . . . L(Gn ) L(Gn 1 ) . . .14Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorialgrammarsand learnabilityLogical Information Systems(LIS) CategorialgrammarsSystemGrammarsand typesLanguagesRefSubclasses and/as LIS AnnexL Lh·,/iNL Lh/,\ipregroupsfrom L or Lh·,/iNL(also NL L, L )pregroupsfrom NLGh1,ni {a 7 p / p ; b 7 q / q ; c 7 Dh1,ni }Gh1, i {a 7 p / p ; b 7 p / p ; c 7 S / (p / p)}Dh1,0i SwhereDh1,ni (Dh1,n 1i / (p / p)) / (q / q)c(b a )nc{a, b} [FN02a]/ onlyorder 2c(b a )nc{a, b} [FN02a]no \′G′h1,ni {a 7 p / p ; c 7 Dh1,ni}G′h1, i {a 7 (p / p) / (p / p) ; c 7 S / (p / p)} ′Dh1,0i Swhere′′Dh1,ni Dh1,n 1i/ (p / p){cak /0 k n}c{a} [FN02c]2/ onlyorder 2c(b a )n {c}[FN02c]no product(by )[For02]“order” 2Gh2,ni {a 7 p / p ; b 7 q / q ; c 7 Dh2,ni }Gh2, i {a 7 A ; b 7 A ; c 7 (S / A)·A}Dh2,0i S, A p / p, B q / qwhereDh2,ni (((Dh2,n 1i / A)·A) / B)·B′G′h3,ni {a 7 p\p ; b 7 q\q ; c 7 Dh3,ni}′G′h2,ni {a 7 p\p ; b 7 q\q ; c 7 Dh2,ni}G′h3, i {a 7 p\p ; b 7 p\p ; c 7 S / (p\p)}8′D S h2,0i′n 1Dh2,ni (S / p)·((p / q)·(q / p))·(p / q)·qwhere:′′Dh3,ni S / (Dh2,ni\S)llGPh1,ni {a 7 pp ; b 7 qq ; c 7 Cn }C0 SwhereCn (Cn 1 )pll pl q ll q lGPh1, i {a 7 ppl ; b 7 ppl ; c 7 Spll pl }89 a 7 A / B (q / (p\q)) / p b 7 DnGn : c 7 E \S;nG {a 7 p / p ; b 7 p ; c 7 p\S}8 A D0 E0 q / (p\q) B pwhere Dn 1 (A / B)\Dn:En 1 (A / A)\En98a 7 qq l ppl [A / B] rr b 7 pp qq qq l p [Dn ] GPn {z } n:;c 7 pr qq r S [E \S]{ak bc/0 k n}[BF03b]no productorder 5a bc{ak bc/0 k n}n[BF03b]“order” 1taln“order” 1/2a bc ffa 7 (pl )n q l ; b 7 qpq l ; c 7 qr ll lr ; e 7 rpn s d 7 rpffll lla 7q;b 7qpq;c 7qrGP ′ d 7 rpr l ; e 7 rsGPn′ .GP {a 7 ppl ; b 7 p ; c 7 pr S}pregroupsc{a, b} {abk cdk e/0 k n}{abk cdk e/k 0}15Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
IntroductionOn categorial grammarsGrammarsand learnability LogicalInformation Systems (LIS) Categorial grammars and/as LIS AnnexSystemand typesGh1,ni {a 7 p / p ; b 7 q / q ; c 7 Dh1,ni }Gh1, i {a 7 p / p ; b 7 p / p ; c 7 S / (p / p)}Dh1,0i SwhereDh1,ni (Dh1,n 1i / (p / p)) / (q / q)L Gh2,ni {a 7 p / p ; b 7 q / q ; c 7 Dh2,ni }Gh2, i {a 7 A ; b 7 A ; c 7 (S / A)·A}Dh2,0i S, A p / p, B q / qwhereDh2,ni (((Dh2,n 1i / A)·A) / B)·BLh·,/iNL Lh/,\i′G′h1,ni {a 7 p / p ; c 7 Dh1,ni}′Gh1, i {a 7 (p / p) / (p / p) ; c 7 S / (p / p)} ′Dh1,0i Swhere′′Dh1,ni Dh1,n 1i/ (p / p)′G′h3,ni {a 7 p\p ; b 7 q\q ; c 7 Dh3,ni}′′Gh2,ni {a 7 p\p ; b 7 q\q ; c 7 Dh2,ni}G′h3, i {a 7 p\p ; b 7 p\p ; c 7 S / (p\p)}8′ Dh2,0i S′Dh2,ni (S / p)·((p / q)·(q / p))n 1 ·(p / q)·qwhere:′′Dh3,ni S / (Dh2,ni\S)llGPh1,ni {a 7 pp ; b 7 qq ; c 7 Cn }C0 SwhereCn (Cn 1 )pll pl q ll q lGPh1, i {a 7 ppl ; b 7 ppl ; c 7 Spll pl }8 France9Annie Foret IRISA & Univ. Rennes,Categorial Inference and LISpregroupsfrom L or Lh·,/i15
CG Acquisition and hierarchiesHierarchy of k-valued categorial grammars.LAB (G ) . 2 valued3 valued.Question: for k valued Lambek languages ?GhAB,ki : k-valued ABk 1, 2. LNL (G ). . . . . ABFact: a strict hierarchy for AB languages LL (G ) . rigidLNL (G ) {Lambek languages} {AB categorial languages} {Context free languages} L (G )L LCDG?, ,. (G. ) L ([G ])PG.LPG?, ([G ]. ) .21 / 31(Buszkowski,Penn 1990, Kanazawa 1998)LACL 2005 – p.2Learnable from FA-structuresLearnable from stringsGhL,ki : k-valued Associative LambekLearnable from full derivation (Bonato,Retore 2001) . . . . . . . . . . .Not learnable from strings (Foret,Le Nir 2002)GhNL,ki : k-valued Non-Associative LambekLearnable from generalized FA-structures (Bechet,Foret 2003)Not Learnable from strings (bracketed) (” ”,” ” ACL03,CG04) “arity”-bounded: learnable from strings (Bechet,Foret 03)GhPG ,ki : k-valued PregroupsNot learnable from strings (2002,ACL03,ICGI04) algorithm from ”features” (Béchet,Foret,Tellier: SL 07) bounded width, learnable from strings (”: ICGI04, SL07)GhCDG ,.i : categorial dependency(Béchet, Foret, Dikovsky -2010)k-valued Not learnable from strings or FA-structures algorithms from dependency structures16
RG on FA-structuresABLearning algorithm example .detailsfrom D {afastfishmanswims},(i) General formGF (D)X1 / X2 , X3 / X4X5 \(X3 \S)X4X2X1 \S, X5(ii) UnificationσX1 X3 , X2 X4X5 X1 \S(iii) Rigid grammarRG (D)X1 / X2(X1 \S)\(X1 \S)X2X2X1 \S.
RG on FA-structuresABLearning algorithm example .detailsfrom D {afastfishmanswims},(i) General formGF (D)X1 / X2 , X3 / X4X5 \(X3 \S)X4X2X1 \S, X5.(ii) UnificationσX1 X3 , X2 X4X5 X1 \S(iii) Rigid grammarRG (D)X1 / X2(X1 \S)\(X1 \S)X2X2X1 \Spossibly incremental, kernel grammar, adaptations17
RG on FA-structuresABLearning algorithm example .detailsfrom D {afastfishmanswims},(i) General formGF (D)X1 / X2 , X3 / X4X5 \(X3 \S)X4X2X1 \S, X5(ii) UnificationσX1 X3 , X2 X4X5 X1 \Spossibly incremental, kernel grammar, adaptationsDi Dj. . . FL(G) RG (Di ) v RG (Dj ) . . . FL((.i )) FL((.j )).(iii) Rigid grammarRG (D)X1 / X2(X1 \S)\(X1 \S)X2X2X1 \SG is AB rigidwhere Gi v G is : σ : σ(Gi ) G17
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS Annexnon-associativeLambekA summary for NL learnabilityAdaptation of RG to NL :A /.xi0 x. i. .A.add the constraint xi xi0Classes of NL grammars from (structured) examples [TCS (Bechet, Foret)]Well bracketedRestriction \ StructurestringsGeneralized S06].k-valued andyesyesyest-arity bounded[CAP 03]corollary of [CAP 03]corollary of [CAP 03]k-valued andyes [TCS 06]yes [TCS 06]yes [TCS 06]FA-arity bounded18Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexTypes,dependenciesand analysis and parsesCDGiteratedtypes, dependenciesexampleexampleWith the lexicon John 7 Nran 7 [N \ S / A ](empty potentials)fast, yesterday 7 ADerivationranfast[N \ S / A ] A[N \ S / A ]Dependency structuresLryesterdayA JohnN[N \ S / A ]IrΩr[N \ S]LlSLlIlΩlH P1 [H\β]P2 [β]P1 P2C P1 [C \β]P2 [C \β]P1 P2[C \β]P [β]PLrIrΩr[β / H]P2 H P1 [β]P2 P1[β / C ]P2 C P1 [β / C ]P2 P1[β / C ]P [β]P23 / 1934Annie Foret IRISA & Univ. Rennes, FranceCategorial Inference and LIS
Introduction On categorial grammars and learnability Logical Information Systems (LIS) Categorial grammars and/as LIS AnnexTypes,dependenciesand analysis and parsesCDGiteratedtypes, dependenciesexampleexampleWith the lexi
1 IntroductionOn categorial grammars and learnabilityLogical Information Systems (LIS)Categorial grammars and/as LISAnnex 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. Ren
Categorial Grammar (CG) is a lexicalized formal grammar well known for its tied connection between syntax and semantics. ariantsV of it (Combinatory Categorial Grammar, CCG, and Categorial ypeT Logic, CTL) have been used to reach wide coverage grammars for Engli
context free grammars, have already been applied to music genera-tion [6]. Categorial grammars are another linguistic formalism that have proved useful to the study of music. 3 Categorial Grammars in Language In 1935, semanticist Kazimierz Ajdukiewicz introduced the concept of categorial gr
Categorial Grammar The term Categorial Grammar names a group of theories of natural language syntax and semantics in which the main responsibility for defining syntactic form is borne by the lexicon. (M. Steedman) Categorial Grammars (CGs) developed as an alternati
Categorial Grammars –Combinatory Categorial Grammar: Constraining surface realisation in OpenCCG . This Lecture The connection between language, sets and logic Semantic Parsing Combinatory Categorial Grammars (CCGs) How to q
This lead to head grammars (Pollard), proven to be weakly equivalent to tree adjoining grammars by Vijay Shankar. Extending the grammar with composition and lift operations was explored extensively in combinatory categorial
hybrid logic dependency structures What do the grammar and lexicon look like? Categorial Grammar Categorial grammars are lexicalised grammars a grammar is just a “dictionary” there are no language-specific grammar rules a grammar is a mapping from words to stru
From CFG to AB grammars Proposition 2 Every "-free Context-Free Grammar in Greibach normal form is strongly equivalent to an AB categorial grammar. Thus every "-free Context-Free grammar is weakly equivalent to an AB categorial grammar. PROOF:Let us consider the following AB grammar: IIts words are the terminals of the CF
ADVANCED ENGINEERING MATHEMATICS By ERWIN KREYSZIG 9TH EDITION This is Downloaded From www.mechanical.tk Visit www.mechanical.tk For More Solution Manuals Hand Books And Much Much More. INSTRUCTOR’S MANUAL FOR ADVANCED ENGINEERING MATHEMATICS imfm.qxd 9/15/05 12:06 PM Page i. imfm.qxd 9/15/05 12:06 PM Page ii. INSTRUCTOR’S MANUAL FOR ADVANCED ENGINEERING MATHEMATICS NINTH EDITION ERWIN .
