Multi Context-Free Tree Grammars and Multi-component Tree Adjoining - PowerPoint PPT Presentation

Multi Context-Free Tree Grammars and Multi-component Tree Adjoining Grammars Joost Engelfriet 1 Andreas Maletti 2 1 LIACS, , Leiden, The Netherlands 2 Institute of Computer Science, , Leipzig, Germany maletti@informatik.uni-leipzig.de Altenberg, Germany September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 1

Motivation Definition Context-free grammar ( N , Σ , S , R ) is in Greibach normal form if each rule ρ ∈ R \ { S → ε } is of the form ρ = A → σ A 1 · · · A n with σ ∈ Σ and A , A 1 , . . . , A n ∈ N September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 2

Motivation Definition Context-free grammar ( N , Σ , S , R ) is in Greibach normal form if each rule ρ ∈ R \ { S → ε } is of the form ρ = A → σ A 1 · · · A n with σ ∈ Σ and A , A 1 , . . . , A n ∈ N Theorem [Greibach 1965] Every CFG can be turned into an equivalent CFG in Greibach normal form September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 2

Motivation Definition CFG ( N , Σ , S , R ) is lexicalized if occ Σ ( r ) � = ∅ for each rule ( A → r ) ∈ R \ { S → ε } September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 3

Motivation Definition CFG ( N , Σ , S , R ) is lexicalized if occ Σ ( r ) � = ∅ for each rule ( A → r ) ∈ R \ { S → ε } CFG in Greibach normal form is lexicalized September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 3

Motivation Definition CFG ( N , Σ , S , R ) is lexicalized if occ Σ ( r ) � = ∅ for each rule ( A → r ) ∈ R \ { S → ε } CFG in Greibach normal form is lexicalized lexicographers (linguists) love lexicalized grammars occurrence of lexical element in a rule is called anchor September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 3

Motivation S NP VP PRP MD VP We must VB PP NP bear IN NP NP PP in NN DT NN IN NP as Community DT NN mind the a whole linguists nowadays care more about the parse tree than the membership of its yield in the (string) language modern grammar formalisms generate tree and string languages September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 4

Motivation Definition For two tree grammars G and G ′ , of which G ′ is lexicalized, G ′ weakly lexicalizes G if yield( L ( G ′ )) = yield( L ( G )) G ′ strongly lexicalizes G if L ( G ′ ) = L ( G ) September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 5

Motivation Definition For two tree grammars G and G ′ , of which G ′ is lexicalized, G ′ weakly lexicalizes G if yield( L ( G ′ )) = yield( L ( G )) G ′ strongly lexicalizes G if L ( G ′ ) = L ( G ) tree language preserved under strong lexicalization string language preserved under weak lexicalization September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 5

Motivation Definition For two tree grammars G and G ′ , of which G ′ is lexicalized, G ′ weakly lexicalizes G if yield( L ( G ′ )) = yield( L ( G )) G ′ strongly lexicalizes G if L ( G ′ ) = L ( G ) tree language preserved under strong lexicalization string language preserved under weak lexicalization lifed to classes C and C ′ as usual C ′ -grammars strongly lexicalize C -grammars if for every G ∈ C there exists a lexicalized G ′ ∈ C ′ such that L ( G ′ ) = L ( G ) September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 5

Motivation Some results: CFGs (local tree grammars) weakly lexicalize themselves [Greibach 1965] Tree adjoining grammars (TAGs) strongly lexicalize CFGs [Joshi, Schabes 1997] September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 6

Motivation Some results: CFGs (local tree grammars) weakly lexicalize themselves [Greibach 1965] Tree adjoining grammars (TAGs) strongly lexicalize CFGs [Joshi, Schabes 1997] TAGs strongly lexicalize themselves [Joshi, Schabes 1997] September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 6

Motivation Some results: CFGs (local tree grammars) weakly lexicalize themselves [Greibach 1965] Tree adjoining grammars (TAGs) strongly lexicalize CFGs [Joshi, Schabes 1997] TAGs strongly lexicalize themselves [Joshi, Schabes 1997] TAGs do not strongly lexicalize themselves [Kuhlmann, Satta 2012] September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 6

Motivation Some results: CFGs (local tree grammars) weakly lexicalize themselves [Greibach 1965] Tree adjoining grammars (TAGs) strongly lexicalize CFGs [Joshi, Schabes 1997] TAGs strongly lexicalize themselves [Joshi, Schabes 1997] TAGs do not strongly lexicalize themselves [Kuhlmann, Satta 2012] Context-free tree grammars (CFTGs) strongly lexicalize TAGs and themselves [Maletti, Engelfriet 2013] September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 6

Contents Motivation 1 Main notion 2 Lexicalization 3 Expressive Power 4 September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 7

Main notion Definition [Engelfriet, Maneth 1998; Kanazawa 2010] Multiple context-free tree grammar (MCFTG) G = ( N , B , Σ , S , R ) finite totally ordered ranked alphabet N (nonterminals) partition B ⊆ P ( N ) of N (big nonterminals) finite ranked alphabet Σ (terminals) S ∈ N ( 0 ) with { S } ∈ B (initial big nonterminal) finite set R of rules of the form A → r with A ∈ B and N -linear forest r ∈ C N ∪ Σ ( X ) + such that rk + ( r ) = rk + ( A ) and B saturates occ N ( r ) September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 8

Main notion Definition [Engelfriet, Maneth 1998; Kanazawa 2010] Multiple context-free tree grammar (MCFTG) G = ( N , B , Σ , S , R ) finite totally ordered ranked alphabet N (nonterminals) partition B ⊆ P ( N ) of N (big nonterminals) finite ranked alphabet Σ (terminals) S ∈ N ( 0 ) with { S } ∈ B (initial big nonterminal) finite set R of rules of the form A → r with A ∈ B and N -linear forest r ∈ C N ∪ Σ ( X ) + such that rk + ( r ) = rk + ( A ) and B saturates occ N ( r ) MCFTGs generalize (linear, nondeleting) CFTGs to multiple components multiple components synchronously applied to “synchronized” nonterminal occurrences September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 8

Main notion Nonterminals S , A , C , C ′ , T 1 , T 2 , T 3 : T 1 γ σ σ σ σ ν C C ′ T 1 T 1 A → → → T 3 → C C ′ C C ′ T 2 T 3 x 1 T 2 α T 3 C x 1 x 1 τ x 1 x 1 A A x 1 T 2 γ T 1 C C ′ S → → x 1 → x 1 T 3 → x 1 T 2 α β x 1 x 1 x 1 A (nonterminals that constitute a big nonterminal connected by splines) September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 9

Main notion γ σ ν T 1 T 1 T 3 → T 2 T 2 α T 3 x 1 τ x 1 nonterminals T 1 , T 2 , T 3 with T 1 < T 2 < T 3 , terminals { γ, τ, σ, α, ν } big nonterminal in lhs and rhs: { T 1 , T 2 , T 3 } of ranks 1 , 0 , 0 3 corresponding rhs contexts with 1 , 0 , 0 variables September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 10

Main notion T 1 γ σ σ σ σ ν T 1 C C ′ T 1 A → → C C ′ → C C ′ T 2 T 3 → T 3 x 1 x 1 T 2 α T 3 C x 1 τ x 1 x 1 A A x 1 T 2 γ T 1 C C ′ → x 1 → x 1 T 3 → x 1 S → T 2 α β x 1 x 1 x 1 A Derivation: γ γ T 1 T 1 T 1 T 1 τ σ σ τ σ σ σ T 3 ⇒ A ⇒ ⇒ σ T 3 ⇒ σ ⇒ σ ν T 3 C σ ν C C ′ T 2 C ′ σ β C ′ T 2 σ C ′ T 3 T 2 A A α α A T 2 α A September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 11

Main notion T 1 γ σ σ σ σ ν T 1 C C ′ T 1 A → → C C ′ → C C ′ T 2 T 3 → T 3 x 1 x 1 T 2 α T 3 C x 1 τ x 1 x 1 A A x 1 T 2 γ T 1 C C ′ → x 1 → x 1 T 3 → x 1 S → T 2 α β x 1 x 1 x 1 A Derivation: γ γ T 1 T 1 T 1 T 1 τ σ σ τ σ σ σ T 3 ⇒ A ⇒ ⇒ σ T 3 ⇒ σ ⇒ σ ν T 3 C σ ν C C ′ T 2 C ′ σ β C ′ T 2 σ C ′ T 3 T 2 A A α α A T 2 α A September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 11

Main notion T 1 γ σ σ σ σ ν T 1 C C ′ T 1 A → → C C ′ → C C ′ T 2 T 3 → T 3 x 1 x 1 T 2 α T 3 C x 1 τ x 1 x 1 A A x 1 T 2 γ T 1 C C ′ → x 1 → x 1 T 3 → x 1 S → T 2 α β x 1 x 1 x 1 A Derivation: γ γ T 1 T 1 T 1 T 1 τ σ σ τ σ σ σ T 3 ⇒ A ⇒ ⇒ σ T 3 ⇒ σ ⇒ σ ν T 3 C σ ν C C ′ T 2 C ′ σ β C ′ T 2 σ C ′ T 3 T 2 A A α α A T 2 α A September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 11

Main notion T 1 γ σ σ σ σ ν T 1 C C ′ T 1 A → → C C ′ → C C ′ T 2 T 3 → T 3 x 1 x 1 T 2 α T 3 C x 1 τ x 1 x 1 A A x 1 T 2 γ T 1 C C ′ → x 1 → x 1 T 3 → x 1 S → T 2 α β x 1 x 1 x 1 A Derivation: γ γ T 1 T 1 T 1 T 1 τ σ σ τ σ σ σ T 3 ⇒ A ⇒ ⇒ σ T 3 ⇒ σ ⇒ σ ν T 3 C σ ν C C ′ T 2 C ′ σ β C ′ T 2 σ C ′ T 3 T 2 A A α α A T 2 α A September 21, 2017 MCFTG and MC-TAG J. Engelfriet, A. Maletti · 11