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Practical problems with Chomsky-Schtzenberger parsing for weighted multiple context-free grammars 1 Tobias Denkinger tobias.denkinger@tu-dresden.de Institute of Theoretical Computer Science Faculty of Computer Science Technische Universitt

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SLIDE 1

Practical problems with Chomsky-Schützenberger parsing for weighted multiple context-free grammars1

Tobias Denkinger

tobias.denkinger@tu-dresden.de

Institute of Theoretical Computer Science Faculty of Computer Science Technische Universität Dresden

WATA, Leipzig, 2018-05-23

1based on T. Denkinger (2017). “Chomsky-Schützenberger parsing for weighted

multiple context-free languages”.

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SLIDE 2

The problem: 𝑙-best parsing

𝑙-best parsing problem [Huang and Chiang 2005]

Input: a (𝒝, ⊙, 𝟚, 𝟙)-weighted grammar ( 𝐻 , wt) a suitable partial order ⊴ on (𝒝, ⊙, 𝟚, 𝟙) a number 𝑙 ∈ ℕ a word 𝑥 Output: a sequence of 𝑙 best derivation s2

  • f 𝑥 in 𝐻

(not unique)

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 2 / 13

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SLIDE 3

The problem: 𝑙-best parsing

𝑙-best parsing problem [Huang and Chiang 2005]

Input: a (𝒝, ⊙, 𝟚, 𝟙)-weighted grammar (𝐻, wt) a suitable partial order ⊴ on (𝒝, ⊙, 𝟚, 𝟙) a number 𝑙 ∈ ℕ a word 𝑥 Output: a sequence of 𝑙 best derivations2 of 𝑥 in 𝐻 (not unique)

2w.r.t. wt and ⊴ (greater is betuer)

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 2 / 13

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SLIDE 4

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [(𝑦, 𝑧) ↦ a𝑦b𝑧 ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟

𝛵∗×𝛵∗→𝛵∗

](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [((𝑦1, 𝑦2), (𝑧1, 𝑧2)) ↦ (a𝑦1𝑧2b, 𝑧1c𝑦2) ⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟

(𝛵∗×𝛵∗)×(𝛵∗×𝛵∗)→(𝛵∗×𝛵∗)

](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

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SLIDE 5

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [(𝑦, 𝑧) ↦ a𝑦b𝑧 ⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟

𝛵∗×𝛵∗→𝛵∗

](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [((𝑦1, 𝑦2), (𝑧1, 𝑧2)) ↦ (a𝑦1𝑧2b, 𝑧1c𝑦2) ⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟

(𝛵∗×𝛵∗)×(𝛵∗×𝛵∗)→(𝛵∗×𝛵∗)

](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

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SLIDE 6

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [ a 𝑦 b 𝑧 ](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [((𝑦1, 𝑦2), (𝑧1, 𝑧2)) ↦ (a𝑦1𝑧2b, 𝑧1c𝑦2) ⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟

(𝛵∗×𝛵∗)×(𝛵∗×𝛵∗)→(𝛵∗×𝛵∗)

](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

slide-7
SLIDE 7

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [ a 𝑦 b 𝑧 ](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [((𝑦1, 𝑦2), (𝑧1, 𝑧2)) ↦ (a𝑦1𝑧2b, 𝑧1c𝑦2) ⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟⏟ ⏟ ⏟ ⏟ ⏟ ⏟ ⏟⏟

(𝛵∗×𝛵∗)×(𝛵∗×𝛵∗)→(𝛵∗×𝛵∗)

](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

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SLIDE 8

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [ a 𝑦 b 𝑧 ](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [ a𝑦1𝑧2b, 𝑧1c𝑦2 ](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

slide-9
SLIDE 9

Multiple context-free grammars

context-free grammars

𝐵 → a𝐵b𝐶 composes strings 𝐵 → [ a 𝑦 b 𝑧 ](𝐵, 𝐶)

multiple context-free grammars [Seki, Matsumura, Fujii, and Kasami 1991]

𝐵 → [ a𝑦1𝑧2b, 𝑧1c𝑦2 ](𝐵, 𝐶) composes tuples of strings ⟹ extra expressive power useful for natural language processing

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 3 / 13

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SLIDE 10

The Chomsky-Schützenberger theorem

CS-theorems [Chomsky and Schützenberger 1963] [Yoshinaka, Kaji, and Seki 2010]

Let 𝑀 be a language. T.f.a.e.

  • 1. ∃

M CFG 𝐻 s.t. 𝑀 = L(𝐻)

  • 2. ∃ regular language 𝑆,

∃ multiple Dyck language 𝐸, ∃ homomorphism ℎ s.t. 𝑀 = ℎ(𝑆 ∩ 𝐸)

Idea [Hulden 2011, for CFGs]

Use the decomposition provided by (1. → 2.) for parsing.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 4 / 13

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SLIDE 11

The Chomsky-Schützenberger theorem

CS-theorems [Chomsky and Schützenberger 1963] [Yoshinaka, Kaji, and Seki 2010]

Let 𝑀 be a language. T.f.a.e.

  • 1. ∃ MCFG 𝐻 s.t. 𝑀 = L(𝐻)
  • 2. ∃ regular language 𝑆,

∃ multiple Dyck language 𝐸, ∃ homomorphism ℎ s.t. 𝑀 = ℎ(𝑆 ∩ 𝐸)

Idea [Hulden 2011, for CFGs]

Use the decomposition provided by (1. → 2.) for parsing.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 4 / 13

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SLIDE 12

The Chomsky-Schützenberger theorem

CS-theorems [Chomsky and Schützenberger 1963] [Yoshinaka, Kaji, and Seki 2010]

Let 𝑀 be a language. T.f.a.e.

  • 1. ∃ MCFG 𝐻 s.t. 𝑀 = L(𝐻)
  • 2. ∃ regular language 𝑆,

∃ multiple Dyck language 𝐸, ∃ homomorphism ℎ s.t. 𝑀 = ℎ(𝑆 ∩ 𝐸)

Idea [Hulden 2011, for CFGs]

Use the decomposition provided by (1. → 2.) for parsing.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 4 / 13

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SLIDE 13

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

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SLIDE 14

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-15
SLIDE 15

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-16
SLIDE 16

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-17
SLIDE 17

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-18
SLIDE 18

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-19
SLIDE 19

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-20
SLIDE 20

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-21
SLIDE 21

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ℎ−1(𝑥))

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-22
SLIDE 22

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥))

enumerate from a weighted finite-state automaton

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 5 / 13

slide-23
SLIDE 23

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

an MCFG 𝐻: 𝛽: 𝑇 → [𝑦1𝑦2](𝐵) 𝛾: 𝐵 → [a𝑦1b, c𝑦2](𝐵) 𝛿: 𝐵 → [𝜁, 𝜁]() L(𝐻) = {a𝑜b𝑜c𝑜 ∣ 𝑜 ∈ ℕ} word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-24
SLIDE 24

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

an MCFG 𝐻: 𝛽: 𝑇 → [𝑦1𝑦2](𝐵) 𝛾: 𝐵 → [a𝑦1b, c𝑦2](𝐵) 𝛿: 𝐵 → [𝜁, 𝜁]() L(𝐻) = {a𝑜b𝑜c𝑜 ∣ 𝑜 ∈ ℕ} word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-25
SLIDE 25

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

an MCFG 𝐻: 𝛽: 𝑇 → [𝑦1𝑦2](𝐵) 𝛾: 𝐵 → [a𝑦1b, c𝑦2](𝐵) 𝛿: 𝐵 → [𝜁, 𝜁]() L(𝐻) = {a𝑜b𝑜c𝑜 ∣ 𝑜 ∈ ℕ} word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-26
SLIDE 26

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-27
SLIDE 27

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-28
SLIDE 28

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-29
SLIDE 29

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-30
SLIDE 30

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-31
SLIDE 31

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = a abbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-32
SLIDE 32

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aa bbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-33
SLIDE 33

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aa bbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-34
SLIDE 34

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aa bbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-35
SLIDE 35

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aa bbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-36
SLIDE 36

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aab bcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-37
SLIDE 37

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-38
SLIDE 38

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 𝜁 a ̄ 𝐵1 𝜁 b 𝐵2 𝜁 c ̄ 𝐵2 𝜁 𝜁 ̄ 𝑇1 𝜁

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-39
SLIDE 39

Derivations and bracket words

𝛽: 𝑇 → [ 𝑦1 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛾: 𝐵 → [ a 𝑦1 𝑐 , 𝑑 𝑦2 ](𝐵) 𝛿: 𝐵 → [ 𝜁 , 𝜁 ]()

𝑇1 ̄ 𝑇1 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2 𝐵1 ̄ 𝐵1 𝐵2 ̄ 𝐵2

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1

[1

𝛾a[1 𝛾,1

̄ 𝐵1 [1

𝛿]1 𝛿

]1

𝛾,1b]1 𝛾

𝐵2 ]1

𝛽,1[1 𝛽,2

[2

𝛾c[2 𝛾,1

̄ 𝐵2 [2

𝛿]2 𝛿

]2

𝛾,1]2 𝛾

̄ 𝑇1 ]1

𝛽,2]1 𝛽

word 𝑥 = aabbcc 𝑥′ = aaabbcc ∉ L(𝐻) word 𝑣 = [1

𝛽[1 𝛽,1 [1 𝛾a[1 𝛾,1 [1 𝛾a[1 𝛾,1 [1 𝛿]1 𝛿 ]1 𝛾,1b]1 𝛾 ]1 𝛾,1b]1 𝛾 ]1 𝛽,1

[1

𝛽,2 [2 𝛾c[2 𝛾,1 [2 𝛾c[2 𝛾,1 [2 𝛿]2 𝛿 ]2 𝛾,1]2 𝛾 ]2 𝛾,1]2 𝛾 ]1 𝛽,2]1 𝛽

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 6 / 13

slide-40
SLIDE 40

From the CS-theorem to CS-parsing

𝑥 ∈ L(𝐻) ⟺ 𝑥 ∈ ℎ(𝑆 ∩ 𝐸) (CS-theorem) ⟺ ∃𝑣 ∈ 𝑆 ∩ 𝐸: ℎ(𝑣) = 𝑥 ⟺ ∃𝑣 ∈ 𝑆 ∩ ℎ−1(𝑥): 𝑣 ∈ 𝐸

Observation

Each 𝑣 ∈ 𝑆 ∩ 𝐸 encodes a derivation of 𝐻.

𝑙-best CS-parsing

parse𝐻,wt,𝑙(𝑥) = (take𝑙 ∘ sortwt

⊴ ∘ toDeriv ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ sortwt

⊴ ∘ filter∩𝐸)(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sortwt

⊴ )(𝑆 ∩ ℎ−1(𝑥))

= (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥))

enumerate from a weighted finite-state automaton

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 7 / 13

slide-41
SLIDE 41

Practical problems

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1

[1

𝛾a[1 𝛾,1

̄ 𝐵1 [1

𝛿]1 𝛿

]1

𝛾,1b]1 𝛾

𝐵2 ]1

𝛽,1[1 𝛽,2

[2

𝛾c[2 𝛾,1

̄ 𝐵2 [2

𝛿]2 𝛿

]2

𝛾,1]2 𝛾

̄ 𝑇1 ]1

𝛽,2]1 𝛽

enumerate 𝑆wt by ascending weight Dijkstra-like algorithm initial idea: atuach weights to [1

𝜏-brackets

problem: loops with weight 𝟚

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 8 / 13

slide-42
SLIDE 42

Practical problems

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1

[1

𝛾a[1 𝛾,1

̄ 𝐵1 [1

𝛿]1 𝛿

]1

𝛾,1b]1 𝛾

𝐵2 ]1

𝛽,1[1 𝛽,2

[2

𝛾c[2 𝛾,1

̄ 𝐵2 [2

𝛿]2 𝛿

]2

𝛾,1]2 𝛾

̄ 𝑇1 ]1

𝛽,2]1 𝛽

enumerate 𝑆wt by ascending weight Dijkstra-like algorithm initial idea: atuach weights to [1

𝜏-brackets

problem: loops with weight 𝟚

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 8 / 13

slide-43
SLIDE 43

Practical problems

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1

[1

𝛾a[1 𝛾,1

̄ 𝐵1 [1

𝛿]1 𝛿

]1

𝛾,1b]1 𝛾

𝐵2 ]1

𝛽,1[1 𝛽,2

[2

𝛾c[2 𝛾,1

̄ 𝐵2 [2

𝛿]2 𝛿

]2

𝛾,1]2 𝛾

̄ 𝑇1 ]1

𝛽,2]1 𝛽

enumerate 𝑆wt by ascending weight Dijkstra-like algorithm initial idea: atuach weights to [1

𝜏-brackets

problem: loops with weight 𝟚

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 8 / 13

slide-44
SLIDE 44

Practical problems

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/wt𝛽

[1

𝛾a[1 𝛾,1/wt𝛾

̄ 𝐵1 [1

𝛿]1 𝛿/wt𝛿

]1

𝛾,1b]1 𝛾/1

𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/1

̄ 𝐵2 [2

𝛿]2 𝛿/1

]2

𝛾,1]2 𝛾/1

̄ 𝑇1 ]1

𝛽,2]1 𝛽/1

enumerate 𝑆wt by ascending weight Dijkstra-like algorithm initial idea: atuach weights to [1

𝜏-brackets

problem: loops with weight 𝟚

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 8 / 13

slide-45
SLIDE 45

Practical problems

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/wt𝛽

[1

𝛾a[1 𝛾,1/wt𝛾

̄ 𝐵1 [1

𝛿]1 𝛿/wt𝛿

]1

𝛾,1b]1 𝛾/1

𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/1

̄ 𝐵2 [2

𝛿]2 𝛿/1

]2

𝛾,1]2 𝛾/1

̄ 𝑇1 ]1

𝛽,2]1 𝛽/1

enumerate 𝑆wt by ascending weight Dijkstra-like algorithm initial idea: atuach weights to [1

𝜏-brackets

problem: loops with weight 𝟚

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 8 / 13

slide-46
SLIDE 46

Solutions and workarounds I

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/wt𝛽

[1

𝛾a[1 𝛾,1/wt𝛾

̄ 𝐵1 [1

𝛿]1 𝛿/wt𝛿

]1

𝛾,1b]1 𝛾/1

𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/1

̄ 𝐵2 [2

𝛿]2 𝛿/1

]2

𝛾,1]2 𝛾/1

̄ 𝑇1 ]1

𝛽,2]1 𝛽/1

assume that wt𝜏 ≠ 𝟚 in loops assume that weights can be factorised distribute factors

  • f

wt𝜏 among transitions with [1

𝜏, ]1 𝜏, [2 𝜏, ]2 𝜏, …

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 9 / 13

slide-47
SLIDE 47

Solutions and workarounds I

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/wt𝛽

[1

𝛾a[1 𝛾,1/wt𝛾

̄ 𝐵1 [1

𝛿]1 𝛿/wt𝛿

]1

𝛾,1b]1 𝛾/1

𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/1

̄ 𝐵2 [2

𝛿]2 𝛿/1

]2

𝛾,1]2 𝛾/1

̄ 𝑇1 ]1

𝛽,2]1 𝛽/1

assume that wt𝜏 ≠ 𝟚 in loops assume that weights can be factorised distribute factors

  • f

wt𝜏 among transitions with [1

𝜏, ]1 𝜏, [2 𝜏, ]2 𝜏, …

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 9 / 13

slide-48
SLIDE 48

Solutions and workarounds I

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/wt𝛽

[1

𝛾a[1 𝛾,1/wt𝛾

̄ 𝐵1 [1

𝛿]1 𝛿/wt𝛿

]1

𝛾,1b]1 𝛾/1

𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/1

̄ 𝐵2 [2

𝛿]2 𝛿/1

]2

𝛾,1]2 𝛾/1

̄ 𝑇1 ]1

𝛽,2]1 𝛽/1

assume that wt𝜏 ≠ 𝟚 in loops assume that weights can be factorised distribute factors

  • f

wt𝜏 among transitions with [1

𝜏, ]1 𝜏, [2 𝜏, ]2 𝜏, …

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 9 / 13

slide-49
SLIDE 49

Solutions and workarounds I

… with the weighted finite state automaton 𝑆wt

𝑇1 start 𝐵1 [1

𝛽[1 𝛽,1/ 2

√wt𝛽 [1

𝛾a[1 𝛾,1/ 4

√wt𝛾 ̄ 𝐵1 [1

𝛿]1 𝛿/ 2

√wt𝛿 ]1

𝛾,1b]1 𝛾/ 4

√wt𝛾 𝐵2 ]1

𝛽,1[1 𝛽,2/1

[2

𝛾c[2 𝛾,1/ 4

√wt𝛾 ̄ 𝐵2 [2

𝛿]2 𝛿/ 2

√wt𝛿 ]2

𝛾,1]2 𝛾/ 4

√wt𝛾 ̄ 𝑇1 ]1

𝛽,2]1 𝛽/ 2

√wt𝛽

assume that wt𝜏 ≠ 𝟚 in loops assume that weights can be factorised distribute factors

  • f

wt𝜏 among transitions with [1

𝜏, ]1 𝜏, [2 𝜏, ]2 𝜏, …

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 9 / 13

slide-50
SLIDE 50

Solutions and workarounds II

Assumption 1: weights wt𝜏 that occur in loops are ≠ 𝟚 ⟹ restrict the grammar

restricted weighted MCFGs:

may not have derivations of the form

𝛽1 𝛽2 ⋮ 𝛽𝑙 𝛽1 ⋮

with wt𝛽1 = … = wt𝛽𝑙 = 𝟚 useful probabilistic MCFGs are al- ways restricted 𝔺-weighted MCFGs can be trans- formed to restricted ℕ-weighted MCFGs with the same support

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 10 / 13

slide-51
SLIDE 51

Solutions and workarounds II

Assumption 1: weights wt𝜏 that occur in loops are ≠ 𝟚 ⟹ restrict the grammar

restricted weighted MCFGs:

may not have derivations of the form

𝛽1 𝛽2 ⋮ 𝛽𝑙 𝛽1 ⋮

with wt𝛽1 = … = wt𝛽𝑙 = 𝟚 useful probabilistic MCFGs are al- ways restricted 𝔺-weighted MCFGs can be trans- formed to restricted ℕ-weighted MCFGs with the same support

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 10 / 13

slide-52
SLIDE 52

Solutions and workarounds II

Assumption 1: weights wt𝜏 that occur in loops are ≠ 𝟚 ⟹ restrict the grammar

restricted weighted MCFGs:

may not have derivations of the form

𝛽1 𝛽2 ⋮ 𝛽𝑙 𝛽1 ⋮

with wt𝛽1 = … = wt𝛽𝑙 = 𝟚 useful probabilistic MCFGs are al- ways restricted 𝔺-weighted MCFGs can be trans- formed to restricted ℕ-weighted MCFGs with the same support

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 10 / 13

slide-53
SLIDE 53

Solutions and workarounds II

Assumption 1: weights wt𝜏 that occur in loops are ≠ 𝟚 ⟹ restrict the grammar

restricted weighted MCFGs:

may not have derivations of the form

𝛽1 𝛽2 ⋮ 𝛽𝑙 𝛽1 ⋮

with wt𝛽1 = … = wt𝛽𝑙 = 𝟚 useful probabilistic MCFGs are al- ways restricted 𝔺-weighted MCFGs can be trans- formed to restricted ℕ-weighted MCFGs with the same support

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 10 / 13

slide-54
SLIDE 54

Solutions and workarounds II

Assumption 1: weights wt𝜏 that occur in loops are ≠ 𝟚 ⟹ restrict the grammar

restricted weighted MCFGs:

may not have derivations of the form

𝛽1 𝛽2 ⋮ 𝛽𝑙 𝛽1 ⋮

with wt𝛽1 = … = wt𝛽𝑙 = 𝟚 useful probabilistic MCFGs are al- ways restricted 𝔺-weighted MCFGs can be trans- formed to restricted ℕ-weighted MCFGs with the same support

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 10 / 13

slide-55
SLIDE 55

Solutions and workarounds III

Assumption 2: weights can be factorised ⟹ restrict weight algebra

factorisable (multiplicative) monoid with zero (𝒝, ⊙, 𝟚, 𝟙)

∀𝑏 ∈ 𝒝 ∖ {𝟙, 𝟚}: ∃𝑏1, 𝑏2 ∈ 𝒝 ∖ {𝟚}: 𝑏1 ⊙ 𝑏2 = 𝑏 two examples from nlp: (𝒝, ⊙, 𝟚, 𝟙) factorisation ([0, 1], ⋅, 1, 0) 𝑏 =

2

√𝑏 ⋅

2

√𝑏 (ℝ−∞

≥0 , +, 0, −∞)

𝑏 = 𝑏/2 + 𝑏/2

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 11 / 13

slide-56
SLIDE 56

Solutions and workarounds III

Assumption 2: weights can be factorised ⟹ restrict weight algebra

factorisable (multiplicative) monoid with zero (𝒝, ⊙, 𝟚, 𝟙)

∀𝑏 ∈ 𝒝 ∖ {𝟙, 𝟚}: ∃𝑏1, 𝑏2 ∈ 𝒝 ∖ {𝟚}: 𝑏1 ⊙ 𝑏2 = 𝑏 two examples from nlp: (𝒝, ⊙, 𝟚, 𝟙) factorisation ([0, 1], ⋅, 1, 0) 𝑏 =

2

√𝑏 ⋅

2

√𝑏 (ℝ−∞

≥0 , +, 0, −∞)

𝑏 = 𝑏/2 + 𝑏/2

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 11 / 13

slide-57
SLIDE 57

Solutions and workarounds III

Assumption 2: weights can be factorised ⟹ restrict weight algebra

factorisable (multiplicative) monoid with zero (𝒝, ⊙, 𝟚, 𝟙)

∀𝑏 ∈ 𝒝 ∖ {𝟙, 𝟚}: ∃𝑏1, 𝑏2 ∈ 𝒝 ∖ {𝟚}: 𝑏1 ⊙ 𝑏2 = 𝑏 two examples from nlp: (𝒝, ⊙, 𝟚, 𝟙) factorisation ([0, 1], ⋅, 1, 0) 𝑏 =

2

√𝑏 ⋅

2

√𝑏 (ℝ−∞

≥0 , +, 0, −∞)

𝑏 = 𝑏/2 + 𝑏/2

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 11 / 13

slide-58
SLIDE 58

Solutions and workarounds III

Assumption 2: weights can be factorised ⟹ restrict weight algebra

factorisable (multiplicative) monoid with zero (𝒝, ⊙, 𝟚, 𝟙)

∀𝑏 ∈ 𝒝 ∖ {𝟙, 𝟚}: ∃𝑏1, 𝑏2 ∈ 𝒝 ∖ {𝟚}: 𝑏1 ⊙ 𝑏2 = 𝑏 two examples from nlp: (𝒝, ⊙, 𝟚, 𝟙) factorisation ([0, 1], ⋅, 1, 0) 𝑏 =

2

√𝑏 ⋅

2

√𝑏 (ℝ−∞

≥0 , +, 0, −∞)

𝑏 = 𝑏/2 + 𝑏/2

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 11 / 13

slide-59
SLIDE 59

Conclusion and outlook

Theorem (𝑙-best parsing)

Let (𝐻, wt) be a restricted weighted MCFG over a factorisable monoid with zero and ⊴ be a suitable partial order on the monoid. Then (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥)) solves the 𝑙-best parsing problem for (𝐻, wt) and a word 𝑥.

Conjecture

The restrictions are not problematic in practice. refined and implemented by T. Ruprecht (in his master thesis) he currently investigates practical viability Thank you for your atuention.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 12 / 13

slide-60
SLIDE 60

Conclusion and outlook

Theorem (𝑙-best parsing)

Let (𝐻, wt) be a restricted weighted MCFG over a factorisable monoid with zero and ⊴ be a suitable partial order on the monoid. Then (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥)) solves the 𝑙-best parsing problem for (𝐻, wt) and a word 𝑥.

Conjecture

The restrictions are not problematic in practice. refined and implemented by T. Ruprecht (in his master thesis) he currently investigates practical viability Thank you for your atuention.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 12 / 13

slide-61
SLIDE 61

Conclusion and outlook

Theorem (𝑙-best parsing)

Let (𝐻, wt) be a restricted weighted MCFG over a factorisable monoid with zero and ⊴ be a suitable partial order on the monoid. Then (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥)) solves the 𝑙-best parsing problem for (𝐻, wt) and a word 𝑥.

Conjecture

The restrictions are not problematic in practice. refined and implemented by T. Ruprecht (in his master thesis) he currently investigates practical viability Thank you for your atuention.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 12 / 13

slide-62
SLIDE 62

Conclusion and outlook

Theorem (𝑙-best parsing)

Let (𝐻, wt) be a restricted weighted MCFG over a factorisable monoid with zero and ⊴ be a suitable partial order on the monoid. Then (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥)) solves the 𝑙-best parsing problem for (𝐻, wt) and a word 𝑥.

Conjecture

The restrictions are not problematic in practice. refined and implemented by T. Ruprecht (in his master thesis) he currently investigates practical viability Thank you for your atuention.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 12 / 13

slide-63
SLIDE 63

Conclusion and outlook

Theorem (𝑙-best parsing)

Let (𝐻, wt) be a restricted weighted MCFG over a factorisable monoid with zero and ⊴ be a suitable partial order on the monoid. Then (toDeriv ∘ take𝑙 ∘ filter∩𝐸 ∘ sort⊴)(𝑆wt ⊲ ℎ−1(𝑥)) solves the 𝑙-best parsing problem for (𝐻, wt) and a word 𝑥.

Conjecture

The restrictions are not problematic in practice. refined and implemented by T. Ruprecht (in his master thesis) he currently investigates practical viability Thank you for your atuention.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 12 / 13

slide-64
SLIDE 64

References

Chomsky, N. and M. P. Schützenberger (1963). “The algebraic theory of context-free languages”. doi: 10.1016/S0049-237X(09)70104-1. Denkinger, T. (2017). “Chomsky-Schützenberger parsing for weighted multiple context-free languages”. doi: 10.15398/jlm.v5i1.159. Huang, L. and D. Chiang (2005). “Betuer k-best Parsing”. Hulden, M. (2011). “Parsing CFGs and PCFGs with a Chomsky-Schützenberger Representation”. doi: 10.1007/978-3-642-20095-3_14. Seki, H., T. Matsumura, M. Fujii, and T. Kasami (1991). “On multiple context-free grammars”. doi: 10.1016/0304-3975(91)90374-B. Yoshinaka, R., Y. Kaji, and H. Seki (2010). “Chomsky-Schützenberger-type characterization of multiple context-free languages”. doi: 10.1007/978-3-642-13089-2_50.

  • T. Denkinger: Practical problems with CS-parsing for wMCFGs

WATA, Leipzig, 2018-05-23 13 / 13