Lambek Calculus Extended with Subexponential and Bracket Modalities - - PowerPoint PPT Presentation

Lambek Calculus Extended with Subexponential and Bracket Modalities

Max Kanovich, Stepan Kuznetsov, Andre Scedrov

Basic Categorial Grammar

John loves Mary

Basic Categorial Grammar

John loves Mary np (np \ s) / np np

Basic Categorial Grammar

John loves Mary np (np \ s) / np np → s

Basic Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s

Basic Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s Non-commutativity: ⊢ np, np \ s → np (“John runs”), but ⊢ np \ s, np → s) (“runs John”).

Basic Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s Non-commutativity: ⊢ np, np \ s → np (“John runs”), but ⊢ np \ s, np → s) (“runs John”). Reduction rules of BCG: A, A \ B → B; B / A, A → B

Basic Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s Non-commutativity: ⊢ np, np \ s → np (“John runs”), but ⊢ np \ s, np → s) (“runs John”). Reduction rules of BCG: A, A \ B → B; B / A, A → B [Ajdukiewicz 1935, Bar-Hillel et al. 1960]

Extending Categorial Grammar

John loves Mary

Extending Categorial Grammar

John loves Mary np (np \ s) / np np

Extending Categorial Grammar

John loves Mary np (np \ s) / np np → s

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves np / n n (n \ n) /(s / np) np (np \ s) / np

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves np / n n (n \ n) /(s / np) np (np \ s) / np

• → s / np

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)).

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)). the boy who loves Mary

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)). the boy who loves Mary np / n n (n \ n) /(np \ s) (np \ s) / np np

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whom John loves ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)). the boy who loves Mary np / n n (n \ n) /(np \ s) (np \ s) / np np

• → np \ s

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whomi John loves ei ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)). the boy who loves Mary ⊢ np / n n (n \ n) /(np \ s) (np \ s) / np np → np

• → np \ s

Extending Categorial Grammar

John loves Mary ⊢ np (np \ s) / np np → s the girl whomi John loves ei ⊢ np / n n (n \ n) /(s / np) np (np \ s) / np → np

• → s / np

Deriving principles like np, (np \ s) / np → s / np requires extra rules (in this particular case, associativity: (A \ B) / C ↔ A \(B / C)). the boy whoi ei loves Mary ⊢ np / n n (n \ n) /(np \ s) (np \ s) / np np → np

• → np \ s

Extending Categorial Grammar (cont.)

Another example (from Italian, see [Moot and Retor´ e 2012]): “She/He watches the train passing” Guarda passare il treno

She/He watches pass the train

Extending Categorial Grammar (cont.)

Another example (from Italian, see [Moot and Retor´ e 2012]): “She/He watches the train passing” Guarda passare il treno

She/He watches pass the train

⊢ s / inf inf / np np / n n → s

Extending Categorial Grammar (cont.)

Another example (from Italian, see [Moot and Retor´ e 2012]): “She/He watches the train passing” Guarda passare il treno

She/He watches pass the train

⊢ s / inf inf / np np / n n → s Transform into a question: “What does she/he watch passing?” Cosa guarda passare ? → s Here we need transitivity: A / B, B / C → A / C.

Extending Categorial Grammar (cont.)

Another example (from Italian, see [Moot and Retor´ e 2012]): “She/He watches the train passing” Guarda passare il treno

She/He watches pass the train

⊢ s / inf inf / np np / n n → s Transform into a question: “What does she/he watch passing?” Cosa guarda passare ? ⊢ q /(s / np) s / inf inf / np → s Here we need transitivity: A / B, B / C → A / C.

Extending Categorial Grammar (cont.)

Another example (from Italian, see [Moot and Retor´ e 2012]): “She/He watches the train passing” Guarda passare il treno

She/He watches pass the train

⊢ s / inf inf / np np / n n → s Transform into a question: “What does she/he watch passing?” Cosa guarda passare ? ⊢ q /(s / np) s / inf inf / np → s

• → s / np

Here we need transitivity: A / B, B / C → A / C.

Extending Categorial Grammar: Two Approaches

• 1. Add necessary principles as extra axioms to BCG

Combinatory Categorial Grammar (CCG) [Steedman 1996]

Extending Categorial Grammar: Two Approaches

• 1. Add necessary principles as extra axioms to BCG

Combinatory Categorial Grammar (CCG) [Steedman 1996]

• 2. One calculus to derive them all! Lambek Grammar

[Lambek 1958]

The Lambek Calculus (L

∗)

A → A Π → A ∆1, B, ∆2 → C ∆1, B / A, Π, ∆2 → C (/ →) Π, A → B Π → B / A (→ /) Π → A ∆1, B, ∆2 → C ∆1, Π, A \ B, ∆2 → C (\ →) A, Π → B Π → A \ B (→ \) [Lambek 1958, 1961, ...]

The Lambek Calculus (L

∗)

A → A Π → A ∆1, B, ∆2 → C ∆1, B / A, Π, ∆2 → C (/ →) Π, A → B Π → B / A (→ /) Π → A ∆1, B, ∆2 → C ∆1, Π, A \ B, ∆2 → C (\ →) A, Π → B Π → A \ B (→ \) [Lambek 1958, 1961, ...] L

∗ ⊢ (A \ B) / C ↔ A \(B / C)

L

∗ ⊢ A / B, B / C → A / C

. . .

Properties of the Lambek Calculus

◮ Lambek grammars generate precisely context-free languages

[Pentus 1993]. his means that formally their expressive power is not greater than the power of BCGs.

Properties of the Lambek Calculus

◮ Lambek grammars generate precisely context-free languages

[Pentus 1993]. This means that formally their expressive power is not greater than the power of BCGs.

Properties of the Lambek Calculus

◮ Lambek grammars generate precisely context-free languages

[Pentus 1993]. This means that formally their expressive power is not greater than the power of BCGs.

◮ The Lambek calculus is NP-complete [Pentus 2006, Savateev

2008]. (Steedman’s CCGs enjoy polynomial-time parsing.)

Properties of the Lambek Calculus

◮ Lambek grammars generate precisely context-free languages

[Pentus 1993]. This means that formally their expressive power is not greater than the power of BCGs.

◮ The Lambek calculus is NP-complete [Pentus 2006, Savateev

2008]. (Steedman’s CCGs enjoy polynomial-time parsing.)

◮ Polynomial-time algorithm for fragments of bounded depth

[Pentus 2010]. (Running time O(2dn4), where n is the length of the sequent and d is the implication nesting depth.)

Unwanted Derivations

book which John laughed without reading

Unwanted Derivations

book which John laughed without reading CN (CN \ CN) /(S / N)

• S / N

Unwanted Derivations

book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N

Unwanted Derivations

book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N

Unwanted Derivations

* book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N

Unwanted Derivations

* book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N * girl who John likes Mary and Pete likes

Unwanted Derivations

* book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N * girl who John likes Mary and Pete likes ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N

Unwanted Derivations

* book which John laughed without reading ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N * girl who John likes Mary and Pete likes ⊢ CN (CN \ CN) /(S / N)

• → CN

S / N

(cf. “John likes Mary and Pete likes Kate” → S; “and” is of type S \ S / S)

The Lambek Calculus with Brackets

[Morrill 1992, Moortgat 1995] A → A Π → A ∆(B) → C ∆(Π, A \ B) → C A, Π → B Π → A \ B Γ(A, B) → C Γ(A · B) → C Π → A ∆(B) → C ∆(B / A, Π) → C Π, A → B Π → B / A Γ → A ∆ → B Γ, ∆ → A · B ∆([A]) → C ∆(A) → C Π → A [Π] → A ∆(A) → C ∆([[]−1A]) → C [Π] → A Π → []−1A

The Lambek Calculus with Brackets

[Morrill 1992, Moortgat 1995] A → A Π → A ∆(B) → C ∆(Π, A \ B) → C A, Π → B Π → A \ B Γ(A, B) → C Γ(A · B) → C Π → A ∆(B) → C ∆(B / A, Π) → C Π, A → B Π → B / A Γ → A ∆ → B Γ, ∆ → A · B ∆([A]) → C ∆(A) → C Π → A [Π] → A ∆(A) → C ∆([[]−1A]) → C [Π] → A Π → []−1A

◮ Brackets introduce controlled non-associativity.

The Lambek Calculus with Brackets

[Morrill 1992, Moortgat 1995] A → A Π → A ∆(B) → C ∆(Π, A \ B) → C A, Π → B Π → A \ B Γ(A, B) → C Γ(A · B) → C Π → A ∆(B) → C ∆(B / A, Π) → C Π, A → B Π → B / A Γ → A ∆ → B Γ, ∆ → A · B ∆([A]) → C ∆(A) → C Π → A [Π] → A ∆(A) → C ∆([[]−1A]) → C [Π] → A Π → []−1A

◮ Brackets introduce controlled non-associativity. ◮ Cut elimination proved by Moortgat [1996].

Islands: Blocking Unwanted Derivations Using Brackets

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed without reading

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading]

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

This sequent is not derivable.

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

This sequent is not derivable.

◮ girl who John likes Mary and Pete likes

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

This sequent is not derivable.

◮ girl who [John likes Mary and Pete likes]

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

This sequent is not derivable.

◮ girl who [John likes Mary and Pete likes] CN, (CN \ CN) /(S / CN), [N, (N \ S) / N, N, (S \[]−1S) / S, N, (N \ S) / N] → CN

Islands: Blocking Unwanted Derivations Using Brackets

◮ book which John laughed [without reading] CN, (CN \ CN) /(S / CN), N, N \ S, [[]−1((N \ S) \(N \ S)) /(N \ S), (N \ S) / N] → CN

This sequent is not derivable.

◮ girl who [John likes Mary and Pete likes] CN, (CN \ CN) /(S / CN), [N, (N \ S) / N, N, (S \[]−1S) / S, N, (N \ S) / N] → CN

Neither is this one.

Subexponential: Medial Extraction

the girl whom John met yesterday

Subexponential: Medial Extraction

the girl whomi John met ei yesterday

Subexponential: Medial Extraction

the girl whomi John met ei yesterday

Subexponential: Medial Extraction

the girl whomi John met ei yesterday

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →)

Subexponential: Medial Extraction

the girl whomi John met ei yesterday

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) N, (N \ S) / N, N, (N \ S) \(N \ S) → S N, (N \ S) / N, !N, (N \ S) \(N \ S) → S (! →) N, (N \ S) / N, (N \ S) \(N \ S), !N → S (perm1) N, (N \ S) / N, (N \ S) \(N \ S) → S / !N (→ /)

Subexponential: Medial Extraction

the girl whomi John met ei yesterday (CN \ CN) /(S / !N)

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) N, (N \ S) / N, N, (N \ S) \(N \ S) → S N, (N \ S) / N, !N, (N \ S) \(N \ S) → S (! →) N, (N \ S) / N, (N \ S) \(N \ S), !N → S (perm1) N, (N \ S) / N, (N \ S) \(N \ S) → S / !N (→ /)

Subexponential: Medial Extraction

the girl whomi John met ei yesterday . . . (CN \ CN) /(S / !N)

• → N

→ S / !N ∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) N, (N \ S) / N, N, (N \ S) \(N \ S) → S N, (N \ S) / N, !N, (N \ S) \(N \ S) → S (! →) N, (N \ S) / N, (N \ S) \(N \ S), !N → S (perm1) N, (N \ S) / N, (N \ S) \(N \ S) → S / !N (→ /)

Subexponential: Parasitic Extraction

the paper that John signed without reading

Subexponential: Parasitic Extraction

the paper thati John signed ei without reading ei

Subexponential: Parasitic Extraction

the paper thati John signed ei without reading ei

• → S / !N

Subexponential: Parasitic Extraction

the paper thati John signed ei without reading ei

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) ∆(!A, !A) → C ∆(!A) → C (contr)

Subexponential: Parasitic Extraction

the paper thati John signed ei [without reading ei]

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) ∆(!A, !A) → C ∆(!A) → C (contr)

Subexponential: Parasitic Extraction

the paper thati John signed ei [without reading ei]

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) ∆(!A1, . . . , !An, [!A1, . . . , !An, Γ]) → B ∆(!A1, . . . , !An, Γ) → B (contrb)

Subexponential: Parasitic Extraction

the paper thati John signed ei [without reading ei]

• → S / !N

∆(!A, Γ) → C ∆(Γ, !A) → C (perm1) ∆(A) → C ∆(!A) → C (! →) ∆(!A1, . . . , !An, [!A1, . . . , !An, Γ]) → B ∆(!A1, . . . , !An, Γ) → B (contrb)

causes undecidability

The Lambek Calculus with Subexponential and Bracket Modalities (!bL

1) A → A Λ → 1 Γ → B ∆(C) → D ∆(C / B, Γ) → D (/ →) Γ, B → C Γ → C / B (→ /) ∆(A, B) → D ∆(A · B) → D (· →) Γ → A ∆(C) → D ∆(Γ, A \ C) → D (\ →) A, Γ → C Γ → A \ C (→ \) Γ1 → A Γ2 → B Γ1, Γ2 → A · B (→ ·) ∆(Λ) → A ∆(1) → A (1 →) ∆([A]) → C ∆(A) → C ( →) Π → A [Π] → A (→ ) Γ(A) → B Γ(!A) → B (! →) ∆(A) → C ∆([[]−1A]) → C ([]−1 →) [Π] → A Π → []−1A (→ []−1) !A1, . . . , !An → A !A1, . . . , !An → !A (→ !) ∆(!A1, . . . , !An, [!A1, . . . , !An, Γ]) → B ∆(!A1, . . . , !An, Γ) → B (contrb) ∆(!A, Γ) → B ∆(Γ, !A) → B (perm1) ∆(Γ, !A) → B ∆(!A, Γ) → B (perm2) Π → A ∆(A) → C ∆(Π) → C (cut)

The Lambek Calculus with Subexponential and Bracket Modalities (!bL

1) A → A Λ → 1 Γ → B ∆(C) → D ∆(C / B, Γ) → D (/ →) Γ, B → C Γ → C / B (→ /) ∆(A, B) → D ∆(A · B) → D (· →) Γ → A ∆(C) → D ∆(Γ, A \ C) → D (\ →) A, Γ → C Γ → A \ C (→ \) Γ1 → A Γ2 → B Γ1, Γ2 → A · B (→ ·) ∆(Λ) → A ∆(1) → A (1 →) ∆([A]) → C ∆(A) → C ( →) Π → A [Π] → A (→ ) Γ(A) → B Γ(!A) → B (! →) ∆(A) → C ∆([[]−1A]) → C ([]−1 →) [Π] → A Π → []−1A (→ []−1) !A1, . . . , !An → A !A1, . . . , !An → !A (→ !) ∆(!A1, . . . , !An, [!A1, . . . , !An, Γ]) → B ∆(!A1, . . . , !An, Γ) → B (contrb) ∆(!A, Γ) → B ∆(Γ, !A) → B (perm1) ∆(Γ, !A) → B ∆(!A, Γ) → B (perm2) Π → A ∆(A) → C ∆(Π) → C (cut)

◮ A fragment of Db!b by Morrill and Valent´

ın, 2015.

The Lambek Calculus with Subexponential and Bracket Modalities (!bL

1) A → A Λ → 1 Γ → B ∆(C) → D ∆(C / B, Γ) → D (/ →) Γ, B → C Γ → C / B (→ /) ∆(A, B) → D ∆(A · B) → D (· →) Γ → A ∆(C) → D ∆(Γ, A \ C) → D (\ →) A, Γ → C Γ → A \ C (→ \) Γ1 → A Γ2 → B Γ1, Γ2 → A · B (→ ·) ∆(Λ) → A ∆(1) → A (1 →) ∆([A]) → C ∆(A) → C ( →) Π → A [Π] → A (→ ) Γ(A) → B Γ(!A) → B (! →) ∆(A) → C ∆([[]−1A]) → C ([]−1 →) [Π] → A Π → []−1A (→ []−1) !A1, . . . , !An → A !A1, . . . , !An → !A (→ !) ∆(!A1, . . . , !An, [!A1, . . . , !An, Γ]) → B ∆(!A1, . . . , !An, Γ) → B (contrb) ∆(!A, Γ) → B ∆(Γ, !A) → B (perm1) ∆(Γ, !A) → B ∆(!A, Γ) → B (perm2) Π → A ∆(A) → C ∆(Π) → C (cut)

◮ A fragment of Db!b by Morrill and Valent´

ın, 2015.

◮ Our analysis of syntactic phenomena is due to Morrill, 2011–2017.

Cut Elimination in !bL

1

We use deep cut elimination strategy (cf. Bra¨ uner and de Paiva 1996).

Cut Elimination in !bL

1

We use deep cut elimination strategy (cf. Bra¨ uner and de Paiva 1996).

Dright ∆1(A) → C1 ∆1(!A) → C1 (! →) ∆2(A) → C2 ∆2(!A) → C2 (! →) ∆3(A) → C3 ∆3(!A) → C3 (! →) !Π → A !Π → !A (→ !) ∆(!A) → C ∆(!Π) → C (cut) (contrb) (contrb) (contrb) . . . Dleft

• !Π → A

∆1(A) → C1 ∆1(!Π) → C1 (cut) !Π → A ∆′

2(A) → C2

∆′

2(!Π) → C2

(cut) !Π → A ∆3(A) → C3 ∆3(!Π) → C3 (cut) ∆(!Π) → C (contrb) (contrb) (contrb) . . . Dleft Dleft Dleft

Algorithmic Results

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

◮ Part of a bigger project:

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

◮ Part of a bigger project:

◮ Kan., Kuz., Sce. FG-2016: undecidability for !L

1 (with !,

without brackets).

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

◮ Part of a bigger project:

◮ Kan., Kuz., Sce. FG-2016: undecidability for !L

1 (with !,

without brackets).

◮ Kan., Kuz., Morrill, Sce. FSCD-2017: pseudo-polynomial

algorithm for Lb (with brackets, without !). (polynomial for formulae of bounded depth)

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

◮ Part of a bigger project:

◮ Kan., Kuz., Sce. FG-2016: undecidability for !L

1 (with !,

without brackets).

◮ Kan., Kuz., Morrill, Sce. FSCD-2017: pseudo-polynomial

algorithm for Lb (with brackets, without !). (polynomial for formulae of bounded depth)

◮ Next step? pseudo-polynomial algorithm for !bL

1 with

restrictions on !.

Algorithmic Results

◮ The derivability problem in !bL 1 is undecidable.

This solves an open question raised by Morrill and Valent´ ın, 2015.

◮ The derivability problem for sequents obeying bracket

non-negative condition belongs to NP.

BNC: any negative occurrence of a !A includes neither a positive

• ccurrence of []−1C, nor a negative occurrence of a C.

Morrill, Valent´ ın 2015: an exp-time algorithm, used in the CatLog parser. NP-complete, as the original Lambek calculus [Pentus 2006].

◮ Part of a bigger project:

◮ Kan., Kuz., Sce. FG-2016: undecidability for !L

1 (with !,

without brackets).

◮ Kan., Kuz., Morrill, Sce. FSCD-2017: pseudo-polynomial

algorithm for Lb (with brackets, without !). (polynomial for formulae of bounded depth)

◮ Next step? pseudo-polynomial algorithm for !bL

1 with

restrictions on !. (open question)

Undecidability Proof Sketch

Encoding type-0 grammar derivations (follow Lincoln et al. 1992):

Undecidability Proof Sketch

Encoding type-0 grammar derivations (follow Lincoln et al. 1992):

Lemma

The following rule is admissible in !bL

1:

∆1, ! []−1B, ∆2, B, ∆3 → C ∆1, ! []−1B, ∆2, ∆3 → C (inst)

Undecidability Proof Sketch

Encoding type-0 grammar derivations (follow Lincoln et al. 1992):

Lemma

The following rule is admissible in !bL

1:

∆1, ! []−1B, ∆2, B, ∆3 → C ∆1, ! []−1B, ∆2, ∆3 → C (inst) Bi = (u1 · . . . · uk) /(v1 · . . . · vm) encodes the i-th rewriting rule.

Undecidability Proof Sketch

Encoding type-0 grammar derivations (follow Lincoln et al. 1992):

Lemma

The following rule is admissible in !bL

1:

∆1, ! []−1B, ∆2, B, ∆3 → C ∆1, ! []−1B, ∆2, ∆3 → C (inst) Bi = (u1 · . . . · uk) /(v1 · . . . · vm) encodes the i-th rewriting rule. !Γ = !B1, . . . , !Bn, ! Γ = ! []−1B1, . . . , ! []−1Bn, !Φ = !(1 /(!B1)), . . . , !(1 /(!Bn)), and ! Φ = !(1 /(! []−1B1)), . . . , !(1 /(! []−1Bn)).

Undecidability Proof Sketch

Lemma

The following are equivalent:

• 1. !bL

1 ⊢!

Φ, ! Γ, a1, . . . , an → s;

Undecidability Proof Sketch

Lemma

The following are equivalent:

• 1. !bL

1 ⊢!

Φ, ! Γ, a1, . . . , an → s;

• 2. !L

1 ⊢!Φ, !Γ, a1, . . . , an → s;

Undecidability Proof Sketch

Lemma

The following are equivalent:

• 1. !bL

1 ⊢!

Φ, ! Γ, a1, . . . , an → s;

• 2. !L

1 ⊢!Φ, !Γ, a1, . . . , an → s;

• 3. !L

1 + (weak) ⊢!Γ, a1, . . . , an → s;

∆1, ∆2 → C ∆1, !A, ∆2 → C (weak)

Undecidability Proof Sketch

Lemma

The following are equivalent:

• 1. !bL

1 ⊢!

Φ, ! Γ, a1, . . . , an → s;

• 2. !L

1 ⊢!Φ, !Γ, a1, . . . , an → s;

• 3. !L

1 + (weak) ⊢!Γ, a1, . . . , an → s;

• 4. s ⇒∗ a1 . . . an in the type-0 grammar.

∆1, ∆2 → C ∆1, !A, ∆2 → C (weak)

Thank you !