Beyond the context-free boundary: generalizing Lambek calculus - - PowerPoint PPT Presentation

Beyond the context-free boundary: generalizing Lambek calculus

Michael Moortgat Flowin’cat 2010 Oxford

Abstract Lambek’s syntactic calculi, both the associative and the non-associative variant, are strictly contextfree. A well-tried strategy to overcome this expressive limitation has been to extend the calculi with unary modalities allowing for controlled forms

• f associativity/commutativity, cf the use of exponentials in linear logic.

Here we pursue an alternative strategy, exploiting the symmetries between resid- uated and Galois connected families of connectives, and between these and their

• duals. Communication between these families takes the form of linear, structure-

preserving distributivity principles. Background reading: Moortgat 2009, Symmetric categorial grammar. JPL, 38 (6) 681-710. Moortgat 2010, Symmetric categorial grammar: residuation and Galois

• connections. Linguistic Analysis. Special issue dedicated to Jim Lambek,

36(1–4), 2010. CoRR 1008.0170.

1. Motivation

Lambek’s syntactic calculus — (N)L, pregroup grammar — is strictly context-free. Expressive limitations Problematic are discontinuous dependencies: ◮ Extraction. Who stole the tarts? vs What did Alice find there? ◮ Infixation. Alice thinks someone is cheating local vs non-local interpretation.

1. Motivation

Lambek’s syntactic calculus — (N)L, pregroup grammar — is strictly context-free. Expressive limitations Problematic are discontinuous dependencies: ◮ Extraction. Who stole the tarts? vs What did Alice find there? ◮ Infixation. Alice thinks someone is cheating local vs non-local interpretation. Stragegies for reconciling form/meaning ◮ NL: controlled structural options, embedding translations; ∼ LL !,? ◮ Lambek-Grishin calculus LG, after Grishin 1983 ⊲ symmetry: residuated, Galois connected operations and their duals ⊲ structural rules logical distributivity principles ⊲ continuation semantics: relieves the burden on syntactic source calculus

2. LG: some results so far

◮ Prooftheoretic formats ⊲ Display sequent calculus, cut-elimination. MM 2007. ⊲ Proof nets: Moot 2007. Tableaux: Bastenhof 2010. ⊲ Focused proof search. Bastenhof (.)

2. LG: some results so far

◮ Prooftheoretic formats ⊲ Display sequent calculus, cut-elimination. MM 2007. ⊲ Proof nets: Moot 2007. Tableaux: Bastenhof 2010. ⊲ Focused proof search. Bastenhof (.) ◮ Models, completeness ⊲ Relational: Kurtonina&MM ’07/’10, Areces ea ’04, Bimbo&Dunn ’09. . . ⊲ Algebraic: Buszkowski 2010; phase semantics: Bastenhof (.)

2. LG: some results so far

◮ Prooftheoretic formats ⊲ Display sequent calculus, cut-elimination. MM 2007. ⊲ Proof nets: Moot 2007. Tableaux: Bastenhof 2010. ⊲ Focused proof search. Bastenhof (.) ◮ Models, completeness ⊲ Relational: Kurtonina&MM ’07/’10, Areces ea ’04, Bimbo&Dunn ’09. . . ⊲ Algebraic: Buszkowski 2010; phase semantics: Bastenhof (.) ◮ Expressivity, complexity ⊲ Without distr: context-free (Bastenhof 2010), polynomial (Capelletti 2007) ⊲ With distr: beyond Mild CS (Melissen 2009), NP-complete (Bransen 2010)

2. LG: some results so far

◮ Prooftheoretic formats ⊲ Display sequent calculus, cut-elimination. MM 2007. ⊲ Proof nets: Moot 2007. Tableaux: Bastenhof 2010. ⊲ Focused proof search. Bastenhof (.) ◮ Models, completeness ⊲ Relational: Kurtonina&MM ’07/’10, Areces ea ’04, Bimbo&Dunn ’09. . . ⊲ Algebraic: Buszkowski 2010; phase semantics: Bastenhof (.) ◮ Expressivity, complexity ⊲ Without distr: context-free (Bastenhof 2010), polynomial (Capelletti 2007) ⊲ With distr: beyond Mild CS (Melissen 2009), NP-complete (Bransen 2010) ◮ Continuation-passing-style interpretation. Bernardi&MM 2007/2010. References See: Categorial type logics. Chapter update. Handbook of Logic and Language, 2nd edition. Elsevier, 2010.

3. Recap: residuated pairs, Galois connections

Basic concepts Posets (X, ≤), (Y, ≤′) with mappings f : X − → Y , g : Y − → X. The pair (f, g) is called a residuated pair (rp), a dual residuated pair (drp), a Galois connection (gc), a dual Galois connection (dgc) depending on which of the following biconditionals holds: (rp) fx ≤′ y ⇔ x ≤ gy (drp) y ≤′ fx ⇔ gy ≤ x (gc) y ≤′ fx ⇔ x ≤ gy (dgc) fx ≤′ y ⇔ gy ≤ x

3. Recap: residuated pairs, Galois connections

Basic concepts Posets (X, ≤), (Y, ≤′) with mappings f : X − → Y , g : Y − → X. The pair (f, g) is called a residuated pair (rp), a dual residuated pair (drp), a Galois connection (gc), a dual Galois connection (dgc) depending on which of the following biconditionals holds: (rp) fx ≤′ y ⇔ x ≤ gy (drp) y ≤′ fx ⇔ gy ≤ x (gc) y ≤′ fx ⇔ x ≤ gy (dgc) fx ≤′ y ⇔ gy ≤ x Alternative characterization in terms of tonicity, compositions (rp) f, g : isotone, x ≤ gfx, fgy ≤′ y (drp) f, g : isotone, gfx ≤ x, y ≤′ fgx (gc) f, g : antitone, x ≤ gfx, y ≤′ fgy (dgc) f, g : antitone, fgx ≤ x, gfy ≤′ y

3. Recap: residuated pairs, Galois connections

Basic concepts Posets (X, ≤), (Y, ≤′) with mappings f : X − → Y , g : Y − → X. The pair (f, g) is called a residuated pair (rp), a dual residuated pair (drp), a Galois connection (gc), a dual Galois connection (dgc) depending on which of the following biconditionals holds: (rp) fx ≤′ y ⇔ x ≤ gy (drp) y ≤′ fx ⇔ gy ≤ x (gc) y ≤′ fx ⇔ x ≤ gy (dgc) fx ≤′ y ⇔ gy ≤ x Alternative characterization in terms of tonicity, compositions (rp) f, g : isotone, x ≤ gfx, fgy ≤′ y (drp) f, g : isotone, gfx ≤ x, y ≤′ fgx (gc) f, g : antitone, x ≤ gfx, y ≤′ fgy (dgc) f, g : antitone, fgx ≤ x, gfy ≤′ y Generalization Residuated triples, etc. Galatos e.a., Dunn.

4. Lambek-Grishin calculus: fusion vs fission

Lambek-Grishin calculus NL has ⊗, left and right division \, / forming a residuated

• triple. LG adds a dual residuated triple: coproduct ⊕, right and left difference ⊘, .

A → C/B ⇔ A ⊗ B → C ⇔ B → A\C B C → A ⇔ C → B ⊕ A ⇔ C ⊘ A → B

4. Lambek-Grishin calculus: fusion vs fission

Lambek-Grishin calculus NL has ⊗, left and right division \, / forming a residuated

• triple. LG adds a dual residuated triple: coproduct ⊕, right and left difference ⊘, .

A → C/B ⇔ A ⊗ B → C ⇔ B → A\C B C → A ⇔ C → B ⊕ A ⇔ C ⊘ A → B Interpretation Algebraic (Ono, Buszkowski); Kripke-style relational (Dunn, Kurton- ina). For the latter: frames (W, R, S), with operations defined on subsets of W. A ⊗ B = {x | ∃yz (Rxyz ∧ y ∈ A ∧ z ∈ B)} C/B = {y | ∀xz ((Rxyz ∧ z ∈ B) ⇒ x ∈ C)} A\C = {z | ∀xy ((Rxyz ∧ y ∈ A) ⇒ x ∈ C)} A ⊕ B = {x | ∀yz (Sxyz ⇒ (y ∈ A ∨ z ∈ B))} C ⊘ B = {y | ∃xz (Sxyz ∧ z ∈ B ∧ x ∈ C)} A C = {z | ∃xy (Sxyz ∧ y ∈ A ∧ x ∈ C)} Note As yet no assumptions about relation between fusion R, fission S.

5. Through the Looking Glass

Two symmetries To the left-right symmetry ·⊲

⊳ of NL, LG adds an arrow reversal

symmetry ·∞. Together with identity and composition: Klein group. A⊲

⊳

f ⊲

⊳

− − → B⊲

⊳

⇔ A f − − → B ⇔ B∞ f ∞ − − → A∞ Translation tables ⊲ ⊳ C/D A ⊗ B B ⊕ A D C D\C B ⊗ A A ⊕ B C ⊘ D ∞ C/B A ⊗ B A\C B C B ⊕ A C ⊘ A

5. Through the Looking Glass

Two symmetries To the left-right symmetry ·⊲

⊳ of NL, LG adds an arrow reversal

symmetry ·∞. Together with identity and composition: Klein group. A⊲

⊳

f ⊲

⊳

− − → B⊲

⊳

⇔ A f − − → B ⇔ B∞ f ∞ − − → A∞ Translation tables ⊲ ⊳ C/D A ⊗ B B ⊕ A D C D\C B ⊗ A A ⊕ B C ⊘ D ∞ C/B A ⊗ B A\C B C B ⊕ A C ⊘ A theorems form quartets: (B ⊘ A) B → A B ⊘ (A B) → A

• ∞
• A → B/(A\B)
• A → (B/A)\B
• ⊲

⊳

6. Distributivity

Interaction fusion, fission Grishin considers two groups of distributivity principles ◮ respecting resources, cf weak/linear distributivities Cockett-Seely, de Paiva ◮ respecting structure: non-associativity/commutativity ⊗/⊕

6. Distributivity

Interaction fusion, fission Grishin considers two groups of distributivity principles ◮ respecting resources, cf weak/linear distributivities Cockett-Seely, de Paiva ◮ respecting structure: non-associativity/commutativity ⊗/⊕ Option A Recipe: select a ⊗/⊕ factor in the premise; simultaneously introduce the residual operations for the remaining two in the conclusion. Note: ·⊲

⊳ symmetry.

A ⊗ B → C ⊕ D C A → D / B A ⊗ B → C ⊕ D B ⊘ D → A \ C A ⊗ B → C ⊕ D C B → A \ D A ⊗ B → C ⊕ D A ⊘ D → C / B Option B Converses of A. Characteristic theorems: (A ⊕ B) ⊗ C → A ⊕ (B ⊗ C) etc Conservativity Adding A or B to the pure residuation logic is conservative; with A+B structure-preservation is lost.

7. Generalizing arity: unary operators

Isotone Residuated pairs: inverse duals wrt interpreting binary relation. A = {x | ∃y (Rxy ∧ y ∈ A)} ′A = {y | ∀x (Rxy ⇒ x ∈ A)} A = {x | ∀y (Sxy ⇒ y ∈ A)} ′A = {y | ∃x (Sxy ∧ x ∈ A)}

7. Generalizing arity: unary operators

Isotone Residuated pairs: inverse duals wrt interpreting binary relation. A = {x | ∃y (Rxy ∧ y ∈ A)} ′A = {y | ∀x (Rxy ⇒ x ∈ A)} A = {x | ∀y (Sxy ⇒ y ∈ A)} ′A = {y | ∃x (Sxy ∧ x ∈ A)} Antitone Galois connection 0·, ·0; dual Galois connection ·1, 1·

0A = {x | ∀y (y ∈ A ⇒ Rxy)}

A0 = {y | ∀x (x ∈ A ⇒ Rxy)} A1 = {y | ∃x (Sxy ∧ x ∈ A)}

1A = {x | ∃y (Sxy ∧ y ∈ A)}

Galois laws: B → A0 ⇔ A → 0B ;

1B → A

⇔ A1 → B Properties of the compositions: A → 0(A0) , A → (0A)0 ; (1A)1 → A ,

1(A1) → A

Composition 0·, ·0 (either order): closure operation (expanding, isotone, idempotent); dually, composition of ·1, 1·: interior operation (contracting, isotone, and idempotent).

8. Distributivity: negations

◮ Grishin: defined negations in terms of multiplicative units A0 A\0, 1A 1⊘A and the ·⊲

⊳ symmetric cases.

◮ Here: primitive negations — multiplicative units overgenerate ◮ But: we use the defined negations to extrapolate for the 0·, ·0,·1, 1· distributivities

8. Distributivity: negations

◮ Grishin: defined negations in terms of multiplicative units A0 A\0, 1A 1⊘A and the ·⊲

⊳ symmetric cases.

◮ Here: primitive negations — multiplicative units overgenerate ◮ But: we use the defined negations to extrapolate for the 0·, ·0,·1, 1· distributivities Illustration Interaction among the (dual) Galois connected operators: A → B 1 ⊗ A → B ⊕ 0 B 1 → 0/A

• A → B

B1 → 0A

8. Distributivity: negations

◮ Grishin: defined negations in terms of multiplicative units A0 A\0, 1A 1⊘A and the ·⊲

⊳ symmetric cases.

◮ Here: primitive negations — multiplicative units overgenerate ◮ But: we use the defined negations to extrapolate for the 0·, ·0,·1, 1· distributivities Illustration Interaction among the (dual) Galois connected operators: A → B 1 ⊗ A → B ⊕ 0 B 1 → 0/A

• A → B

B1 → 0A Interaction between Galois connected and residuated families: A → B ⊕ C 1 ⊗ A → B ⊕ C B 1 → C/A

• A → B ⊕ C

B1 → C/A

9. Display sequent calculus

Motivation At first sight, it looks like the Grishin distributivity laws could be absorbed in Lambek’s logical sequent rules. Compare (∆[B\A]: B\A in a structural ⊕ context): NL: (B, Γ) ⊢ A Γ ⊢ B\A \R

?

• LG:

(B, Γ) ⊢ ∆[A] Γ ⊢ ∆[B\A] \R But, this LG rule (and the ⊲ ⊳, ∞ duals) is incomplete: no cut-free derivations below. (a, (c ⊘ ((a\b) c))) ⊢ b b ⊢ (((c/(b ⊘ a))\c), a)

9. Display sequent calculus

Motivation At first sight, it looks like the Grishin distributivity laws could be absorbed in Lambek’s logical sequent rules. Compare (∆[B\A]: B\A in a structural ⊕ context): NL: (B, Γ) ⊢ A Γ ⊢ B\A \R

?

• LG:

(B, Γ) ⊢ ∆[A] Γ ⊢ ∆[B\A] \R But, this LG rule (and the ⊲ ⊳, ∞ duals) is incomplete: no cut-free derivations below. (a, (c ⊘ ((a\b) c))) ⊢ b b ⊢ (((c/(b ⊘ a))\c), a) Display sequent calculus Gor´ e 1999, MM 2007. ◮ structural punctuation for every logical connective ◮ (dual) residuation, Galois principles: structural rules, display equivalences ◮ Grishin’s distributivity laws: structural too

10. LG display calculus: structural rules

Sequents Arrows A → B to sequents X ⊢ Y , with X (Y ) input (output) structures. I ::= x : A | I · ⊗ · I | I · ⊘ · O | O · · I | 1·O | O ·1 O ::= α : A | O · ⊕ · O | I · \ · O | O · / · I | I ·0 | 0·I

10. LG display calculus: structural rules

Sequents Arrows A → B to sequents X ⊢ Y , with X (Y ) input (output) structures. I ::= x : A | I · ⊗ · I | I · ⊘ · O | O · · I | 1·O | O ·1 O ::= α : A | O · ⊕ · O | I · \ · O | O · / · I | I ·0 | 0·I Axiom, formula cut A ⊢ A X ⊢ A A ⊢ Y X ⊢ Y Residuation, Galois laws Display equivalences. For example: A → C/B A ⊗ B → C

• X ⊢ Z · / · Y

X · ⊗ · Y ⊢ Z ;

1B → A

A1 → B

• 1·Y → X

X ·1 → Y Distributivity laws All operations are structural. For example: A ⊗ B → C ⊕ D C A → D/B

• X · ⊗ · Y ⊢ Z · ⊕ · W

Z · · X ⊢ W · / · Y

11. LG display calculus: logical rules

Each connective has a left and a right introduction rule. They fall in two groups. Rewrite rules

• Reversible. Toggle between logical, structural operation.

A · ⊘ · B ⊢ Y A ⊘ B ⊢ Y ⊘L A ·1 ⊢ Y A1 ⊢ Y ·1L

11. LG display calculus: logical rules

Each connective has a left and a right introduction rule. They fall in two groups. Rewrite rules

• Reversible. Toggle between logical, structural operation.

A · ⊘ · B ⊢ Y A ⊘ B ⊢ Y ⊘L A ·1 ⊢ Y A1 ⊢ Y ·1L Monotonicity rules X ⊢ A B ⊢ Y X · ⊘ · Y ⊢ A ⊘ B ⊘R A ⊢ Y Y ·1 ⊢ A1 ·1R Complete the picture using the ⊲ ⊳ and ∞ symmetries (exercise).

11. LG display calculus: logical rules

Each connective has a left and a right introduction rule. They fall in two groups. Rewrite rules

• Reversible. Toggle between logical, structural operation.

A · ⊘ · B ⊢ Y A ⊘ B ⊢ Y ⊘L A ·1 ⊢ Y A1 ⊢ Y ·1L Monotonicity rules X ⊢ A B ⊢ Y X · ⊘ · Y ⊢ A ⊘ B ⊘R A ⊢ Y Y ·1 ⊢ A1 ·1R Complete the picture using the ⊲ ⊳ and ∞ symmetries (exercise). Cut elimination MM 2007.

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / /

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b · ⊘ · a ⊢ (c/(b ⊘ a))\c b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / b · ⊘ · a ⊢ b ⊘ a / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / b · ⊘ · a ⊢ b ⊘ a ⊘R / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. a ⊢ a / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. a ⊢ a / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. a ⊢ a / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c ⊢ c c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. a ⊢ a / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c ⊢ c c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr

12. Cut-free derivation

Display property Any formula component of a sequent can be displayed as the sole antecedent or succedent part, where the logical rules are applicable. a ⊢ a / b ⊢ b b · ⊘ · a ⊢ b ⊘ a ⊘R / c ⊢ c c/(b ⊘ a) ⊢ c · / · (b · ⊘ · a) /L b · ⊘ · a ⊢ (c/(b ⊘ a)) · \ · c r b · ⊘ · a ⊢ (c/(b ⊘ a))\c \R b ⊢ ((c/(b ⊘ a))\c) · ⊕ · a dr Observe This sequent is problematic for a standard Gentzen presentation, with only structural punctuation for ⊗ and ⊕.

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj.

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj. ◮ Proof structure. Set of links over finite set of frm’s s.t. every frm is at most once premise and at most once conclusion of a link. ⊲ hypotheses: ¬ conclusion of any link ⊲ conclusions: ¬ premise of any link ⊲ axioms: ¬ main formula of any link

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj. ◮ Proof structure. Set of links over finite set of frm’s s.t. every frm is at most once premise and at most once conclusion of a link. ⊲ hypotheses: ¬ conclusion of any link ⊲ conclusions: ¬ premise of any link ⊲ axioms: ¬ main formula of any link ◮ Abstract proof structure: PS with formulas at internal nodes erased.

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj. ◮ Proof structure. Set of links over finite set of frm’s s.t. every frm is at most once premise and at most once conclusion of a link. ⊲ hypotheses: ¬ conclusion of any link ⊲ conclusions: ¬ premise of any link ⊲ axioms: ¬ main formula of any link ◮ Abstract proof structure: PS with formulas at internal nodes erased. ◮ Rewriting: logical and structural conversions next slides

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj. ◮ Proof structure. Set of links over finite set of frm’s s.t. every frm is at most once premise and at most once conclusion of a link. ⊲ hypotheses: ¬ conclusion of any link ⊲ conclusions: ¬ premise of any link ⊲ axioms: ¬ main formula of any link ◮ Abstract proof structure: PS with formulas at internal nodes erased. ◮ Rewriting: logical and structural conversions next slides ◮ Proof net: APS converting to a tensor tree (possibly unrooted)

13. Lambek-Grishin proof nets

◮ Links. Type: (co)tensor; premises P1, . . . , Pn, conclusions C1, . . . , Cm, 0 ≤ n, m. Main formula: empty or one of the Pi, Cj. ◮ Proof structure. Set of links over finite set of frm’s s.t. every frm is at most once premise and at most once conclusion of a link. ⊲ hypotheses: ¬ conclusion of any link ⊲ conclusions: ¬ premise of any link ⊲ axioms: ¬ main formula of any link ◮ Abstract proof structure: PS with formulas at internal nodes erased. ◮ Rewriting: logical and structural conversions next slides ◮ Proof net: APS converting to a tensor tree (possibly unrooted) References Moot & Puite (2002), Proof nets for the multimodal Lambek calculus. Studia Logica 71(3). Moot (2007), Proof nets for display logic. CoRR 0711.2444.

14. Binary links, contractions: tensor

A / B B A A / B B A A B A ⊗ B A B A ⊗ B A A \ B B A A \ B B

• C
• H
• C
• H
• C
• H
• [R\]

[L⊗] [R/]

15. Binary links, contractions: tensor∞

A B A ⊕ B A B B A A A B B A B A ⊕ B A B B A A A B B

• H
• C
• H
• C
• H
• C
• [L]

[R⊕] [L]

16. Structural rewriting

Example Two of Grishin’s distributivity laws.

• X
• V
• W
• Y

←Gr1

• X
• Y
• V
• W
• X
• W
• V
• Y
• →Gr2

X · · V ⊢ Y · / · W

Gr1

⇐ V · ⊗ · W ⊢ X · ⊕ · Y

Gr2

⇒ X · · W ⊢ V · \ · Y

17. Lexicon: partial proof trees

John np np np \ s s (np \ s) / s s believes np np \ s s left (s s) np np someone s s s s

Clicking these together, one can produce sentences such as: John left, someone left, J believes someone left, someone believes J left, . . .

18. Rewriting: structural

John

• believes
• left
• someone
• s

→Gr2 John

• believes
• left
• someone
• s
• →Gr2
• s

John believes

• left
• someone
• . . .

19. Rewriting: structural, logical

→Gr2

• s

John believes

• left
• someone
• →Gr1
• s

John believes

• left
• someone
• →L
• s

John believes

• left

someone

• Final result

Tensor tree, root s. Yield: recognized string. Reading: wide scope ∃.

20. The syntax-semantics mapping

The standard view Compositional mapping, cf Montague’s Universal Grammar. (N)L{n,np,s}

/,\

(·)′ − − − − − − − − − − − − → LP{e,t}

→

(MILL) syntactic calculus homomorphism semantic calculus np′ = e ; s′ = t ; n′ = e → t ; (A\B)′ = (B/A)′ = A′ → B′ desirable recipes for meaning assembly are lost in translation.

20. The syntax-semantics mapping

The standard view Compositional mapping, cf Montague’s Universal Grammar. (N)L{n,np,s}

/,\

(·)′ − − − − − − − − − − − − → LP{e,t}

→

(MILL) syntactic calculus homomorphism semantic calculus np′ = e ; s′ = t ; n′ = e → t ; (A\B)′ = (B/A)′ = A′ → B′ desirable recipes for meaning assembly are lost in translation. Continuation semantics More structured view of the syn/sem mapping: ◮ distinction: values vs continuations: functions from values to answer type ◮ evaluation context explicit part of interpretation process ◮ more balanced division of labour between source and target calculus

21. LG: continuation semantics

Bernardi & MM 2007, 2010, after Curien/Herbelin, Lengrand. LGA

/,\,⊘,,·1,1·,·0,0·

⌈·⌉ − − − − → LPA∪{r}

→

· − − − − → IL{e,t}

→

Two-step interpretation ◮ ⌈·⌉ : double-negation/continuation-passing-style translation ⊲ maps multiple conclusion source logic to intuitionistic linear logic ⊲ introduces special response type r

21. LG: continuation semantics

Bernardi & MM 2007, 2010, after Curien/Herbelin, Lengrand. LGA

/,\,⊘,,·1,1·,·0,0·

⌈·⌉ − − − − → LPA∪{r}

→

· − − − − → IL{e,t}

→

Two-step interpretation ◮ ⌈·⌉ : double-negation/continuation-passing-style translation ⊲ maps multiple conclusion source logic to intuitionistic linear logic ⊲ introduces special response type r ◮ · : combining lexical with derivational semantics ⊲ atomic types: np = e, s = r = t ⊲ terms: nonlinearity restricted to constants; (M N) = (M N) ; λx.M = λ x.M

21. LG: continuation semantics

Bernardi & MM 2007, 2010, after Curien/Herbelin, Lengrand. LGA

/,\,⊘,,·1,1·,·0,0·

⌈·⌉ − − − − → LPA∪{r}

→

· − − − − → IL{e,t}

→

Two-step interpretation ◮ ⌈·⌉ : double-negation/continuation-passing-style translation ⊲ maps multiple conclusion source logic to intuitionistic linear logic ⊲ introduces special response type r ◮ · : combining lexical with derivational semantics ⊲ atomic types: np = e, s = r = t ⊲ terms: nonlinearity restricted to constants; (M N) = (M N) ; λx.M = λ x.M ◮ target interpretation: composition · ◦ ⌈·⌉

22. CPS translation

Below the call-by-value ⌈·⌉ version. Call by name: ⌊A⌋ = ⌈A∞⌉. CPS mapping ◮ Source: LG display sequent calculus. ◮ Target: fragment of natural deduction LP (MILL) with response type r; all functions have head type r. ◮ Notation: A⊥ A → r.

22. CPS translation

Below the call-by-value ⌈·⌉ version. Call by name: ⌊A⌋ = ⌈A∞⌉. CPS mapping ◮ Source: LG display sequent calculus. ◮ Target: fragment of natural deduction LP (MILL) with response type r; all functions have head type r. ◮ Notation: A⊥ A → r. Types For source types A, the target calculus makes a distinction between values: ⌈A⌉, continuations: ⌈A⌉⊥, and computations: ⌈A⌉⊥⊥. ◮ For p atomic, ⌈p⌉ = p. ◮ Target is non-directional: ⌈A⊲

⊳⌉ = ⌈A⌉.

◮ Duality (co)implication: ⌈A\B⌉ = ⌈B⌉⊥ → ⌈A⌉⊥ ; ⌈A ⊘ B⌉ = ⌈A\B⌉⊥ ◮ Negations: ⌈A0⌉ = ⌈1A⌉ = ⌈A⌉⊥

23. Translation (cont’d)

Structures LG structures are translated into MILL linear typing environments: ◮ Atomic structures: ⌈x : A⌉ = { x : ⌈A⌉} ; ⌈α : A⌉ = { α : ⌈A⌉⊥} ◮ Composite, for n-place structure building operations f: ⌈f(X1, . . . , Xn)⌉ =

n

• i=1

⌈Xi⌉

23. Translation (cont’d)

Structures LG structures are translated into MILL linear typing environments: ◮ Atomic structures: ⌈x : A⌉ = { x : ⌈A⌉} ; ⌈α : A⌉ = { α : ⌈A⌉⊥} ◮ Composite, for n-place structure building operations f: ⌈f(X1, . . . , Xn)⌉ =

n

• i=1

⌈Xi⌉ Sequents Neutral (commands); active output (terms) or input (contexts) formula. Invariants of the translation source: LGA

/,\,⊘,,·1,1·,·0,0· ⌈·⌉

− − − − → CPS target: LPA ∪{r}

→

terms X ⊢ B contexts A ⊢ Y commands X ⊢ Y ⌈X⌉ ⊢ M : ⌈B⌉⊥⊥ ⌈Y ⌉ ⊢ K : ⌈A⌉⊥ ⌈X⌉ ∪ ⌈Y ⌉ ⊢ S : r

24. Translation (cont’d)

Identity Axiom, co-axiom; cut. x : A ⊢ A Ax X ⊢ A A ⊢ Y X ⊢ Y Cut A ⊢ α : A Co-Ax ⌈Ax⌉ = λk.(k x) : ⌈A⌉⊥⊥ ⌈Cut⌉ = (M ⌈A⌉⊥⊥ K⌈A⌉⊥) : r ⌈Co-Ax⌉ = α : ⌈A⌉⊥

24. Translation (cont’d)

Identity Axiom, co-axiom; cut. x : A ⊢ A Ax X ⊢ A A ⊢ Y X ⊢ Y Cut A ⊢ α : A Co-Ax ⌈Ax⌉ = λk.(k x) : ⌈A⌉⊥⊥ ⌈Cut⌉ = (M ⌈A⌉⊥⊥ K⌈A⌉⊥) : r ⌈Co-Ax⌉ = α : ⌈A⌉⊥ Activate a passive formula New wrt the neutral sequent presentation. X ⊢ α : A X ⊢ A µ x : A ⊢ Y A ⊢ Y

• µ

⌈µ⌉ = λ α.Sr : ⌈A⌉⊥⊥ ⌈ µ⌉ = λ x.Sr : ⌈A⌉⊥ Deactivate cuts with a (co)axiom as premise.

25. LG: logical rules

Monotonicity X ⊢ A B ⊢ Y A\B ⊢ X · \ · Y \L X ⊢ A B ⊢ Y X · ⊘ · Y ⊢ A ⊘ B ⊘R ⌈\L⌉ = λu.(M ⌈A⌉⊥⊥(u K⌈B⌉⊥)) : ⌈A\B⌉⊥ ⌈⊘R⌉ = λk.(k ⌈\L⌉) : ⌈A ⊘ B⌉⊥⊥

25. LG: logical rules

Monotonicity X ⊢ A B ⊢ Y A\B ⊢ X · \ · Y \L X ⊢ A B ⊢ Y X · ⊘ · Y ⊢ A ⊘ B ⊘R ⌈\L⌉ = λu.(M ⌈A⌉⊥⊥(u K⌈B⌉⊥)) : ⌈A\B⌉⊥ ⌈⊘R⌉ = λk.(k ⌈\L⌉) : ⌈A ⊘ B⌉⊥⊥ Rewrites reversible X ⊢ x : A · \ · β : B X ⊢ A\B \R x : A · ⊘ · β : B ⊢ X A ⊘ B ⊢ X ⊘L ⌈⊘L⌉ = ⌈\R⌉ = λh.(h λ βλ x.Sr) : ⌈A\B⌉⊥⊥ = ⌈A ⊘ B⌉⊥

26. Logical rules: negations

Galois X ⊢ 0·(x : A) X ⊢ 0A

0· R

λk.(k λ x.Sr) : ⌈0A⌉⊥⊥ X ⊢ A

0A ⊢ 0·X 0· L

M : ⌈0A⌉⊥ = ⌈A⌉⊥⊥

26. Logical rules: negations

Galois X ⊢ 0·(x : A) X ⊢ 0A

0· R

λk.(k λ x.Sr) : ⌈0A⌉⊥⊥ X ⊢ A

0A ⊢ 0·X 0· L

M : ⌈0A⌉⊥ = ⌈A⌉⊥⊥ Dual Galois A ⊢ Y Y ·1 ⊢ A1 ·1R λk.(k K⌈A⌉⊥) : ⌈A1⌉⊥⊥ (α : A)·1 ⊢ Y A1 ⊢ Y ·1L λ α.Sr : ⌈A1⌉⊥

27. Scope ambiguity in base logic

We use the interior operation 1( · 1) to obtain scope ambiguities. This example does not involve distributivity principles. Compare the (†) and (‡) derivations. np ⊢ · np · np·1 ⊢ np1 ·1R

1·(np1) ⊢ np

⇀ ⇁ s

α

⊢ · s · np\s ⊢ 1·(np1) · \ · s \L · np ·

y

⊢ np (np\s)/np ⊢ (1·(np1) · \ · s) · / · np /L np ⊢ (np\s)/np · \ · (1·(np1) · \ · s) ↼ ↽ † ((np\s)/np · \ · (1·(np1) · \ · s))·1 ⊢ np1 ·1R (direct object)

1(np1) ⊢ (np\s)/np · \ · (1·(np1) · \ · s) 1· L 1(np1) ⊢ s · / · ((np\s)/np · ⊗ · 1(np1)) 1· L 1(np1) su

· ⊗ ·((np\s)/np

• tv

· ⊗ · 1(np1)

do

) ⊢ s ⇌ ⌈·⌉ translation: λ α.(do λ y.((tv λu.(su (u α))) y))

28. Object wide scope: step by step

·1R λk.(k β) : ⌈np1⌉⊥⊥ ⇀ ⇁ λ β.( γ β) = γ : ⌈np1⌉⊥ = ⌈np⌉⊥⊥ \L λu.( γ (u α)) : ⌈np\s⌉⊥ /L λu′.(u′ λu.( γ (u α)) y) : ⌈(np\s)/np⌉⊥ ↼ ↽ λ y.(tv λu.( γ (u α)) y) : ⌈np⌉⊥ ·1R λk.(k λ y.(tv λu.( γ (u α)) y)) : ⌈np1⌉⊥⊥

1· L

λ κ.( κ λ y.(tv λu.( γ (u α)) y)) : ⌈1(np1)⌉⊥

1· L

λ γ.(do λ y.(tv λu.( γ (u α)) y)) : ⌈1(np1)⌉⊥ ⇌ λ α.(do λ y.((tv λu.(su (u α))) y)) : ⌈s⌉⊥⊥

29. Subject wide scope

Below an alternative derivation, targeting first the subject, rather than the direct object. · np ·

x

⊢ np s

α

⊢ · s · np\s ⊢ np · \ · s \L . . .

1·(np1) ⊢ np

⇀ ⇁ (np\s)/np ⊢ (np · \ · s) · / · 1·(np1) /L np ⊢ s · / · ((np\s)/np · ⊗ · 1·(np1)) ↼ ↽ ‡ (s · / · ((np\s)/np · ⊗ · 1·(np1)))·1 ⊢ np1 ·1R (subject) . . . ⌈·⌉ translation: λ α.(su λ x.(do (tv λu.((u α) x))))

30. Translating the lexical constants

The table below gives the · translation of the constants, for the sample sentence ‘everyone saw something’, assuming np = e, s = r = t, and a target constant ‘see’ of type e → e → t. source · translation everyone : ⌈np⌉⊥⊥ ∀ : (e → t) → t someone : ⌈np⌉⊥⊥ ∃ : (e → t) → t saw : (⌈s⌉⊥ → ⌈np⌉⊥)⊥ → ⌈np⌉⊥ λvλy.(v λcλx.(c ((see y) x))) : (((t → t) → e → t) → t) → e → t

30. Translating the lexical constants

The table below gives the · translation of the constants, for the sample sentence ‘everyone saw something’, assuming np = e, s = r = t, and a target constant ‘see’ of type e → e → t. source · translation everyone : ⌈np⌉⊥⊥ ∀ : (e → t) → t someone : ⌈np⌉⊥⊥ ∃ : (e → t) → t saw : (⌈s⌉⊥ → ⌈np⌉⊥)⊥ → ⌈np⌉⊥ λvλy.(v λcλx.(c ((see y) x))) : (((t → t) → e → t) → t) → e → t Final result Composition · ◦ ⌈·⌉, and an evaluation step, providing the identity function λp.p for the abstraction over the parameter c of type t → t. λ α.(do λ y.((tv λu.(su (u α))) y)) = λc.(∃ λy.(∀ λx.(c ((see y) x)))) λ α.(su λ x.(do (tv λu.((u α) x)))) = λc.(∀ λx.(∃ λy.(c ((see y) x))))

31. Comparison

1(np1) is an optimization w.r.t. the following typing alternatives.

◮ (B ⊘ C) A modeling in situ binder q(A, B, C), Bernardi/Moortgat ⊲ behaves locally as A within domain B; acts as function mapping B into C. ⊲ semantically: λh.((h λu.(u M ⌈B\C⌉)) N ⌈A⌉) ⊲ someone(s⊘s)np = λh.(∃ λx.((h λu.(u id)

• ⌈s⊘s⌉⊥

) x)) : s ⊘ s vacuous ◮ 1(np1) versus s ⊘ (np s) ⊲ syntax: same behaviour in source calculus ⊲ semantics: vacuous contribution from s component in s ⊘ (np s) ⌈s ⊘ (np s)⌉ = (⌈np s⌉⊥ → ⌈s⌉⊥)⊥ someone = λh.(h λg.(g λke→tλpt.((∃ k) ∧ p))

• ⌈nps⌉⊥

⊤

• ⌈s⌉

)

32. Conclusions

The symmetric Lambek-Grishin calculus offers some strategies to tackle the expressive limitations of the original Lambek calculi: ◮ Form ⊲ logical distributivity laws relating dual families ⊲ narrows the options for structural reasoning: preservation properties ◮ Meaning ⊲ continuation semantics for multiple-conclusion source calculus ⊲ optimizes division of labour between syntax and semantics

32. Conclusions

The symmetric Lambek-Grishin calculus offers some strategies to tackle the expressive limitations of the original Lambek calculi: ◮ Form ⊲ logical distributivity laws relating dual families ⊲ narrows the options for structural reasoning: preservation properties ◮ Meaning ⊲ continuation semantics for multiple-conclusion source calculus ⊲ optimizes division of labour between syntax and semantics More to explore ESSLLI 2007 course wiki http://symcg.pbworks.com/