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 of 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 Posets ( X, ≤ ) , ( Y, ≤ ′ ) with mappings f : X − Basic concepts → 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: fx ≤ ′ y ( rp ) ⇔ x ≤ gy y ≤ ′ fx ( drp ) ⇔ gy ≤ x y ≤ ′ fx ( gc ) ⇔ x ≤ gy fx ≤ ′ y ( dgc ) ⇔ gy ≤ x
3. Recap: residuated pairs, Galois connections Posets ( X, ≤ ) , ( Y, ≤ ′ ) with mappings f : X − Basic concepts → 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: fx ≤ ′ y ( rp ) ⇔ x ≤ gy y ≤ ′ fx ( drp ) ⇔ gy ≤ x y ≤ ′ fx ( gc ) ⇔ x ≤ gy fx ≤ ′ y ( dgc ) ⇔ gy ≤ x Alternative characterization in terms of tonicity, compositions fgy ≤ ′ y ( rp ) f, g : isotone , x ≤ gfx, y ≤ ′ fgx ( drp ) f, g : isotone , gfx ≤ x, y ≤ ′ fgy ( gc ) f, g : antitone , x ≤ gfx, gfy ≤ ′ y ( dgc ) f, g : antitone , fgx ≤ x,
3. Recap: residuated pairs, Galois connections Posets ( X, ≤ ) , ( Y, ≤ ′ ) with mappings f : X − Basic concepts → 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: fx ≤ ′ y ( rp ) ⇔ x ≤ gy y ≤ ′ fx ( drp ) ⇔ gy ≤ x y ≤ ′ fx ( gc ) ⇔ x ≤ gy fx ≤ ′ y ( dgc ) ⇔ gy ≤ x Alternative characterization in terms of tonicity, compositions fgy ≤ ′ y ( rp ) f, g : isotone , x ≤ gfx, y ≤ ′ fgx ( drp ) f, g : isotone , gfx ≤ x, y ≤ ′ fgy ( gc ) f, g : antitone , x ≤ gfx, gfy ≤ ′ y ( dgc ) f, g : antitone , fgx ≤ x, 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 ⊳ of NL , LG adds an arrow reversal To the left-right symmetry · ⊲ Two symmetries symmetry · ∞ . Together with identity and composition: Klein group. f ⊲ ⊳ B ∞ f ∞ f A ⊲ ⊳ → B ⊲ ⊳ → A ∞ − − ⇔ A − − → B ⇔ − − Translation tables C/D A ⊗ B B ⊕ A D � C C/B A ⊗ B A \ C ⊲ ⊳ ∞ D \ C B ⊗ A A ⊕ B C ⊘ D B � C B ⊕ A C ⊘ A
� � � � � � 5. Through the Looking Glass ⊳ of NL , LG adds an arrow reversal To the left-right symmetry · ⊲ Two symmetries symmetry · ∞ . Together with identity and composition: Klein group. f ⊲ ⊳ B ∞ f ∞ f A ⊲ ⊳ → B ⊲ ⊳ → A ∞ − − ⇔ A − − → B ⇔ − − Translation tables C/D A ⊗ B B ⊕ A D � C C/B A ⊗ B A \ C ⊲ ⊳ ∞ D \ C B ⊗ A A ⊕ B C ⊘ D 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 ⊳ symmetry. residual operations for the remaining two in the conclusion. Note: · ⊲ A ⊗ B → C ⊕ D A ⊗ B → C ⊕ D C � A → D / B B ⊘ D → A \ C A ⊗ B → C ⊕ D A ⊗ B → C ⊕ D C � B → A \ 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 = { x | ∀ y ( Sxy ⇒ y ∈ A ) } � ′ A = { y | ∀ x ( Rxy ⇒ x ∈ A ) } � ′ A = { y | ∃ x ( Sxy ∧ x ∈ A ) }
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