Symmetric categorial grammar Michael Moortgat British Logic Colloquium, Nottingham, Sept 2008
Abstract Fifty years ago, Jim Lambek published the seminal ”Mathematics of sentence structure”. In that paper, the familiar parts of speech (nouns, verbs, ...) take the form of formulas of a substructural logic; determining whether a phrase is well-formed, and assigning it an interpretation, i.e. parsing, can then be seen as a process of deduction in the grammar logic. The original Syntactic Calculus has turned out to be strictly context free, and it has an NP-complete decision problem. The goal of recent extensions of Lambek-style categorial type logics is to find a balance between expressivity and computational tractability: can one combine the ability to recognize patterns be- yond context-free with polynomial parsing algorithms? In the talk, I show how symmetric categorial grammar answers that challenge. In addition to the Lambek connectives (product, for phrasal composition, and resid- ual left and right division) one considers a dual family: coproduct with residual left and right difference operations. The two families interact via structure-preserving distributivity principles originally studied by Grishin in 1983. I discuss modelthe- oretic and prooftheoretic properties of the resulting Lambek-Grishin calculus. I show how its derivations can be given a proofs-as-programs interpretation in the continuation passing style.
1. The Lambek program from: Lambek 1958
2. Categorial grammar A categorial grammar in the tradition of Lambek consists of two components ◮ universal: syntactic calculus freely generated from a given set of basic types ◮ language-specific: a lexicon, associating each word with a finite number of types Language generated string w 1 · · · w n is assigned type B by a categorial grammar if ◮ there are A i such that ( w i , A i ) is in the lexicon, and ◮ there is a ⊗ -tree X with yield A 1 , . . . , A n such that X → B is derivable in the syntactic calculus. Expressivity, tractability Searching for the appropriate syntactic calculus, we want to find a balance between ◮ expressivity: ability to handle patterns beyond CF ◮ computational tractability: polynomial parsing problem
3. Outline of the talk In its original formulation, Lambek’s calculus falls short of the set aims. ◮ Lacking expressivity: L is strictly context-free (Pentus 1993) ◮ Computational complexity: L is NP-complete (Pentus 2006) I compare two strategies to address these problems. ◮ ♦ , � Modalities ⊲ Controlled structural rules ◮ ⊗ , ⊕ Symmetric categorial grammar (Grishin 1983) ⊲ Structure-preserving distributivity principles Both strategies combine mild CS expressivity with polynomial parsing (Moot 2008).
Part I. Structural control
4. Vocabulary The parts of speech are turned into ◮ formulas: logical perspective ◮ types: computational perspective p | A, B ::= atoms: s sentence, np noun phrase, . . . ♦ A | � A | features: key, lock A ⊗ B | A/B | B \ A fusion, right vs left selection
5. Structural semantics Modal logic: ‘logic of structures’. Logic of language: grammatical structures. ◮ Frames F = � W, R 2 ♦ , R 3 ⊗ � ⊲ W : ‘signs’, linguistic resources, expressions ⊲ R 3 ⊗ : ‘Merge’, grammatical composition ⊲ R 2 ♦ : ‘feature checking’, structural control ◮ Models M = � F, V � ◮ Valuation V : F �→ P ( W ) : types as sets of expressions Remark The language is purely modal — no Boolean operations.
6. Interpretation of the constants ⊗ and ♦ as existential multiplicative modalities; slashes and � as Inverse duality duals with respect to the rotations of R ⊗ and R ♦ x � ♦ A ∃ y.R ♦ xy and y � A iff y � � B ∀ x.R ♦ xy implies x � B iff x � A ⊗ B ∃ yz.R ⊗ xyz and y � A and z � B iff y � C/B ∀ xz. ( R ⊗ xyz and z � B ) implies x � C iff z � A \ C ∀ xy. ( R ⊗ xyz and y � A ) implies x � C iff For ♦ , � : � F � , [ P ] in minimal temporal logic; for ⊗ and its residuals: fusion Compare in relevant logics. p �→ � F � p [ P ] p �→ p � F � [ P ] p → p → [ P ] � F � p
7. The pure logic of residuation The minimal grammar logic is given by the preorder laws for derivability (reflexivity A → A ) and transitivity: from A → B and B → C , deduce A → C ), together with the residuation laws below. Residuation laws relating pairs of opposites (inverse duals): A → � B ( res-1 ) ♦ A → B iff ( res-l ) A ⊗ B → C iff A → C/B ( res-r ) A ⊗ B → C iff B → A \ C Completeness With no constraints on the interpretation of Merge/Check we have A → B is provable iff ∀ F, V, V ( A ) ⊆ V ( B ) Invariants The laws of the base logic hold no matter what the structural particularities of individual languages are
8. Emergence of grammatical notions Grammatical notions and their properties, rather than being postulated, emerge from the type structure. Some examples: ◮ Subcategorization, valency. Intransitive np \ s , transitive ( np \ s ) /np , ditransitive (( np \ s ) /np ) /np , etc ◮ Case. Subject s/ ( np \ s ) , direct object (( np \ s ) /np ) \ ( np \ s ) , prepositional object ( pp/np ) \ pp , etc ◮ Complements versus modifiers. Compare exocentric A/B with A � = B versus endocentric A/A categories. Optionality of the latter follows. ◮ Filler-gap dependencies (rudimentary!). Nested implications C/ ( A \ B ) signal withdrawal of a gap hypothesis A in a domain B .
9. Soving type equations Inducing the lexicon from structured data (Buszkowski/Penn, Kanazawa) SU ⊗ ( TV ⊗ OBJ ) Lewis likes Alice np ( np \ s ) /np np He likes Alice s/ ( np \ s ) ( np \ s ) /np np ( np \ s ) /np (( np \ s ) /np ) \ ( np \ s ) Lewis likes her np Who likes Alice? wh/ ( np \ s ) ( np \ s ) /np np Limitation One cannot reconcile semantic uniformity with structural disparity: Claudia ◦ (( lo ◦ presta ) ◦ a Fabio ) Claudia ◦ (( lo ◦ vuole ) ◦ ( prestare ◦ a Fabio )) Claudia ◦ ( vuole ◦ (( prestar ◦ lo ) ◦ a Fabio ))
10. Discontinuous dependencies In a term ( M λx A .N B ) C , which positions can the A hy- Restricted λ abstraction pothesis (gap) occupy? Two kinds of discontinuity problems: ◮ Extraction: syntactic displacement. Example: wh “movement”. Leopold knows what wh/ ( np ??? s ) Molly suggested np to Mulligan ◮ Infixation: non-local semantic construal. Examples: wh “in situ”; scope. Molly thinks someone s ???( np ??? s ) is cheating Needed Logical tools to express under what structural deformations the form-meaning correspondence is preserved.
11. Constants for structural control Instead of hard-wired options with a global effect ( ⊗ associativity, commutativity), languages use controlled structural reasoning, anchored in lexical type assignment. Residuated pair ♦ , � : ♦ A → B iff A → � B New forms of Structural modalities expressivity: ◮ Subtyping via ♦ � A → A → � ♦ A ◮ ♦ controlled structural rules: left versus right extraction ( wh : q/ ( s/ ♦ � np ) ) ( P 1) ♦ A ⊗ ( B ⊗ C ) → ( ♦ A ⊗ B ) ⊗ C ( C ⊗ B ) ⊗ ♦ A → C ⊗ ( B ⊗ ♦ A ) ( P 3) ( P 2) ♦ A ⊗ ( B ⊗ C ) → B ⊗ ( ♦ A ⊗ C ) ( C ⊗ B ) ⊗ ♦ A → ( C ⊗ ♦ A ) ⊗ B ( P 4) Embeddings The expressivity of LP (implication/fusion fragment of intuitionistic Linear Logic) is regained through embedding translations (Kurtonina/Moortgat 1997).
12. Structural modalities: some results ◮ PhD theses @ OTS ⊲ Kurtonina 1995, Frames & Labels. ⊲ Moot 2002, Proof Nets for Linguistic Analysis. ⊲ Bernardi 2002, Reasoning with Polarity in Categorial Type Logic. ⊲ Vermaat 2006, The logic of variation. A cross-linguistic account of wh- question formation in type logical grammar. ◮ Moortgat 1997, Categorial type logics. Handbook of Logic and Language, Chap- ter 2. Elsevier/MIT Press.
Part II. Symmetry
13. Lambek and Grishin Existing proposals to extend the syntactic calculus beyond CF model derivability as an asymmetric relation A 1 , . . . , A n → B (the “intuitionistic” restriction). Pure residua- tion logic plus non-logical axioms for structural flexibility. Complementary strategy We restore the symmetry. ◮ LG = symmetric NL + structure preserving distributivity principles ◮ Symmetry: A 1 ⊗ · · · ⊗ A n → B 1 ⊕ · · · ⊕ B m ◮ Distributivities: respecting word order and phrase structure LG stands for Lambek-Grishin calculus; it is based on Grishin 1983 ersion " Об одном обобщении системы Айдукевича – Ламбека " non-classical logics and formal systems , Nauka, Moscow 1983,
14. Symmetric CG: results so far ◮ Kripke relational semantics, soundness/completeness. Kurtonina & MM 2007. Decidability. MM 2007. ◮ Proof nets. Moot 2007, generalizing Moot & Puite 2002. ◮ Complexity. LG is is mildly context-sensitive, polynomially parseable. Moot 2008. ⊲ The result applies also to the ♦ controlled extraction postulates. ◮ Grouptheoretic characterization of LG type similarity. Moortgat & Pentus FG07. ◮ Continuation semantics (Moortgat & Bernardi WoLLIC07). More To Explore ESSLLI07 course materials at http://symcg.pbwiki.com
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