A fundamental restriction: non empty antecedents a simple exercise a very simple exercise *a very exercise npnn nnnn n,nnnnnn/),//()/(,/,/,/ nnn... nnnn //,/ nn,,, npnn nnnn nnnn),//()/(/ //... nn /
What sequent calculus reveals to us… cf. classical logic (some rules) (note the symmetries)
but also: (on the two sides) + axiom and cut-rule Permutation Contraction Weakening
Lambek calculus = intuitionistic logic WITHOUT A, C, P Intuitionistic multiplicative linear logic (+ restriction on non empty antecedents)
subformula property AABB B/A A/B A\B B\A AB A B
le livre que Pierre lit (the book that Peter reads) lelivrequePierrelit
General properties of Lambek grammars Weak equivalence with CFGs : –A result by M. Pentus (1993) No strong equivalence with CFGs : –A result by H. J. Tiede (1998) Polynomiality? No result yet… –probably NP complete
Limitations They are numerous: –only peripheral extraction The girl who I met : OK The girl who I met yesterday (or on the beach) : not OK –coordination and polymorphic types The mathematician whom Gottlob admired and Kazimierz detested : OK *The mathematician whom Gottlob admired Jim and Kazimierz detested : also OK!
–parasitic gaps The book John filed _ without reading _ (linearity properties) –empty signs The book John read (cf. non empty antecedents)
Extensions Multimodal Categorial Grammar (Moortgat, Oehrle, Morrill and their students) –ref. Categorial Type Logics in Van Benthem and ter Meulen (HLL) see further…
A « cousin » Minimalist Grammars: –Inspired by Chomskys minimalist program –Ed. Stabler they have also type-logical formulations: –W. Vermaat –Retoré – Lecomte see further… or another day…
to sum up We get rid of « syntactic » rules… by means of a logic which accepts a natural deduction presentation (because intuitionist) –proofs are -terms and also a sequent calculus –convenient for the proof search a logic which is linear (resource sensitive)
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