By Glyn V. Morrill
This ebook offers a cutting-edge creation to categorial grammar, a kind of formal grammar which analyzes expressions as features or in keeping with a function-argument dating. The book's concentration is on linguistic, computational, and psycholinguistic features of logical categorial grammar, i.e. enriched Lambek Calculus. Glyn Morrill opens with the heritage and notation of Lambek Calculus and its software to syntax, semantics, and processing. Successive chapters expand the grammar to a few major syntactic and semantic houses of traditional language. the ultimate half applies Morrill's account to numerous present matters in processing and parsing, thought of from either a mental and a computational point of view. The e-book deals a rigorous and considerate research of 1 of the most strains of analysis within the formal and mathematical thought of grammar, and may be appropriate for college kids of linguistics and cognitive technology from complicated undergraduate point upwards.
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Additional info for Categorial Grammar: Logical Syntax, Semantics, and Processing
The ‚- and Á-proof reductions for conjunction are as shown in Figs. 6 respectively and the ‚- and Á-proof reductions for implication are as shown in Figs. 8 respectively. 5. 6. 7. 8. 3 Semantic readings of the Lambek calculus (derivational semantics) Let us call a (semantic) type map a mapping T from syntactic types to semantic types satisfying: (8) T (A\B) = T (A) → T (B) T (B/A) = T (A) → T (B) T (A•B) = T (A)&T (B) The semantic reading of a Lambek derivation is its reading as an intuitionistic proof under this homomorphism.
For example, for John loves Mary: (26) loves John (N\S)/N N\S N S Mary N E/ E\ For The cat slept we get one (normal form) natural deduction derivation, in contrast to the two Cut-free sequent derivations: (27) the cat N/CN CN N E/ S slept N\S E\ For I have set my bow in the cloud there is the natural deduction derivation given in Fig. 15. 6. Give natural deduction proofs of backward type lifting, backward composition, forward and backward division, and associativity and other laws such as those in Fig.
E. f. g. h. i. j. k. l. m. n. o. p. 3. Normalize the following lambda terms (Carpenter, 1996). a. (Îx(walk 1 (x, y)) a) b. (Îx(walk 2 (x, y)) a) c. , 1989) is that intuitionistic natural deduction and typed lambda calculus are isomorphic. This formulasas-types and proofs-as-programs correspondence exists at the following three levels: (7) intuitionistic natural deduction typed lambda calculus formulas: A→B A∧ B types: Ù 1 → Ù2 Ù1 &Ù2 proofs: E(limination of) → I(introduction of) → E(limination of) ∧ I(ntroduction of) ∧ terms: functional application functional abstraction projection ordered pair formation normalization: elimination of detours computation: lambda-reduction Overall, the laws of lambda-reduction are the same laws as the natural deduction proof normalizations of Prawitz (1965).