By Richard Moot
This booklet is meant for college kids in machine technology, formal linguistics, mathematical common sense and to colleagues attracted to categorial grammars and their logical foundations. those lecture notes current categorial grammars as deductive platforms, within the process referred to as parsing-as-deduction, and the ebook comprises special proofs in their major houses. The papers are prepared in topical sections on AB grammars, Lambek’s syntactic calculus, Lambek calculus and montague grammar, non-associative Lambek calculus, multimodal Lambek calculus, Lambek calculus, linear common sense and facts nets and evidence nets for the multimodal Lambek calculus.
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Extra resources for The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics
Example text
The redex e which is not reduced possesses a unique trace e¯ in the reduced proof d . Symmetrically if we reduce e the redex e which is not reduced possesses a unique trace e” ¯ in d . If in d we reduce e¯ we obtain a proof d but if in d we reduce e¯ we also obtain d . We will now show, by an easy induction on the proofs, that whenever a natural deduction is normal (that is without such configuration) each formula is a subformula of a free hypothesis or of the conclusion. More precisely. In order to establish this, let us introduce the notion of principal branch.
An C (sequents), with a comma-separated list of formulae (the antecedent, or the hypotheses of the statement) on the left hand side of the turnstile and a single formula on the right hand side (the succedent or the conclusion of the statement). Variables Γ , Δ , . . range over (possibly empty) lists of formulae which we will call contexts, so we can write a sequent as Γ C, or a sequent containing a formula A as Γ , A, Δ C. We will call the sequents above the horizontal line of a rule its premises and the single sequent below the horizontal line its conclusion.
2. 10? 3. 2. 6. 4. 7. Consider the following context-free grammar, which is already in Greibach normal form. S → aA S → bB S → aSA S → bSB A→a B→b 1. What is the language generated by this grammar? 2. Transform the grammar into an AB grammar. : Die syntaktische Konnexit¨at. : A quasi arithmetical notation for syntactic description. : On categorial and phrase-structure grammars. : Identification of reversible dependency tree languages. In: Popel´ınsk´y and Nepil, pp. : Uno studio sull’apprendibilit`a delle grammatiche di Lambek rigide — a study on learnability for rigid Lambek grammars.