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**1 - 7**of**7**### A Savateev-style parsing algorithm for pregroup grammars, manuscript

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### Parsing pregroup grammars in polynomial time

"... Abstract—We consider polynomial time recognition algorithm and parsing procedures for pregroup grammars. In particular, we present a cubic parsing algorithm for ambiguous pregroup grammars. It modifies the recognition algorithm of Savateev [13] for categorial grammars based on L \. We present a Java ..."

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Abstract—We consider polynomial time recognition algorithm and parsing procedures for pregroup grammars. In particular, we present a cubic parsing algorithm for ambiguous pregroup grammars. It modifies the recognition algorithm of Savateev [13] for categorial grammars based on L \. We present a Java application that uses the algorithm for parsing natural language sentences. We apply metarules to keep the lexicon reasonably small. I. INTRODUCTION AND PRELIMINARIES PREGROUPS were introduced by Lambek [8] as an algebraic tool for the syntactical analysis of sentences. The idea is that syntactical properties of words can be described by a finite set of pregroup types. A string of words is assigned

### Pregroup Grammars with Letter Promotions

"... We study pregroup grammars with letter promotions p(m) ⇒ q(n). We show that the Letter Promotion Problem for pregroups is solvable in polynomial time, if the size of p(n) is counted as |n | + 1. In Mater and Fix [9], the problem is shown NP-complete, but that proof assumes the binary (or decimal, e ..."

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We study pregroup grammars with letter promotions p(m) ⇒ q(n). We show that the Letter Promotion Problem for pregroups is solvable in polynomial time, if the size of p(n) is counted as |n | + 1. In Mater and Fix [9], the problem is shown NP-complete, but that proof assumes the binary (or decimal, etc.) representation of n in p(n), which seems less natural for applications. We reduce the problem to a graph-theoretic one, and the latter to the emptiness problem for context-free languages. As a consequence, the word problem for pregroups with letter promotions is polynomial time decidable, and similarly for the membership problem for pregroup grammars with letter promotions. 1 Introduction and

### Letter Promotion Problem in Pregroups

"... We show that Letter Promotion Problem in pregroups is solvable in polynomial time, if one represents p(n) as pl...l or pr...r. The problem was formulated in [6] with a proof of its NP-completeness, but this proof assumed that n in p(n) is represented as a binary string. We reduce the problem to a gr ..."

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We show that Letter Promotion Problem in pregroups is solvable in polynomial time, if one represents p(n) as pl...l or pr...r. The problem was formulated in [6] with a proof of its NP-completeness, but this proof assumed that n in p(n) is represented as a binary string. We reduce the problem to a graph-theoretic problem and show that the latter problem is P-TIME, using some standard algorithms of formal language theory. Pregroups, introduced in Lambek [4], are partially ordered monoids with two unary operations l, r, fulfilling the following conditions: ala ≤ 1 ≤ aal, aar ≤ 1 ≤ ara, (1) for all elements a. The element al (resp. ar) is called the left (resp. right) adjoint of a. Pregroups are special algebras of Bilinear Logic (Noncommutative MLL) in which ⊗ and ⊕, whence 0 and 1, collapse. The logic of pregroups is called Compact Bilinear Logic [4]. CBL can be formalized as follows. (P,≤) is a nonempty finite poset. Ele-

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"... Categorial Grammar comprises a family of lexicalized theories of grammar characterized by very tight coupling of syntactic derivation and semantic composition, having their origin in the work of Frege. Some ver-sions of CG have extremely restricted expressive power, corresponding to the smallest kno ..."

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Categorial Grammar comprises a family of lexicalized theories of grammar characterized by very tight coupling of syntactic derivation and semantic composition, having their origin in the work of Frege. Some ver-sions of CG have extremely restricted expressive power, corresponding to the smallest known natural family of formal languages that properly in-cludes the context-free. Nevertheless, they are also strongly adequate to the capture of a wide range of cross-linguistically attested non-context-free constructions. For these reasons, categorial grammars have been quite widely applied, not only to linguistic analysis of challenging phenomena such as coordination and unbounded dependency, but to computational lin-guistics and psycholinguistic modeling. 1. INTRODUCTION. Categorial Grammar (CG) is a “strictly ” lexicalized theory of natural language grammar, in which the linear order of constituents and their interpre-tation in the sentences of a language are entirely defined by the lexical entries for the

### Cyclic Pregroups and Natural Language: a Computational Algebraic Analysis

"... Abstract. The calculus of pregroups is introduced by Lambek [1999] as an algebraic computational system for the grammatical analysis of natural languages. Pregroups are non commutative structures, but the syntax of natural languages shows a diffuse presence of cyclic patterns exhibited in different ..."

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Abstract. The calculus of pregroups is introduced by Lambek [1999] as an algebraic computational system for the grammatical analysis of natural languages. Pregroups are non commutative structures, but the syntax of natural languages shows a diffuse presence of cyclic patterns exhibited in different kinds of word order changes. The need of cyclic operations or transformations was envisaged both by Z. Harris and N. Chomsky, in the framework of generative transformational grammar. In this paper we propose an extension of the calculus of pregroups by introducing appropriate cyclic rules that will allow the grammar to formally analyze and compute word order and movement phenomena in different languages such as Persian, French, Italian, Dutch and Hungarian. This cross-linguistic analysis, although necessarily limited and not at all exhaustive, will allow the reader to grasp the essentials of a pregroup grammar, with particular reference to its straightforward way of computing linguistic information. 1