Lambek's computational approach to conjugation

Abstract

“For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory... Jim Lambek’s ideas keep inspiring upcoming generations of scholars." (Festschrift on the occasion of Lambek’s 90th birthday[3]). This is an overview of the work of Joackim, Jim, Lambek on formal grammars for verb conjugation in several languages English, French, Latin, Turkish, Arabic, Hebrew and partly Serbian and Croatian. We will focus on his early work ([8, 9]), which has been further developed and extended by Lambek and his co-authors to Turkish, Arabicand (Biblical) Hebrew ([1, 2, 11]). Lambek’s approach was applied to Serbian and Croatian in [6] and to Japanese in [4].The Latin verb has 3×5×6 = 90 finite conjugational forms (inflected forms) corresponding to three patterns, five (simple) tenses and six persons. The production grammar given in Lambek [9] associates three matrices (patterns) of the Latin verb (present-active, perfect-active, present-passive) with 30 inflected forms (5 simpletenses × 6 persons). The French verb has 7×6 = 42 finite conjugational forms (inflected forms) corresponding to seven (simple) tenses and six persons. A production grammar of the French verb which generates the 42 inflected forms is presented in Lambek [8]. In Biblic Hebrew each verb has 7×2×10 = 140 finite conjugational forms corresponding to seven patterns, two tenses and ten persons. A production grammar that generates 140 inflected forms of the Biblic Hebrew verb is given in [11]. The Serbian and Croatian verb has 4×6 = 24 conjugational forms corresponding to four simple tenses and six person-numbers. there are two patterns, however they conjugate in the same way. As verb may also, as in Latin, be regarded as one-word sentences. A production grammar of Spanish presenting 54 forms of the Spanish verb is given in Mel’ˇcuk [12]. A study of Russian conjugation is given in Jakobson [7]. Lambek’s production grammar is a simple computational method for generating these conjugational forms step by step. The mathematical structure involved is the finitely generated partially ordered semi-group, also called “semi-Thue system" in mathematics, “rewriting system" in computer science and “production grammar" or Chomsky’s Type zero language ([5]) in linguistics. With each verb V, there is associated ap×n×mmatrix of conjugational verb-forms, Ckij(V). The index i= 1,...n represents the (simple) tense, and the index j= 1,...,m represents the person-number and the index k= 1,...,p represents the pattern. We shall only consider simple tenses here, and shall disregard participles and compound tenses. A production grammar, in general, provides a method for calculating Ckij(V) for a given (i,j,k,V). We shall present a simple production grammar developed in [6] for generating these 24 verb forms of Serbian and Croatian verb. This work was supported by the Social Sciences and Humanities Research council of Canada within a project on Mathematical Linguistics led by Jim Lambek during the spring of 1993 at McGill University, Montréal.