Symmetries in Natural Language Syntax and Semantics: The Lambek-Grishin Calculus

In this paper, we explore the Lambek-Grishin calculus LG: a symmetric version of categorial grammar based on the generalizations of Lambek calculus studied in Grishin [1]. The vocabulary of LG complements the Lambek product and its left and right residuals with a dual family of type-forming operations: coproduct, left and right difference. The two families interact by means of structure-preserving distributivity principles. We present an axiomatization of LG in the style of Curry's combinatory logic and establish its decidability. We discuss Kripke models and Curry-Howard interpretation for LG and characterize its notion of type similarity in comparison with the other categorial systems. From the linguistic point of view, we show that LG naturally accommodates non-local semantic construal and displacement -- phenomena that are problematic for the original Lambek calculi.

[1]  Clye ClY.  A BRIEF SURVEY OF FRAMES FOR THE LAMBEK CALCULUS , 2006 .

[2]  J. Michael Dunn,et al.  Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation, Implication, and Various Logical Operations , 1990, JELIA.

[3]  Hugo Herbelin,et al.  The duality of computation , 2000, ICFP '00.

[4]  Matteo Capelletti,et al.  Parsing with Structure Preserving Categorial Grammars , 2007 .

[5]  Gerard Allwein,et al.  Kripke models for linear logic , 1993, Journal of Symbolic Logic.

[6]  François Lamarche,et al.  Classical Non-Associative Lambek Calculus , 2002, Stud Logica.

[7]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[8]  Michael Moortgat.  Categorial Type Logics , 1997, Handbook of Logic and Language.

[9]  Michael Moortgat,et al.  Continuation Semantics for Symmetric Categorial Grammar , 2007, WoLLIC.

[10]  Haskell B. Curry,et al.  Some logical aspects of grammatical structure , 1961 .

[11]  Philippe de Groote,et al.  The Non-Associative Lambek Calculus with Product in Polynomial Time , 1999, TABLEAUX.

[12]  Joachim Lambek,et al.  On the Calculus of Syntactic Types , 1961 .

[13]  Willemien Katrien Vermaat,et al.  The logic of variation : A cross-linguistic account of wh-question formation , 2005 .

[14]  Michael Moortgat,et al.  Multimodal linguistic inference , 1995, J. Log. Lang. Inf..

[15]  Mati Pentus.  The conjoinability relation in Lambek calculus and linear logic , 1994, J. Log. Lang. Inf..

[16]  Richard Moot.  Proof nets for lingusitic analysis , 2002 .

[17]  Johan van Benthem,et al.  Handbook of Logic and Language , 1996 .

[18]  Mati Pentus,et al.  Lambek calculus is NP-complete , 2006, Theor. Comput. Sci..

[19]  Michael Moortgat,et al.  Categorial Type Logics , 1997, Handbook of Logic and Language.

[20]  Michael Moortgat,et al.  Relational Semantics for the Lambek-Grishin Calculus , 2009, MOL.

[21]  Roman Jakobson,et al.  Structure of Language and Its Mathematical Aspects , 1961 .

[22]  Roy Dyckhoff.  Automated Reasoning with Analytic Tableaux and Related Methods , 2000, Lecture Notes in Computer Science.

[23]  Michael Moortgat,et al.  Structural control , 1997 .

[24]  N. Kurtonina,et al.  Frames and Labels , 1995 .

[25]  Annie Foret,et al.  On the computation of joins for non associative Lambek categorial grammars , 2003 .

[26]  R. Bernardi.  Reasoning with Polarity in Categorial Type Logic , 2002 .

[27]  N. Kurtonina,et al.  Frames and Labels. A modal analysis of categorial inference , 1995 .

[28]  Maarten de Rijke,et al.  Specifying syntactic structures , 1997 .

[29]  Rajeev Goré,et al.  Substructural Logics on Display , 1998, Log. J. IGPL.

[30]  Gerd Wagner,et al.  Logics in AI , 1992, Lecture Notes in Computer Science.