Language Models for Some Extensions of the Lambek Calculus

07/31/2020
by   Max Kanovich, et al.
0

We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exissts a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/01/2020

Complexity of the Infinitary Lambek Calculus with Kleene Star

We consider the Lambek calculus, or non-commutative multiplicative intui...
research
11/17/2017

Eliminating the unit constant in the Lambek calculus with brackets

We present a translation of the Lambek calculus with brackets and the un...
research
04/14/2022

Normalization by Evaluation for the Lambek Calculus

The syntactic calculus of Lambek is a deductive system for the multiplic...
research
09/22/2017

On the Recognizing Power of the Lambek Calculus with Brackets

Every language recognized by the Lambek calculus with brackets is contex...
research
05/23/2021

Diagrammatic Polyhedral Algebra

We extend the theory of Interacting Hopf algebras with an order primitiv...
research
07/31/2020

The Multiplicative-Additive Lambek Calculus with Subexponential and Bracket Modalities

We give a proof-theoretic and algorithmic complexity analysis for system...
research
02/07/2020

Intuitionistic Euler-Venn Diagrams (extended)

We present an intuitionistic interpretation of Euler-Venn diagrams with ...

Please sign up or login with your details

Forgot password? Click here to reset