Formalized Lambek Calculus in Higher Order Logic (HOL4)

TL;DR

This paper formalizes Lambek Calculus in HOL4, translating from Coq, defining multiple deduction systems, proving their equivalence, and establishing foundational theorems to support language parsing applications.

Contribution

It introduces a comprehensive formalization of Lambek Calculus in HOL4, including multiple deduction systems and their equivalence, with improvements over previous Coq implementations.

Findings

Formalization of three deduction systems in HOL4

Proof of equivalence between systems

Establishment of sub-formula properties in cut-free proofs

Abstract

In this project, a rather complete proof-theoretical formalization of Lambek Calculus (non-associative with arbitrary extensions) has been ported from Coq proof assistent to HOL4 theorem prover, with some improvements and new theorems. Three deduction systems (Syntactic Calculus, Natural Deduction and Sequent Calculus) of Lambek Calculus are defined with many related theorems proved. The equivalance between these systems are formally proved. Finally, a formalization of Sequent Calculus proofs (where Coq has built-in supports) has been designed and implemented in HOL4. Some basic results including the sub-formula properties of the so-called "cut-free" proofs are formally proved. This work can be considered as the preliminary work towards a language parser based on category grammars which is not multimodal but still has ability to support context-sensitive languages through customized extensions.