Formalising Type-Logical Grammars in Agda

TL;DR

This paper formalizes the Lambek-Grishin calculus in Agda, providing verified proofs and a translation to programs, enhancing the readability and practical application of formal type-logical grammar systems.

Contribution

It models the Lambek-Grishin calculus in Agda, implements a verified cut elimination procedure, and demonstrates a translation to Agda programs, improving formal proof clarity.

Findings

Verified cut elimination procedure in Lambek-Grishin calculus

A CPS translation from proofs to Agda programs

Application to analyze a simple sentence

Abstract

In recent years, the interest in using proof assistants to formalise and reason about mathematics and programming languages has grown. Type-logical grammars, being closely related to type theories and systems used in functional programming, are a perfect candidate to next apply this curiosity to. The advantages of using proof assistants is that they allow one to write formally verified proofs about one's type-logical systems, and that any theory, once implemented, can immediately be computed with. The downside is that in many cases the formal proofs are written as an afterthought, are incomplete, or use obtuse syntax. This makes it that the verified proofs are often much more difficult to read than the pen-and-paper proofs, and almost never directly published. In this paper, we will try to remedy that by example. Concretely, we use Agda to model the Lambek-Grishin calculus, a grammar logic with a rich vocabulary of type-forming operations. We then present a verified procedure for cut elimination in this system. Then we briefly outline a CPS translation from proofs in the Lambek-Grishin calculus to programs in Agda. And finally, we will put our system to use in the analysis of a simple example sentence.