Formalized Lambek Calculus in Higher Order Logic (HOL4)
This incremental work provides a formal foundation for developing context-sensitive language parsers based on category grammars, targeting researchers in computational linguistics and theorem proving.
The authors formalized Lambek Calculus in the HOL4 theorem prover, porting and extending a Coq implementation to define three deduction systems and prove their equivalence, along with new theorems like sub-formula properties for cut-free proofs.
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.