Deriving Theorems in Implicational Linear Logic, Declaratively

Paul Tarau
(University of North Texas)
Valeria de Paiva
(Topos Institute)

The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable.

Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

In Francesco Ricca, Alessandra Russo, Sergio Greco, Nicola Leone, Alexander Artikis, Gerhard Friedrich, Paul Fodor, Angelika Kimmig, Francesca Lisi, Marco Maratea, Alessandra Mileo and Fabrizio Riguzzi: Proceedings 36th International Conference on Logic Programming (Technical Communications) (ICLP 2020), UNICAL, Rende (CS), Italy, 18-24th September 2020, Electronic Proceedings in Theoretical Computer Science 325, pp. 110–123.
Published: 19th September 2020.

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