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.

中文翻译：

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推导隐含线性逻辑中的定理，声明式

我们要解决的问题是如何在命题直觉线性逻辑的蕴涵片段中生成所有给定大小的定理。我们首先过滤线性关系的证明项，这些证明项与我们的基于 Prolog 的定理证明器相关联，用于蕴涵直觉逻辑。这是可行的，但是对于每个公式，使用 PSPACE 完整算法会将其限制为非常小的公式。我们在由 Curry-Howard 同构提供的证明项和定理之间的桥梁上来回走动，并逐步推导出一个有效的算法，每个生成的定理需要低多项式工作量。生成的 Prolog 程序在 O(N) 空间中运行，大小为 N，并在几个小时内生成 7,566,084,686 个线性直觉逻辑的蕴涵片段中的定​​理及其标准形式的证明项。作为应用程序，我们生成数据集用于线性逻辑定理证明器的正确性和可扩展性测试，以及用于应对定理证明挑战的神经网络的训练数据。论文中的结果被组织成一个有文化的 Prolog 程序，是完全可复制的。关键词：给定大小的可证明公式的组合生成，直觉和线性逻辑定理证明器，命题线性直觉逻辑的蕴涵片段定理，Curry-Howard 同构，有效生成范式中的线性 lambda 项，lambda 项的 Prolog 程序生成和定理证明。组织为一个有文化的 Prolog 程序，是完全可复制的。关键词：给定大小的可证明公式的组合生成，直觉和线性逻辑定理证明器，命题线性直觉逻辑的蕴涵片段定理，Curry-Howard 同构，有效生成范式中的线性 lambda 项，lambda 项的 Prolog 程序生成和定理证明。组织为一个有文化的 Prolog 程序，是完全可复制的。关键词：给定大小的可证明公式的组合生成，直觉和线性逻辑定理证明器，命题线性直觉逻辑的蕴涵片段定理，Curry-Howard 同构，有效生成范式中的线性 lambda 项，lambda 项的 Prolog 程序生成和定理证明。