First-Order Intuitionistic Linear Logic and Hypergraph Languages

2025-02-09

AI Summary

This work investigates deep connections between first-order intuitionistic linear logic (ILL1/MILL1) and hypergraph grammar theory. To address this, we introduce the *first-order hypergraph logical categorial grammar*, construct an MILL1-based hypergraph language model grounded in HR-algebra, and establish completeness for its universal-implicational fragment. We resolve Moot’s (2014) open problem by proving that string-based MILL1 grammars generate non-semilinear and NP-complete languages. We further show that MILL1 grammars are strictly more expressive than Lambek categorial grammars over strings, yet equivalent to linear-time hypergraph transformation systems over hypergraphs. Finally, we prove that hypergraph ILL1 grammars precisely characterize the class of recursively enumerable hypergraph languages. These results unify logical semantics, categorial grammar, and hypergraph language generation, yielding a novel semantic framework for formal languages and structured reasoning.

Technology Category

Knowledge Representation and Reasoning: Knowledge Representation LanguagesMachine Learning: Statistical Relational/Logic LearningNatural Language Processing: Lexical Semantics and Morphology

Application Category

Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphsSemantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The Lambek calculus is a substructural logic known to be closely related to the formal language theory: on the one hand, it is used for generating formal languages by means of categorial grammars and, on the other hand, it is sound and complete with respect to formal language semantics. This paper studies a similar relation between first-order intuitionistic linear logic ILL1 along with its multiplicative fragment MILL1 and the hypergraph grammar theory. In the first part, we introduce a novel concept of hypergraph first-order logic categorial grammar, which is a generalisation of string MILL1 grammars studied in (Moot, 2014). We prove that hypergraph ILL1 grammars generate all recursively enumerable hypergraph languages and that hypergraph MILL1 grammars are as powerful as linear-time hypergraph transformation systems. Using these results, we solve an open problem from the article (Moot, 2014), which asks whether string MILL1 grammars generate exactly multiple context-free languages. We show that the class of languages generated by string MILL1 grammars is closed under intersection and that it includes a non-semilinar language as well as an NP-complete one. This shows how powerful string MILL1 grammars are as compared to Lambek categorial grammars. In the second part, we develop hypergraph language models for MILL1. In such models, formulae of the logic are interpreted as hypergraph languages and multiplicative conjunction is interpreted using parallel composition, which is one of the operations of HR-algebras introduced by Courcelle. We prove completeness of the universal-implicative fragment of MILL1 with respect to these models and thus present a new kind of semantics for a fragment of first-order linear logic.

Problem

• Explores ILL1 and MILL1 in hypergraph grammars.
• Solves open problem on string MILL1 grammars.
• Develops hypergraph language models for MILL1.

Innovation

• Hypergraph first-order logic categorial grammar
• Hypergraph ILL1 grammars generate languages
• Hypergraph language models for MILL1