Abstract
This paper introduces the Linear Logic Recurrent Neural Network (LLRNN), a theoretical framework for sequence modeling that embeds the denotational semantics of linear logic into the operational dynamics of recurrent neural networks. The primary contribution is formal: we define a class of architectures in which the update operator applied to the hidden state is the denotation of a linear logic proof, and we establish that this construction yields smooth (differentiable) functions suitable for gradient-based optimization. We further show, through explicit mappings, that several established architectures, including second-order RNNs, multiplicative RNNs, and Neural Turing Machines, can be represented as special cases of the LLRNN framework. The paper also provides proof-of-concept examples demonstrating how logical constructs such as integer and binary integer types can generate structured memory access patterns. We emphasize that the current work is theoretical and illustrative: it does not report large-scale empirical results, and claims regarding practical performance enhancement remain hypotheses to be tested. The paper's contribution is to offer a principled, logic-based perspective on neural computation and to lay the groundwork for future empirical investigation.