In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with nonstandard analysis in the context of (semi)intuitionistic realizability, focusing on the Lightstone-Robinson construction of a model for nonstandard analysis through an ultrapower. In particular, we consider an extension of the $\lambda$-calculus with a memory cell, that contains an integer (the state), in order to indicate in which slice of the ultrapower $\cal{M}^{\mathbb{N}}$ the computation is being done. We pay attention to the nonstandard principles (and their computational content) obtainable in this setting. In particular, we give non-trivial realizers to Idealization and a non-standard version of the LLPO principle. We then discuss how to quotient this product to mimic the Lightstone-Robinson construction.

@article{paperbot2221,
  title = {Stateful Realizers for Nonstandard Analysis},
  author = {Bruno Dinis and Étienne Miquey},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 19, Issue 2},
  year = {2023},
  doi = {10.46298/lmcs-19(2:7)2023}
}