Medvedev logic is undecidable
Abstract
We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct -- in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields -- for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes. The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5. A detailed methodology section outlines how such results were obtained.
Related
BibTeX
@misc{almeida2026medvedev,
title = {{Medvedev logic is undecidable}},
author = {Rodrigo Nicolau Almeida and Søren Brinck Knudstorp},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13359},
eprint = {2609.13359},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13359},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Medvedev logic is undecidable
AU - Rodrigo Nicolau Almeida
AU - Søren Brinck Knudstorp
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13359
UR - https://arxiv.org/abs/2609.13359
AB - We show that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable. The key method is a reduction from the periodic tiling problem to non-theoremhood in Medvedev's logic. This settles a longstanding open problem. Using similar techniques, but reducing instead to the ordinary tiling problem, we likewise obtain undecidability of Skvortsov's logic of infinite problems, and the fact that the two logics are distinct -- in fact, they are separated by any aperiodic tiling of the plane. Due to the fact that Medvedev's logic figures in so many different areas, these results have implications for several fields -- for example, the study of schematic fragments of logics such as propositional dependence logic, or the study of internal logics of toposes. The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5. A detailed methodology section outlines how such results were obtained.
ER -