Cypher is Turing-Complete: A Formal Proof via 2-Counter Machine Simulation
Computational Complexity
2026-05-20 v1 Formal Languages and Automata Theory
Logic in Computer Science
Programming Languages
Abstract
We prove that Cypher 25, the graph query language of Neo4j, is Turing-complete. The proof shows that a single RETURN statement using reduce(), CASE expressions, and list comprehensions can simulate any 2-counter machine (Minsky 1967). We address the bounded-step objection via two complementary resolutions and present a third graph-native simulation using quantified path patterns (QPP) with allReduce. All three constructions were verified on a live Neo4j Aura instance.
Cite
@article{arxiv.2605.18757,
title = {Cypher is Turing-Complete: A Formal Proof via 2-Counter Machine Simulation},
author = {Pierre Halftermeyer},
journal= {arXiv preprint arXiv:2605.18757},
year = {2026}
}
Comments
Submitted to the GRADES-NDA 2026 workshop (collocated with SIGMOD). Preprint available on HAL