Cypher is Turing-Complete: A Formal Proof via 2-Counter Machine Simulation

Pierre Halftermeyer

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arXiv · 2605.18757

Cypher is Turing-Complete: A Formal Proof via 2-Counter Machine Simulation

Pierre Halftermeyer

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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.

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BibTeX RIS

Pierre Halftermeyer. 2026-03-24. Cypher is Turing-Complete: A Formal Proof via 2-Counter Machine Simulation. https://arxiv.org/abs/2605.18757

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