Hypergraph formalisms · Széchenyi István University
HyMeKo
A language, a machine and a field formalism for knowledge that is genuinely n-ary.
Graphs and trees encode a relation among three or more things by inventing an intermediate node. HyMeKo takes the signed, directed hyperedge as its primitive — and rests on a computational model and a geometric formalism built for exactly that.
Hover a ring
The HyMeKo Mandala — the eleven levels of G-SPHF drawn as concentric incidence rings.
−50%
Smaller than URDF
kinematic + dynamic models, Applied Mechanics 2025
−20%
Smaller than MuJoCo-XML
same benchmark set, no loss of semantics
O(1)
Label-directed lookup
in k-ary relations; linear in Schönhage's SMM
15k
Lines of Rust
LALR(1) compiler, templates, tensor export
01 · The problem
Flat formats pay a tax on every n-ary relation
URDF, SDF, MuJoCo-XML, RDF/OWL — each of them can only say that three or more things belong together by inventing an intermediate node and a fan of binary links.
That intermediate node is not in the physics. It is bookkeeping, and it multiplies: a closed kinematic chain, a tendon coupling four joints, a multi-physics constraint binding an electrical and a mechanical quantity — each one becomes a small subgraph whose meaning lives in a naming convention rather than in the structure.
HyMeKo makes the hyperedge primitive. One edge, n endpoints, with sign and direction, plus templates and inheritance so that a family of mechanisms is written once. The structural complexity analysis in the Acta Polytechnica Hungarica paper quantifies the gain against RDF reification; the Applied Mechanics benchmarks measure it on real robot models.
The models stay readable by large language models — the compression is in the structure, not in an opaque binary encoding.
Model size reduction
Reduction in model description size for the same mechanism, measured on simple robotic arms and a quarter-vehicle model (Applied Mechanics 6(4):74, 2025).
02 · The stack
Four layers, from field theory to a compiler
HyMeKo is the surface a user writes. Underneath it sits a machine model with a proven cost structure, and under that a geometric formalism that says what a hypergraph patch is in the first place.
Generalized Signed Patch Hypergraph Field
An eleven-level architecture built on the GGK kernel K = (B, G, µ, r) with axioms K1–K4 — representation-agnostic, replacing NURBS as the geometric substrate. Friedler’s P-graph axioms fall out as a constrained projection of it.
Hypergraph Storage Modification Machine
A nine-tuple extension of Schönhage’s SMM over signed directed hypergraphs: NAV, NEWV, NEWE, ATT, DET, MOVC. Turing-complete, with SMM programs embeddable in its binary-deterministic subset.
The modeling language
A formal markup language with an LALR(1) grammar and a Rust workspace of roughly fifteen thousand lines. Template-based modeling, inheritance, and a tensor representation for numerical and learning pipelines.
Robotics, CPS, machine memory
Kinematic and dynamic structures with closed chains and tendon drives; state representation for industrial robots; hypergraph-backed memory for AI systems in place of flat vector stores.
03 · Results so far
What has been established
Three kinds of result: theorems about the formalism, measured performance, and hardware and tooling that runs.
Theory
- Turing completenessHSMM is Turing-complete; SMM programs translate into its binary-deterministic subset.
- Expressiveness hierarchyTransformer ⊂ xLSTM ⊂ HSMM, placing sequence models inside the hypergraph machine.
- P-graph as a projectionFriedler's canonical axioms recovered as a constrained projection of G-SPHF.
- GMDH bridgeIvakhnenko's 1968 self-organizing networks read as the first hypergraph neural networks.
- Temporal logicsHTL, HSTL and hypergraph hybrid automata for structures that change over time.
Measured
- Constant-time lookupLabel-directed access in k-ary relations in O(1), against linear scanning in the original SMM.
- Model compressionHalf the size of URDF and a fifth smaller than MuJoCo-XML on the same mechanisms.
- Cheaper than reificationStructural complexity gain over RDF reification for n-ary relations, analysed formally.
Engineering
- Compiler≈15,000 lines of Rust: LALR(1) front end, template expansion, tensor export.
- Silicon822 lines of SystemVerilog RTL for an HSMM core targeting Zynq UltraScale+.
- Signed-incidence networksSignedKAN and IncidenceKAN — Kolmogorov–Arnold networks over incidence structures.
04 · Publications
Peer-reviewed record
The formalism is published in the open literature; preprints and reference implementations accompany it.
Dissertation1
2026
Hypergraph-Based Semantic Models in Cognitive and Robotic Systems
Cs. Hajdu · Széchenyi István University, Multidisciplinary Doctoral School of Engineering Sciences
10.15477/SZE.MMTDI.2026.008open access
Journal articles21
2026
HyMeKo Language
Cs. Hajdu, Á. B. Csapó · Acta Polytechnica Hungarica 23(5), 227–246
10.12700/aph.23.5.2026.5.12open access
2026
Hypergraph Formalism for Fuzzy Signature-Based Robot Environment Representation
A. M. Karadeniz, Cs. Hajdu, T. L. Kóczy · Journal of Artificial Intelligence and Soft Computing Research 16(1), 91–100
2026
Multi-Contextual State Representation for Industrial Robots: A Hypergraph-Based Modeling Framework
Z. Szilágyi, Cs. Hajdu, B. K. Farkas, P. Galambos, K. Széll · Technologies 14(6), 1–39
10.3390/technologies14060332open access
2026
Novel Middleware Framework for Integrating Extended Reality into Robotic Manufacturing Processes
Z. Szilágyi, Cs. Hajdu, K. Széll, P. Galambos · Journal of Manufacturing and Materials Processing 10(2), 46
10.3390/jmmp10020046open access
2025
Evaluating the Impact of Aggregation Operators on Fuzzy Signatures for Robot Path Planning
A. M. Karadeniz, Cs. Hajdu, Á. Ballagi, T. L. Kóczy · Sensors 25(23)
10.3390/s25237342open access
2025
Mobile Robot Environment Representation Through Fuzzy Signatures-Integrated Quadtrees
A. M. Karadeniz, Cs. Hajdu, T. L. Kóczy · Romanian Journal of Information Science and Technology 28(1), 103–116
2025
Modeling Kinematic and Dynamic Structures with Hypergraph-Based Formalism
Cs. Hajdu, N. Hegyi · Applied Mechanics 6(4), 74
2024
Az erdőtűz talajra gyakorolt hatásainak vizsgálata számítógépes szimulációval I. - Modellek vizsgálata
F. Hajdu, D. Beke, Cs. Hajdu, L. Környei, R. Kuti · Tűzvédelem 31(1), 5–8
2024
Comparison of different geometry trees in fire simulation
F. Hajdu, Cs. Hajdu, L. Környei, D. Beke, R. Kuti · Pollack Periodica: an International Journal for Engineering and Information Sciences 19(3), 34–39
10.1556/606.2024.01022open access
2024
Dataset creation for the artificial intelligence-based video analysis of Yellow Hungarian chicken flocks
É. Rampasek, Cs. Hajdu, B. Tüű-Szabó, K. Tempfli, L. Környei · Georgikon for Agriculture: a Multidisciplinary Journal in Agricultural Sciences 28(Suppl. 2), 145–152
2024
Erdőtűz talajra gyakorolt hatásainak vizsgálata számítógépes szimulációval II. - Nyári tűz
F. Hajdu, D. Beke, Cs. Hajdu, L. Környei, R. Kuti · Tűzvédelem 2024(2), 4–7
2024
Erdőtűz talajra gyakorolt hatásainak vizsgálata számítógépes szimulációval III. - Tavaszi és téli tűz
F. Hajdu, D. Beke, Cs. Hajdu, L. Környei, R. Kuti · Tűzvédelem 31(3), 6–10
2023
Analysis of Human-Robot Interactions as a Sustainability Factor
Cs. Hajdu, Á. B. Csapó · Chemical Engineering Transactions 107, 85–90
2023
Calibration Measurements and Computational Models of Sensors Used in Autonomous Vehicles
Cs. Hajdu, I. Lakatos · Periodica Polytechnica Transportation Engineering 51(3), 230–241
2023
Examination of effects of indoor fires on building structures and people
R. Kuti, G. Zólyomi, G. László, Cs. Hajdu, L. Környei, F. Hajdu · Heliyon 9(1), 1–14
10.1016/j.heliyon.2022.e12720open access
2023
Numerical Examination of a Forest Area Fire
F. Hajdu, Cs. Hajdu, D. Beke, L. Környei, R. Kuti · Chemical Engineering Transactions 107, 61–66
2023
Sustainability Indicators in Industrial Robotic Systems
Z. Szilágyi, Cs. Hajdu, Á. B. Csapó, K. Széll, P. Galambos · Chemical Engineering Transactions 107, 79–84
2021
Overview of Human-Machine Interaction of Pneumatically Actuated Industrial Exoskeletons
Cs. Hajdu, F. Hajdu, R. Kuti · Hidraulica 2021(3), 7–14
2020
LIDAR-based Collision-Free Space Estimation Approach
M. Unger, E. Horváth, Cs. Hajdu · Hungarian Journal of Industry and Chemistry 48(1), 25–31
10.33927/hjic-2020-05open access
2020
Theoretical background and application of multiple goal pursuit trajectory follower
E. Horváth, P. Kőrös, Cs. Hajdu, Á. Ballagi · Hungarian Journal of Industry and Chemistry 48(1), 11–17
10.33927/hjic-2020-03open access
2018
Designing Complex Technical Rescues with a Proprietary Application (Computer Program)
Cs. Hajdu, R. Kuti · Academic and Applied Research in Military and Public Management Science 17(1), 45–52
10.32565/aarms.2018.1.5open access
Preprints1
2026
The Hypergraph Storage Modification Machine: A Computational Model for Signed Directed Hypergraphs
Cs. Hajdu, Á. B. Csapó, K. Kovács · Zenodo
10.5281/zenodo.19646224open access
Conference papers and book chapters are not listed here; the complete record is at MTMT (76) · ORCID
05 · Current directions
Where the work is going
Four open threads, from silicon to pedagogy.
HSMM as a portable ISA
A virtual instruction set for hypergraph memory, with an FPGA core underneath it — hypergraph navigation as a machine operation rather than a library call.
Autoholonomic training
A biologically motivated training scheme combining global pooling, entropy feedback and delayed backpropagation over signed incidence networks.
Hypergraph-backed machine memory
A Rust-native AI workspace whose memory is a typed hypergraph rather than a flat vector store, with HyMeKo as the interchange format.
A mathematical companion
A reference volume covering the prerequisites of G-SPHF end to end, from incidence algebra through to sheaf theory.
06 · Contact
Csaba Hajdu, PhD
ハイドゥ・チャバ
Széchenyi István University, Győr — Department of Informatics · Óbuda University, Alba Regia Faculty
Research stay 2026: Nagoya Institute of Technology, Kato Laboratory (two months, Pannónia Scholarship)
Collaboration, student projects and industrial pilots around hypergraph modeling are welcome. Write in any of the six languages on this page.
- ORCID
- 0000-0002-1058-7525
- MTMT
- 10067362
- ZENODO
- 10.5281/zenodo.19646224