ATAP · Axiomatic Theories, Algorithms and Proofs · Gary Welz · CUNY Graduate Center

Formal mathematical structure as visual graphs

Algorithms, axiomatic theories, and proofs expressed as labeled directed graphs using Mermaid Markdown — revealing structure that prose alone cannot show.

17,153
Mathematics papers in corpus

ATAP represents algorithms, axiomatic theories, and proofs as labeled directed graphs using Mermaid Markdown — a single graph-based representation across mathematically distant domains, from classical geometry and number theory to mathematical logic and theoretical computer science. The same representation that makes an algorithm's steps and branch points legible also makes a proof's justification structure and an axiom system's dependency chain legible, in the same visual language. That structural regularity — the same kind of graph doing the same kind of explanatory work across unrelated mathematical fields — is what the corpus is built to make visible.

The Euclidean Algorithm — one of the oldest algorithms in recorded mathematics (~300 BC)

graph TD A["Input: two integers a, b"] --> B{Is b = 0?} B -->|Yes| C["Return a — this is the GCD"] B -->|No| D["Compute remainder: r = a mod b"] D --> E["Set a = b, b = r"] E --> B style A fill:#ddd8f5,stroke:#3d2d8e,color:#000 style B fill:#e1bee7,stroke:#4a148c,color:#000 style C fill:#c8e6c9,stroke:#1b5e20,color:#000 style D fill:#ddd8f5,stroke:#3d2d8e,color:#000 style E fill:#ddd8f5,stroke:#3d2d8e,color:#000
Indigo = computation step Purple = decision Green = terminal result

What this is

ATAP represents algorithms, axiomatic theories, and proofs as labeled directed graphs using Mermaid Markdown. The same representation that makes an algorithm's steps legible also makes a proof's justification structure legible. Early encodings suggest regularity in algorithm capsules across distant domains — a preliminary pattern, not yet an established finding. The corpus spans classical geometry, number theory, algebra, set theory, mathematical logic, and theoretical computer science.

Browse and explore

Collection of Directed Graphs representing Axiomatic Theories, Algorithms and Proofs

Flowcharts of computational procedures, dependency graphs of logical structures, and proof graphs encoding justification structure — sorting and searching algorithms, Euclid's Elements, Peano arithmetic, ZFC set theory, group theory, category theory, the diagonalization family, incompleteness, and independence results.

Papers and resources

Papers and manuscripts

Resources

ATAP in the suite

ATAP is the other working engine of the CopernicusAI Knowledge Engine, beside GLMP. Same method, disjoint subject matter.