History

“Hi-de-hi, ho-de-ho — discovery begins with a call, and an echo.” — Inspired by Cab Calloway, Minnie the Moocher (1931)

The dream of machines that conjecture like mathematicians has a long and fascinating history. TxGraffiti continues this tradition by building upon generations of symbolic, heuristic, and data-driven systems that have shaped the evolving field of automated mathematical discovery.

Origins: Symbolic and Heuristic Beginnings

  • Wang’s Program II (1959) One of the earliest documented attempts to generate mathematical statements automatically. It produced thousands of logical formulas but lacked mechanisms to identify meaningful ones.

  • Lenat’s AM (1976–77) Simulated mathematical creativity through hundreds of hand-coded heuristics. Rediscovered fundamental notions like primes, divisibility, and proposed versions of famous conjectures.

  • Epstein’s Graph Theorist (1980s) Focused on symbolic graph reasoning and the automation of proofs, using algebraic definitions and transformations to uncover property relationships.

The Graffiti Era

  • Fajtlowicz’s Graffiti (1980s–2000s) A foundational system that generated conjectures by evaluating inequalities on graph invariants. Introduced IRIN, CNCL, and the now-famous Dalmatian heuristic to refine results. Its conjectures led to dozens of published theorems.

  • DeLaViña’s Graffiti.pc (1990s–2010s) Extended Graffiti into a graphical environment for research and education. Introduced the Sophie heuristic for structural inference and integrated conjecture discovery into undergraduate learning.

Optimization and Geometry

  • AutoGraphiX (2000s) Recast conjecturing as an optimization problem, searching graph space using Variable Neighborhood Search to minimize or maximize target expressions.

  • GraPHedron & PHOEG Applied geometric and polyhedral analysis to identify conjecture boundaries, using convex hulls of graph invariant vectors to discover inequality facets.

The Modern Era: TxGraffiti and Beyond

  • TxGraffiti (2017–present) A hybrid system that merges linear optimization with heuristic filters. It generates inequalities over tabular data, ranks them by touch number, and applies Dalmatian-style pruning. TxGraffiti has independently rediscovered known theorems and produced novel open problems in graph theory, and now powers both a public Python package and an interactive web application.

Data-Driven and Neural Approaches

  • The Ramanujan Machine (2019–present) Uses symbolic expression search and numerical precision to conjecture continued fraction identities for mathematical constants like π and ζ(3).

  • Learning Algebraic Varieties (2018–present) Infers defining polynomials and geometric structure from sample points using tools from algebraic geometry and topological data analysis.

  • DeepMind’s Neural Mathematician (2021–present) Trains neural networks on mathematical datasets to predict invariants and discover new theorems in areas such as knot theory and representation theory.

A New Paradigm: Agent-Based Discovery

  • The Optimist–Pessimist Model (2024–present) Formalizes the interaction between conjecture generation (Optimist) and counterexample search (Pessimist). The Optimist agent (powered by TxGraffiti) generates inequalities with supporting heuristics. The Pessimist agent uses reinforcement learning to explore graph space and find violations, forming a feedback loop of empirical refinement.

Why It Matters

From Wang’s early logic engine to TxGraffiti’s optimization pipelines and DeepMind’s learning-guided insights, the landscape of automated conjecturing has evolved into a rich, multi-agent, multi-modal field.

Today’s systems no longer simply enumerate formulas — they dream, doubt, refine, and learn. TxGraffiti carries this legacy forward with modern tools for automated reasoning, bridging symbolic mathematics with interactive AI.