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Automated conjecturing in mathematics with \emph{TxGraffiti}

Randy Davila

TL;DR

The design and core principles ofTxGraffiti are presented, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing, and the techniques demonstrated extend to other areas of mathematics.

Abstract

\emph{TxGraffiti} is a data-driven, heuristic-based computer program developed to automate the process of generating conjectures across various mathematical domains. Since its creation in 2017, \emph{TxGraffiti} has contributed to numerous mathematical publications, particularly in graph theory. In this paper, we present the design and core principles of \emph{TxGraffiti}, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing. We describe the data collection process, the generation of plausible conjectures, and methods such as the \emph{Dalmatian} heuristic for filtering out redundant or transitive conjectures. Additionally, we highlight its contributions to the mathematical literature and introduce a new web-based interface that allows users to explore conjectures interactively. While we focus on graph theory, the techniques demonstrated extend to other areas of mathematics.

Automated conjecturing in mathematics with \emph{TxGraffiti}

TL;DR

The design and core principles ofTxGraffiti are presented, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing, and the techniques demonstrated extend to other areas of mathematics.

Abstract

\emph{TxGraffiti} is a data-driven, heuristic-based computer program developed to automate the process of generating conjectures across various mathematical domains. Since its creation in 2017, \emph{TxGraffiti} has contributed to numerous mathematical publications, particularly in graph theory. In this paper, we present the design and core principles of \emph{TxGraffiti}, including its roots in the original \emph{Graffiti} program, which pioneered the automation of mathematical conjecturing. We describe the data collection process, the generation of plausible conjectures, and methods such as the \emph{Dalmatian} heuristic for filtering out redundant or transitive conjectures. Additionally, we highlight its contributions to the mathematical literature and introduce a new web-based interface that allows users to explore conjectures interactively. While we focus on graph theory, the techniques demonstrated extend to other areas of mathematics.
Paper Structure (20 sections, 1 theorem, 9 equations, 3 figures, 2 tables)

This paper contains 20 sections, 1 theorem, 9 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fajtlowicz's Graffiti
  3. Bingo!
  4. Graffiti.pc, Counterexamples, and Data Quality
  5. Other Notable Systems
  6. TxGraffiti Methods
  7. Generating Inequalities
  8. Limitations in the Forms of Inequalities Generated by TxGraffiti
  9. Sorting Conjectures
  10. The Static-Dalmatian Heuristic
  11. Full TxGraffiti Conjecture Generation
  12. Evaluating and Refining Conjectures in TxGraffiti
  13. Identifying Equality from Upper and Lower Bounds
  14. Pepper's Rule
  15. TicTac
  16. ...and 5 more sections

Key Result

Theorem 1

If $G$ is an $r$-regular graph with $r > 0$, then and this bound is sharp.

Figures (3)

  • Figure 1: A dataset of 9 connected graphs: ($G_1$) the path graph on 3 vertices, ($G_2$) the cycle graph on 3 vertices, ($G_3$) the cycle graph on 4 vertices, ($G_4$) the diamond graph, ($G_5$) the complete graph $K_4$, ($G_6$) the complete bipartite graph $K_{4,4}$, ($G_7$) the star graph $K_{1,3}$, ($G_8$) the double star graph $S(2,2)$, ($G_9$) and graph obtained by two triangles joined by an edge.
  • Figure 2: A plot of the independence number $\alpha$ vs matching number $\mu$ under 4 different boolean restrictions found in our dataset. The red lines show the solution to each linear optimization model presented in (\ref{['eq:model1']}).
  • Figure 3: A plot of the independence number $\alpha$ vs matching number $\mu$ under 4 different boolean restrictions found in our dataset. The red lines show the solution to each of the linear optimization models presented in (\ref{['eq:model2']}).

Theorems & Definitions (51)

  • Theorem 1: Caro et al. CaDaPe2020
  • Conjecture A1
  • Conjecture A2
  • Conjecture A3
  • Conjecture A4
  • Conjecture A5
  • Conjecture A6
  • Conjecture A7
  • Conjecture A8
  • Conjecture A9
  • ...and 41 more