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The \emph{Optimist}: Towards Fully Automated Graph Theory Research

Randy Davila

TL;DR

The Optimist is introduced, an autonomous system developed to advance automated conjecture generation in graph theory that generates conjectures that both rediscover established theorems and propose novel inequalities.

Abstract

This paper introduces the \emph{Optimist}, an autonomous system developed to advance automated conjecture generation in graph theory. Leveraging mixed-integer programming (MIP) and heuristic methods, the \emph{Optimist} generates conjectures that both rediscover established theorems and propose novel inequalities. Through a combination of memory-based computation and agent-like adaptability, the \emph{Optimist} iteratively refines its conjectures by integrating new data, enabling a feedback process with minimal human (\emph{or machine}) intervention. Initial experiments reveal the \emph{Optimist}'s potential to uncover foundational results in graph theory, as well as to produce conjectures of interest for future exploration. This work also outlines the \emph{Optimist}'s evolving integration with a counterpart agent, the \emph{Pessimist} (a human \emph{or machine} agent), to establish a dueling system that will drive fully automated graph theory research.

The \emph{Optimist}: Towards Fully Automated Graph Theory Research

TL;DR

The Optimist is introduced, an autonomous system developed to advance automated conjecture generation in graph theory that generates conjectures that both rediscover established theorems and propose novel inequalities.

Abstract

This paper introduces the \emph{Optimist}, an autonomous system developed to advance automated conjecture generation in graph theory. Leveraging mixed-integer programming (MIP) and heuristic methods, the \emph{Optimist} generates conjectures that both rediscover established theorems and propose novel inequalities. Through a combination of memory-based computation and agent-like adaptability, the \emph{Optimist} iteratively refines its conjectures by integrating new data, enabling a feedback process with minimal human (\emph{or machine}) intervention. Initial experiments reveal the \emph{Optimist}'s potential to uncover foundational results in graph theory, as well as to produce conjectures of interest for future exploration. This work also outlines the \emph{Optimist}'s evolving integration with a counterpart agent, the \emph{Pessimist} (a human \emph{or machine} agent), to establish a dueling system that will drive fully automated graph theory research.

Paper Structure

This paper contains 23 sections, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Fajtlowicz's Graffiti and the Dalmatian Heuristic
  4. Extensions of the Dalmatian Heuristic: Graffiti.pc and Conjecturing
  5. TxGraffiti
  6. The Optimist: An Automated Conjecturing Agent
  7. Initial Knowledge and Setup
  8. Constructing a Knowledge Base from Graph Invariants
  9. Mixed-Integer Programming for Conjecture Generation
  10. Systematic Generation of Conjectures
  11. The Hazel Heuristic
  12. The Morgan Heuristic
  13. The Strong-Smokey and Weak-Smokey Heuristics
  14. Conjecture Generation, Filtering, and Adaptive Knowledge Update
  15. Knowledge Update and Counterexample Handling
  16. ...and 8 more sections

Figures (4)

  • Figure 1: A diagram illustrating the Optimist agent's process for generating, filtering, and updating conjectures when targeting the invariant $\alpha$.
  • Figure 2: The initial set of graphs in Optimist’s knowledge base. (a) The complete graph $K_2$ (also the path $P_2$), (b) the complete graph $K_3$ (also the cycle graph $C_3$), (c) The path graph $P_3$.
  • Figure 3: The set of counterexamples introduced to the Optimist agent during the iterative feedback process of working with a human Pessimist -- notably some of the counterexamples being nontrivial (see graphs (c) and (e))
  • Figure 4: The feedback loop within the GraphMind framework, illustrating the iterative interaction between the Optimist (conjecture generation) and Pessimist (counterexample search) agents, and the refinement of the knowledge base.