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An 'Experimental Mathematics' Approach to Stolarsky Interspersions via Automata Theory

Jeffrey Shallit

TL;DR

The paper presents an experimental mathematics framework for Stolarsky interspersions by combining Zeckendorf representations, finite automata, and the Walnut decision procedure. It demonstrates how to guess and verify automata that compute column entries, derive closed-form column formulas, and prove intricate combinatorial identities and Kimberling-style questions, often with significantly less effort than traditional proofs. The authors apply the method to the Wythoff, Stolarsky, and dual Wythoff arrays, as well as generalized interspersions like EFC and ESC, obtaining new results (e.g., classification sequences) and verified automata-driven proofs. The approach offers a versatile, verifiable toolkit for discovering and proving properties of interspersions, though it faces limitations when automata for certain periodic classifications are elusive or when state counts grow large, inviting further theoretical development and broader exploration.

Abstract

We look at the Stolarsky interspersions (such as the Wythoff array) one more time, this time using tools from automata theory. These tools allow easy verification of many of the published results on these arrays, as well as proofs of new results.

An 'Experimental Mathematics' Approach to Stolarsky Interspersions via Automata Theory

TL;DR

The paper presents an experimental mathematics framework for Stolarsky interspersions by combining Zeckendorf representations, finite automata, and the Walnut decision procedure. It demonstrates how to guess and verify automata that compute column entries, derive closed-form column formulas, and prove intricate combinatorial identities and Kimberling-style questions, often with significantly less effort than traditional proofs. The authors apply the method to the Wythoff, Stolarsky, and dual Wythoff arrays, as well as generalized interspersions like EFC and ESC, obtaining new results (e.g., classification sequences) and verified automata-driven proofs. The approach offers a versatile, verifiable toolkit for discovering and proving properties of interspersions, though it faces limitations when automata for certain periodic classifications are elusive or when state counts grow large, inviting further theoretical development and broader exploration.

Abstract

We look at the Stolarsky interspersions (such as the Wythoff array) one more time, this time using tools from automata theory. These tools allow easy verification of many of the published results on these arrays, as well as proofs of new results.

Paper Structure

This paper contains 11 sections, 12 theorems, 13 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Zeckendorf representation
  3. Automata, logic, and Walnut
  4. The experimental approach to Stolarsky interspersions
  5. The Wythoff array
  6. The Stolarsky array
  7. The dual Wythoff array
  8. More general interspersions
  9. The EFC array
  10. The ESC array
  11. Final remarks

Key Result

Lemma 1

Suppose $(a)_F = x$ and $(b)_F = x0$. Then $(a+b)_F = x00$.

Figures (8)

  • Figure 1: Automaton accepting the set of odd natural numbers.
  • Figure 2: Automaton for the function $f(x) = 2x+1$.
  • Figure 3: Automaton for the shift.
  • Figure 4: Guessed automaton for the function $n \rightarrow W_{n,1}$.
  • Figure 5: Automaton computing the function $n \rightarrow W_{n,2}$.
  • ...and 3 more figures

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 13 more