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Computing $\sqrt{2}$ with FRACTRAN

Khushi Kaushik, Tommy Murphy, David Weed

TL;DR

The paper addresses computing the decimal expansion of $sqrt{2}$ within the FRACTRAN framework by presenting two constructions: a Catalan-product-based program ($sqrt{2}$GAME) and a Newton–Raphson-based program (NR$sqrt{2}$GAME). It shows a simpler proof strategy that parallels Conway's PIGAME, yielding digit extraction with a clear truncation bound and a transferable approach to $pi$ via PIGAME. The results include a detailed flowchart-to-FRACTRAN translation and an error-analysis argument ensuring that the truncated approximations determine the correct digits. The work highlights how embedding standard numerical algorithms in FRACTRAN exposes a clean state-encoding and paves the way for more efficient or alternative encodings of classical computations within a minimalistic, Turing-complete language.

Abstract

The FRACTRAN programs $\sqrt{2}$GAME and NR$\sqrt{2}$GAME are presented, both of which compute the decimal expansion of $\sqrt{2}$. Our $\sqrt{2}$GAME is analogous to Conway's PIGAME program. In fact, our proof carries over to PIGAME to produce a simpler proof of Conway's theorem as well as highlight how the efficiency of the program can be improved. NR$\sqrt{2}$GAME encodes the canonical example of the Newton--Raphson method in FRACTRAN.

Computing $\sqrt{2}$ with FRACTRAN

TL;DR

The paper addresses computing the decimal expansion of within the FRACTRAN framework by presenting two constructions: a Catalan-product-based program (GAME) and a Newton–Raphson-based program (NRGAME). It shows a simpler proof strategy that parallels Conway's PIGAME, yielding digit extraction with a clear truncation bound and a transferable approach to via PIGAME. The results include a detailed flowchart-to-FRACTRAN translation and an error-analysis argument ensuring that the truncated approximations determine the correct digits. The work highlights how embedding standard numerical algorithms in FRACTRAN exposes a clean state-encoding and paves the way for more efficient or alternative encodings of classical computations within a minimalistic, Turing-complete language.

Abstract

The FRACTRAN programs GAME and NRGAME are presented, both of which compute the decimal expansion of . Our GAME is analogous to Conway's PIGAME program. In fact, our proof carries over to PIGAME to produce a simpler proof of Conway's theorem as well as highlight how the efficiency of the program can be improved. NRGAME encodes the canonical example of the Newton--Raphson method in FRACTRAN.

Paper Structure

This paper contains 9 sections, 3 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Rules of the game
  4. Initial Comments
  5. Conway's PIGAME
  6. Establishing Equation (\ref{['e1']})
  7. $\sqrt{2}$GAME
  8. NRsqrt2 game
  9. Description

Key Result

Theorem 3.1

(PIGAME conway) When started at $2^n\cdot89$, the FRACTRAN code will terminate at $2^{\pi(n)}$, where $\pi(n)$ is the $n$-th digit in the decimal expansion of $\pi$.

Figures (5)

  • Figure 1: Hierarchy of arrows
  • Figure 2: A flow-chart for addition
  • Figure 3: The flowchart for PIGAME
  • Figure 4: The flowchart for $\sqrt{2}$GAME
  • Figure 5: The flowchart for NR$\sqrt{2}$GAME

Theorems & Definitions (6)

  • Theorem 3.1
  • Remark
  • Remark
  • Lemma 4.1
  • Remark
  • Lemma 6.1