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Specifying and Verifying the Convergence Stairs of the Collatz Program

Ali Ebnenasir

TL;DR

This work reframes the Collatz conjecture as a self-stabilization problem for a two-process concurrent program and introduces convergence stairs, a hierarchical, verifiable partition of natural numbers by the number of Collatz steps needed to reach an invariant. It defines the j-th stair $S_j$ as the set of numbers that reach the finite invariant $I_{cltz} = \{1,2,4\}$ (and also considers the unbounded invariant $\mathcal{I}_u$) in exactly $j$ steps, and provides algorithmic construction without full backward-tree expansion. A Binary Verification Code is attached to generated nodes to certify their validity, and a software tool implements the stair-generation approach. While the authors prove intra-subtree correctness and discuss complexity and parallelization, the overarching Collatz conjecture reduces to proving that $\bigcup_{j≥0} S_j = \mathbb{N}$, an open problem that invites further exploration of stair-structure completeness and global coverage.

Abstract

This paper presents an algorithmic method that, given a positive integer $j$, generates the $j$-th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly $j$ applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the $j$-th convergence stair for any $j > 0$. The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. To the best of our knowledge, this is the first time that such a general specification is provided, which can have significant applications in analyzing and testing the behaviors of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.

Specifying and Verifying the Convergence Stairs of the Collatz Program

TL;DR

This work reframes the Collatz conjecture as a self-stabilization problem for a two-process concurrent program and introduces convergence stairs, a hierarchical, verifiable partition of natural numbers by the number of Collatz steps needed to reach an invariant. It defines the j-th stair as the set of numbers that reach the finite invariant (and also considers the unbounded invariant ) in exactly steps, and provides algorithmic construction without full backward-tree expansion. A Binary Verification Code is attached to generated nodes to certify their validity, and a software tool implements the stair-generation approach. While the authors prove intra-subtree correctness and discuss complexity and parallelization, the overarching Collatz conjecture reduces to proving that , an open problem that invites further exploration of stair-structure completeness and global coverage.

Abstract

This paper presents an algorithmic method that, given a positive integer , generates the -th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the -th convergence stair for any . The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. To the best of our knowledge, this is the first time that such a general specification is provided, which can have significant applications in analyzing and testing the behaviors of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.
Paper Structure (7 sections, 15 theorems, 1 equation, 3 figures, 2 algorithms)

This paper contains 7 sections, 15 theorems, 1 equation, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Collatz Program, Its Invariant and the Verification Problem
  3. Convergence Stairs of the Collatz Program
  4. Convergence Stairs With Respect to $I_{cltz}$
  5. Convergence Stairs With Respect to $\mathcal{I}_u$
  6. Related Work
  7. Conclusions and Future Work

Key Result

Lemma 2

All values in stair $k+1$ are backward reachable by $R(x)$ from all values in the $k$-th stair, where $k \geq 1$. That is, for any value $y$ in the $(k+1)$-th stair, there is some value $x$ in the $k$-th stair such that $y \in R(x)$. (Proof straightforward; hence omitted.)

Figures (3)

  • Figure 1: Collatz computation tree and convergence stairs with respect to $I_{cltz}$.
  • Figure 2: Collatz computation tree and convergence stairs with respect to $\mathcal{I}_u = \{1, 2, 4\} \cup \{2^k \mid k \in \mathcal{N} \wedge (k > 2) \}$. Green states represent Stair 1, Blue states capture Stair 2, and Red states illustrate Stair 3 with respect to $\mathcal{I}_u$.
  • Figure 3: Structure of a subtree rooted at $Y_k/3$, labelled with binary verification codes.

Theorems & Definitions (27)

  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • Lemma 10
  • ...and 17 more