Simulation of the Collatz 3x+1 function by Turing machines

TL;DR

This paper introduces new Turing machines that efficiently simulate the Collatz 3x+1 function, including a non-halting machine with fewer states and symbols, and machines that halt on the final loop, advancing computational models of the problem.

Contribution

It presents the first known small Turing machines that simulate the Collatz function, including a non-halting machine with 3 states and 4 symbols, and halting machines with various configurations.

Findings

A 3-state, 4-symbol Turing machine that never halts.

Turing machines that halt on the Collatz final loop with minimal states and symbols.

Improved bounds on Turing machine complexity for Collatz simulation.

Abstract

We give new Turing machines that simulate the iteration of the Collatz 3x+1 function. First, a never halting Turing machine with 3 states and 4 symbols, improving the known 3x5 and 4x4 Turing machines. Second, Turing machines that halt on the final loop, in the classes 3x10, 4x6, 5x4, and 13x2.