The complexity of small universal Turing machines: a survey

TL;DR

This survey reviews recent advances in small universal Turing machines and cellular automata, highlighting efficient simulation capabilities and minimal program sizes, and discusses various models and future research directions.

Contribution

It compiles and analyzes recent results demonstrating efficient simulation and minimal size of small universal Turing machines and cellular automata, and explores diverse models and open questions.

Findings

Small universal Turing machines can simulate others efficiently.

Rule 110 cellular automaton is shown to be efficiently universal.

Minimal program size for universal machines has been improved.

Abstract

We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. In addition, another related result shows that Rule 110, a well-known elementary cellular automaton, is efficiently universal. We also discuss some old and new universal program size results, including the smallest known universal Turing machines. We finish the survey with results on generalised and restricted Turing machine models including machines with a periodic background on the tape (instead of a blank symbol), multiple tapes, multiple dimensions, and machines that never write to their tape. We then discuss some ideas for future work.