A relatively small turing machine whose behavior is independent of set theory

Since the definition of the Busy Beaver function by Radó in 1962, an interesting open question has been what the smallest value of n for which BB(n) is independent of ZFC. Is this n approximately 10, or closer to 1,000,000, or is it unfathomably large? In this thesis, I show that it is at most 340,943 by presenting an explicit description of a 340,943-state Turing machine Z with 1 tape and a 2-symbol alphabet whose behavior cannot be proved in ZFC, assuming ZFC is consistent. The machine is based on work of Harvey Friedman on independent statements involving order-invariant graphs. Ill In doing so, I give the first known upper bound on the highest provable Busy Beaver number in ZFC. I also present an explicit description of a 7,902-state Turing machine G that halts if and only if there's a counterexample to Goldbach's conjecture, and an explicit description of a 36,146-state Turing machine R that halts if and only if the Riemann hypothesis is false. In the process of creating G, R, and Z, I define a higher-level language, TMD, which is much more convenient than direct state manipulation, and explain in great detail the process of compiling this language down to a Turing machine description. TMD is a well-documented language that is optimized for parsimony over efficiency. This makes TMD a uniquely useful tool for creating small Turing machines that encode mathematical statements.

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2015

Author dc:creator

• Yedidia, Adam

Advisor dc:contributor.advisor

• Scott Aaronson.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/100680

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/100680