# What are the limits of computation in this universe?

I understand that Turing completeness requires unbounded memory and unbounded time.

However there is a finite amount of atoms in this universe, thus making memory bounded. For example even though $$\pi$$ is irrational there is no way to store more than a certain number of digits even if all the atoms in the universe were used for this purpose.

What then are the limits of computability of an implemented Turing machine (which could use all the resources of the universe but no more) based on the limits of the universe? What is the maximum number of digits of $$\pi$$? Are there any papers on this subject that might be interesting to read?

• Possibly related topic: Can a computer simulate itself as part of a simulated world. Feb 26 '11 at 8:32
• There is a fun essay by Scott Aaronson on this: scottaaronson.com/writings/finite.html Feb 26 '11 at 8:57
• – user4379
Mar 24 '11 at 23:37
• You might be interested in the discussion around this question. Yarosloav Bulatov posted a link to a popular-science version of the Lloyd paper Peter Shor linked to below, but, unfortunately, the link appears to be broken now. I read that paper at the time, but didn't save it, and don't remember the exact title. Mar 26 '11 at 17:11
• How do you know there is a bounded amount of atoms in the universe? Aug 17 '20 at 13:13