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# DrCeasium

• ## The value of Rado's sigma function for 5-state TMs

September 2, 2017 by DrCeasium

Hi everyone, I'm back from the dead (again...), no promises about how long I'll stay active this time. I've seen some comments about FAN and stuff that people have made recently, you could well be right, I haven't really looked at any of my array stuff for quite a while.

Anyway, the reason for this post (as you can tell from the title) is to tentatively say I think I've proved all the hardly non-regular TMs run forever, so showing that . I'm still in the process of writing it up, and it's not finished yet; some of the 'proofs' are still just collections of ideas, but I'm pretty confident I can stick them all together.

The reason that I'm posting this before I've actually finished sorting all the proofs out is that hopefully, this will encour…

• ## HAN: multidimensional arrays

July 6, 2013 by DrCeasium

By request, I am going to write a blog post about how multidimensional arrays work (and the dreaded w(k)/ operator). If there is anything you feel could need a bit (or a lot) more work, tell me in the comments and i'll work on it. To start off with, we can look at the w(k)/ operator:

The w(k)/ operator is formally(ish) defined as follows:

• ◆ can be anything
• ◇ contains only 1's and separators
• ○ either starts with a separator or a ']'.
• ▮ represents any number of '['s
• ▲ any w/ chain
• ▼ an array with something before the first (k) divider
• ▽ an array without anything before the first (k) divider
• ▬ a string of ▽w(x)/'s (any x) and empty arrays ([1]s).
• R1: ◆[▬▽w(k)/[q◆]▲]◆ = ◆[▬[1(k)1(k)1(k)...(k)1(k)2▽]w(k)/[1◆]▲]◆, where there are q 1's. This rule means th…

• ## HAN: better definitions

June 8, 2013 by DrCeasium

I have decided to change the defintion of my notation for rows after the first one to make it a lot simpler, and only a little less powerful. The change is for all of the rows to now behave like the first one, meaning the limit of multidimensional arrays is now only the LVO (I shall put another post out to show this soon). This will actually change surprisingly little of the evaluation actually on the wiki, but some of my previous posts need to be re-done. Also, the apocalypxul no longer exists (see current numbers here). After a suggestion by FB100Z, I have replaced the @ symbols with unicode geometric shapes. The w(k)/ operator is still around with the same definition. These are the complete definitions:

Shapes

• ◆ can be anything
• ◇ contains…
• ## Hyperfactorial array notation: Analysis part 3

May 29, 2013 by DrCeasium

This has been corrected for new definitions. This post will look at extended hyperfactorial array notation. It is simply definied as [k@1[k+11]@2] = [k@1[k@1[k...[k@1[k@1@2]...]@2]@2] with n nests.  This is surprisingly powerful. As a small sub rule, if the type-k brackets don't exist, put them in around the type-k+1. This was designed to reach the TFB ordinal, and the type-k brackets work in pretty much the same way to $$\Omega_k$$. In the following comparisons I have put the type-k brackets in the second row just to kick start it a bit. Quite a lot of the evaluation relies on the type-k brackets doing to $$\Omega_k$$ exactly as type-1 brackets do to $$\omega$$. See the type-1 bracket comparisons here and here.

One thing I have noticed late…