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  • 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…

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  • 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…
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  • 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…

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  • DrCeasium

    This is a continuation of the work of Cloudy176 here. the w/ operator means that m...m in the array before it is evaluated as (the value to the right) repeats of m. For example, [1...1,2]w/[1] = [1...1,2]w/n will have n 1's in the first array. If they are chained together, they are solved from right to left. Correct it when needed:



    Hyperfactorial array (without the n!) FGH ordinal
    [1]w/[1] 1,1,1,1,2]].
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  • DrCeasium

    Ignore this blog post: I realised just after publication that this has been done before. Twice. And it was really nothing new.

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