10,100 Pages

# LittlePeng9

## aka Penguin!

My favorite wikis
• I live in Poland
• I was born on May 23
• My occupation is http://youtu.be/sDj72zqZakE
• I am your local penguin
• ## Long hierarchies of functions (on my other blog)

October 31, 2017 by LittlePeng9

I thought this might be of interest to some of you, so I'm going to mention it here.

Yesterday I've put up a new blog post on my personal blog, which you can find here. It shows existence of hierarchies of functions somewhat akin to SGH which however have length exceeding $$\omega_1$$. To be more precise, I show existence of hierarchies of any length below $$\omega_2$$ in which functions higher up eventually dominate functions below them. There is also a section about computable functions containing such hierarchies of any length below $$\omega_1$$.

If you have any questions or comments feel free to post them here or on my blog.

• ## Hard-coding long binary strings

October 29, 2017 by LittlePeng9

Today I have discovered a way of hard-coding binary strings into Turing machines with asymptotically fewer states. More precisely, I've found the following:

For any string of $$2^k\cdot l$$ bits, there is a Turing machine with three colors (0, 1 and blank) and $$2^k+k+4\cdot 2^l+1$$ states which, on blank input, writes that string of bits and halts with its head on the first character of the string.

By simulating a three-color Turing machine with a two-color one and adding a few more steps to take care of deciphering the string, we can show the following (I don't bother with establishing the constants):

For any $$l$$ and large enough $$k$$ (depending on $$l$$), for any string of $$2^k\cdot l$$ there is a Turing machine with at most $$2^k$$ stâ€¦
• ## Fast growing hierarchy of analytic functions

June 12, 2017 by LittlePeng9

Extending the hyperoperators to all real or complex values is a known open problem. Some time ago I had a somewhat converse idea on how to approach this - instead of starting with a fast growing function(s) on natural numbers and extending it to more inputs, we can directly construct a sequence of fast growing functions defined for all complex values. I do not claim this is of as much interest as extending hyperoperators, but might nevertheless be interesting for some.

Suppose we have chosen a system of fundamental sequences for countable limit ordinals up to some bound. We now construct a sequence $$f_\alpha:\mathbb C\rightarrow\mathbb C$$ which satisfies the following propertes:

1. $$f_\alpha$$ is an ,