User blog:Bubby3/Difficulty of comparison of functions and notations.

This blog post talks about how hard it is to compare two functions. The difficulty is determined by two factors, which are how related to each other they are, and how strong tthe weakest of the two is. So, here is the scale

1/10 (Trivial): This is where the two notations are almost identical in form, but with some very minor differences, that don't affect how the function is evaulated, just only affecting how the function is displayed (Example: Kirby-Paris hydra and primative sequence system)

2/10 (Very easy): The two functions are still very similar, but some minor differences between them appear here that may affect the form of the function more significantly than a 1/10 would, and some contain things that change the behavior in a minor way. (Example: Kirby-paris hydra and Hardy-hierarchy below epsilon-zero)

3/10 (Easy): This level contains more siginficant behavioral differences between the two functions, but they only "shift" the ordinals in the construction, for example, adding one more omega to a power tower, or multlyplying by omega. (Example: Cascading-E vs HH)

4/10 (Moderate): At this level, the functions have less relation between them, with you having to make a transformation to make them alike, that is not trivial. (Example: R function linear array and FGH with inacessible cardinal OCFs)

5/10 (Challenging): At this level, functions don't look anything like each other, and take a very different form. Milestone ordinals in one notation may not necessarly corresponds with milestone ordinals in the other notation, and making transformation rules will be very complicated. (Example: Hyperfactorial array notation and FGH)

6/10 (Difficult): At this point, therse functions get even more different to compare. This is like the last level of difficulty, but taken to a whole new level. Main sequence expressions in one notation often will not equal main sequence expressions in the other notation. Sometimes, multiple expressions will correspond to one ordinal, so you have to do some re-tracing your steps, but with another way to expression. (Example: Bucholoz hydra and FGH with OCFs)

7/10 (Very difficult): At this point, patterns show up, but they cease to hold at some point, which means that it is very easy to mess up your comparisons, so make sure you're comparisons are very detailed, and you don't make unjust extralopations, even those that seem obvious. At this level and above, some of the comparisons are unfinished, or even  unmade, so it is unknown how these functions compare. (Example: BMS above pair-sequences and FGH wtih OCFs)

8/10 (Insanely difficult): At this level, comparisons need to become VERY detailed, some very long expressions are shown, the comparisons don't make sense to a layman or even a person who knows quite a bit about both functions. A lot of the time, milestone epressions in one notation may correspond to expressions which aren't near a milestone expressions, (Example: SAN and Taranovsky's ordinal notation, particularly above \(C(C(\Omega_2,0),0)\)

9/10 (Extremely difficult): In these notations, there are many, possibly infintely many expressions which correspond to one ordinal. Unlike the lower levels, it is hard to find what path in one notation that is the strongest. A lot of the time, one of the functions acts like an uncomputable function, diagnolizing over a language and how many symbols. Analysis people have done only goes relatively low, or even non at all. (Example: CoC and any other notation)

10/10 (Hellish): These functions are nearly impossible to compare. They combine several elements from the lower levels of difficulty. Comparing these functions would be by far the biggest thing googology has ever done, taking a truly enormous amoount of work to even get to relatively low ordinals, let alone finish it. However, such comparison is highly worth it, since it would provide new insight into googology that was never gained before. (Examples: Currently none, but there might be some in the future)

Infinite/10 (Impossible): These functions are literally impossible to compare. These involve uncomputable functions where one function can't clearly simulate the other. Comparing these functions would involve calculating when Turing machines halt, or even higher in the uncomputable hierarchy. In order to compare these functions with an algorthim, we need infinite time, and then some. We won't be able to make comparisons beyond very low level expressions unless we gain some form of omnipotence. (Example: Comparing different types of busy beaver functions)