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Evaluation of Analysis

P進大好きbot
P進大好きbot • 20 November 2018 • User blog:P進大好きbot

I introduce a way to evaluate an analysis in googology, e.g. explanation of the termination of a given computation process, the growth rate of a given function, and the correspondence from a given system of notations to ordinals.

I list two simple factors which affect the score of analysis in the following:

  1. If the analysis contains an unspecified or ill-defined mathematical stuff e.g. function, notation, or ordinal, then the explanation below its first appearance is worth nothing in this evaluation.
  2. If an analysis is just given as a table with finitely many examples without a complete definition of the correspondence from a given system of notations to ordinals, then the result is not reliable because it is really difficult for others to check the result.

This evaluation just scales the reproductibility of your analysis, and hence does not concern whether it is correct or not. On the other hand, it scales neither how difficult the analysis is nor how you much days and energy you devoted to the analysis. Therefore please do not feel upset by the evaluation of your great "analysis". Please tell me if you want me to add your analysis in this blog post.

For example, if you write analysis containing unspecified stuffs, then it is even not wrong, because you can reject any feedbacks by stating "The unspecified right hand side is just defined as the left hand side! All strings appearing there are just placeholders!" Therefore others have no way to judge whether it is correct or not. Arguments with ambiguity is often worse than incorrect arguments without ambiguity in this sense.

If you just write a table with finitely many examples without giving a complete definition of the correspondence from a given system of notations to ordinals, it is impossible for others to guess ordinals correspinding to intermediate values. In that case, it is quite difficult to point out any contradiction, even if you intentionally shift the table, because pointing out a contradiction requires an argument based on a proof, which requires a complete analysis up to the term causing the contradiction. In other words, pointing out a contradiction is much harder than writing a table. For example, even if you intentionally write a wrong table on terms corresponding to very big ordinals beyond experts' ability of analysis, it is essentially impossible to point out a contradiction. In order to ensure the reproducibility, you need to give a complete definition of the correspondence which is surjective onto a countable ordinal. Analysis based on a table without a full definition of the correspondence is not reproducible.


Level 0[]

  1. The target of the analysis is unspecified.


You are not even standing at a start point of analysis.


Fiction Example[]

I completely analysed BMS! See the following table! (omit)


Without specifying the version of BMS and (a link to) its definition, no one could understand your result. This is just an analysis-like whatchamacallit. This accident occurs if you are an innocent beginner who do not know the significance of the reproducibility.


Real Example[]


Level 1[]

  1. The target of the analysis is specified, but contains rough intuition-based description in the definition.


You might be ready for analysis, but the result is still non-sense.


Fiction Example 1[]

This is analysis of a new version of BMS! It expands like the system of weighted hydras of weighted hydras, whose weights are also nested weighted hydras. Here is an example of the expansion rule, which might be helpful to grasp a portion of the strength! (omit) The following is my analysis table! (omit)


You can cheat any results using unspecified rules. If someone point out something strange, then you can say "No, it does not work in that way! My function is obviously well-defined at any rate, if you know elementary arithmetic! Please read it more deeply!" Therefore it is far from the reproducibility of your result. This accident occurs if mathematical stuffs in the definition of the target is far beyond your comprehension.


Fiction Example 2[]

This is analysis of a new version of BMS! Ok, I give you a right to access a portion of the expansion rule! (omit) The following is my analysis table! (omit)


Maybe you have a full description of your algorithm. However, showing a portion of the definition is not sufficient for others to understand your result. That is why your result lacks the reproducibility. This is not an accident, but sometimes occurs if you need to hide the full definition in order to beat others in a googology competition.


Fiction Example 3[]

This is analysis of BM2! The following is my analysis table! (omit)


Declaring a version of BMS does not necessary tell others a common definition, because some of them are not defined in terms of algorithm or mathematics. (Especially, BM2 is not defined in terms of algorithm or mathematics. See more details on this issue here.) In order to avoid the ambiguity, write an explicit definition or an explicit link to it.


Real Examples[]


Level 2[]

  1. The target of the analysis is specified, and is actually defined in terms of mathematics, e.g. a full description of the algorithm to compute it.
  2. The description of the result of the analysis mainly consists of unspecified or ill-defined stuffs.


You might have done a good job, but other specialists could not precisely understand what you have actually done.


Fiction Example 1[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) The following is my analysis table!

matrix \(M\) ordinal \(\alpha\)
\(()\) \(0\)
\((0)\) \(\psi_0(0)\)
(omit) (omit)


Specify the meaning of \(\psi\) in this context, because there are many distinct \(\psi\)-functions. Such ambiguity prevent others checking your result. Therefore it lacks the reproducibility.


Fiction Example 2[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) The following is my analysis table, where \(\psi\) denotes the mixture of UNOCF and Bashicu's OCF (omit):


Using undefined stuffs such as UNOCF and Bashicu's OCF prevents others checking your results, because there is no common rule to expand associated notations. Therefore it lacks the reproducibility.


Real Examples[]


Level 3[]

  1. The target of the analysis is specified, and is actually defined in terms of mathematics, e.g. a full description of the algorithm to compute it.
  2. The description of the result of the analysis consists of well-defined stuffs.
  3. For analysis of a notation system, the description of a correspondence from terms to ordinals lacks a reasoning of the surjectivity in the case where the analysis includes a statement on the strength or lacks an explanation of the compatibility of the expansions in the case where the analysis includes a statement on the well-foundedness. For analysis of a large function, the estimation with a function hierarchy lacks a reasoning of inequalities.


Your result is written well without ambiguity so that other specialists can precisely understand what you have done, but there is no reason for them to believe that you are correct.


Fiction Example 1[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) The following is my analysis table! Here \(\psi\) denotes Buchholz's OCF!

matrix \(M\) ordinal \(\alpha\)
\(()\) \(0\)
\((0)\) \(\psi_0(0) = 1\)
\((0)(0)\) \(2\)
\((0)(0)(0)\) \(3\)
\((0)(1)\) \(\psi_0(1) = \omega\)
(omit) (omit)
\((0,0)(1,1)\) \(\psi_0(\psi_1(0)) = \psi_0(\Omega)\)
\((0,0)(1,1)(2,2)\) \(\psi_0(\psi_2(0)) = \psi_0(\Omega_2)\)
\((0,0)(1,1)(2,2)(3,3)\) \(\psi_0(\psi_3(0)) = \psi_0(\Omega_3)\)
\((0,0,0)(1,1,1)\) \(\psi_0(\psi_{\omega}(0)) = \psi_0(\Omega_{\omega})\)
(omit) (omit)


If an ordinal \(\alpha\) appears in a correspondence (restricted to the subsystem which gives the limit of the notation), you need to show the surjectivity below \(\alpha\). Here, \(\psi_0(\psi_{\omega}(0))\) appears in this table, while \(\psi_0(\psi_3(0) + \psi_2(\psi_3(0)) + \psi_1(\psi_3(0) + \psi_2(\psi_3(0)) + \psi_1(0)))\) does not. How could others understand the surjectivity between \(\psi_0(\psi_3(0))\) and \(\psi_0(\psi_{\omega})\)?
If you are talking about the surjectivity between \(3\) and \(\omega\), you might not need to care about such a problem, because you can easily write the correspondence restricted to the segment. For example, you can clearly say "\(n\) times \((0)\) corresponds to \(n\)". On the other hand, the surjectivity on segments containing ordinals described by OCFs are not so easy to write down.
It is actually a matter of degree, but it is obvious that you have no way to explain the surjectivity with one or two sentenses. If you completely know the algorithm which sends a term to the corresponding ordinal, then it suffices to just write it. Maybe even you do not have an idea to describe the algorithm to execute the correspondence, because you hesitate to write it down.
Then you just guessed the intermediate values of your undefined "correspondence". It heavily lacks the reproducibility, because even you are not able to interpolate the unwritten facts. In this case, you should go back to analyses of easier notations within your comprehension.


Fiction Example 2[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) The following is my analysis table! Here \(\psi\) denotes Madore's OCF!

matrix \(M\) ordinal \(\alpha\)
\((0,0,0)(1,1,1)\) \(\psi(\Omega)\)
\((0,0,0)(1,1,1)(0,0,0)\) \(\psi(\Omega) + 1\)
\((0,0,0)(1,1,1)(0,0,0)(0,0,0)\) \(\psi(\Omega) + 2\)
\((0,0,0)(1,1,1)\underbrace{(0,0,0) \cdots (0,0,0)}_{n}\) \(\psi(\Omega) + n\)
\((0,0,0)(1,1,1)(0,0,0)(1,0,0)\) \(\psi(\Omega) + \omega\)
(omit) (omit)


Why did you start from the intermediate value? It does not ensure the surjectivity below \(\psi(\Omega)\) unless you give a link to a previous result dealing with it.


Fiction Example 3[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) The following is my analysis table! Here \(\psi\) denotes my OCF defined in this way! (omit)

matrix \(M\) ordinal \(\alpha\)
\(()\) 0
(omit) (omit)
\((0,0,0)(1,1,1)\) \(\psi(\Omega)\)
\((0,0,0,0)(1,1,1,1)\) \(\textrm{PTO}(\textrm{KP} + \textrm{"existence of recursively huge cardinal"})\)
\((0,0,0,0,0)(1,1,1,1,1)\) \(\textrm{PTO}(Z_2)\)
\((0,0,0,0,0,0)(1,1,1,1,1,1)\) \(\textrm{PTO}(\textrm{ZFC})\)
(omit) (omit)


See the serious gaps on the surjectivity. You may laugh at this table, but the tables above are so bad as this.


Real Examples[]


Level 4[]

  1. The target of the analysis is specified, and is actually defined in terms of mathematics, e.g. a full description of the algorithm to compute it.
  2. The description of the result of the analysis consists of well-defined stuffs.
  3. For analysis of a notation system, the description of a correspondence from terms to ordinals contains an explanation of the surjectivity in the case where the analysis includes a statement on the strength, and contains an explanation of the compatibility of the expansions in the case where the analysis includes a statement on the well-foundedness. For analysis of a large function, the estimation with a function hierarchy contains a reasoning of inequalities.
  4. The analysis lacks a sufficient proof.


Your analysis is sufficiently reproducible so that other specialists can believe your results, but you need to write down a proof in order to deal with your assertion as a fact.


Fiction Example[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) A suspected limit is \((0,0)(1,1)\), which corresponds to \(\varepsilon_0\)! The following is my analysis table below \((0,0)(1,1)\)!

matrix \(M\) ordinal \(\alpha\)
\(()\) \(0\)
\((0)\) \(1\)
\((0)(0)\) \(2\)
\(\underbrace{(0) \cdots (0)}_n\) \(n\)
\((0)(1)\) \(\omega\)
\((0)(1)(0)\) \(\omega+1\)
\((0)(1) \underbrace{(0) \cdots (0)}_{n}\) \(\omega+n\)
\((0)(1)(0)(1)\) \(\omega \times 2\)
\(\underbrace{(0)(1) \cdots (0)(1)}_m \underbrace{(0) \cdots (0)}_n\) \(\omega \times m + n\)
\((0)(1)(1)\) \(\omega^2\)
\((0)(1)(1) \underbrace{(0)(1) \cdots (0)(1)}_m \underbrace{(0) \cdots (0)}_n\) \(\omega^2 + \omega \times m + n\)
\((0)(1)(1)(0)(1)(1)\) \(\omega^2 \times 2\)
\(\underbrace{(0)(1)(1) \cdots (0)(1)(1)}_{n_2} \underbrace{(0)(1) \cdots (0)(1)}_{n_1} \underbrace{(0) \cdots (0)}_{n_0}\) \(\omega^2 \times n_2 + \omega \times n_1 + n_0\)
\((0)(1)(1)(1)\) \(\omega^3\)
(omit) (omit)
\((0,0)(1,1)\) \(\varepsilon_0\)
(omit) (omit)


This is one example to state the surjectivity by filling all gaps by inserting oridnals described by suitable variables according to a fixed ordinal notation system. If you have no idea to fill your gaps in such a recursive explicit way, then it means that you have not completed your analysis because there is no reason why the terms appearing your table actually corresponds to the written ordinal without skipping. If you could not even understand why you need to avoid skipping, read this.


Real Examples[]


Level 5[]

  1. The target of the analysis is specified, and is actually defined in terms of mathematics, e.g. a full description of the algorithm to compute it.
  2. The description of the result of the analysis consists of well-defined stuffs.
  3. For analysis of a notation system, the description of a correspondence from terms to ordinals contains an explanation of the surjectivity in the case where the analysis includes a statement on the strength, and contains an explanation of the compatibility of the expansions in the case where the analysis includes a statement on the well-foundedness. For analysis of a large function, the estimation with a function hierarchy contains a reasoning of inequalities.
  4. The analysis contains a sufficient proof.


You must be treated as an annoying pedant who prevent the "development" of this community.


Fiction Example[]

This is analysis of a new version of BMS! It expands in the following explicit algorithm! (omit) This is the proof of the termination, in which I used the following recursive correspondence to ordinals! (omit) The following is my analysis table which might be helpful for readers to understand the correspondence more deeply! (omit)


If you want to submit your result to a mathematical journal, it is usually better to write it in English.


Real Examples[]

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