User blog:Rpakr/Converting 2nd System Expressions in Taranovsky's C to Labelled Hydras

Taranovsky's C is a ordinal notation by Dmytro Taranovsky, and even the 2nd system of the notation is believed to be very strong. However, this notation, unlike many other ordinal notations, uses lexicographical ordering and something Taranovsky calls "n-built from below", which makes the analysis extremely difficult. Among the people that understands the notation well, it is mostly accepted that \(C(C(\Omega_22,0),0)=\), but people still do not have an agreed-upon opinion on the behaviour of the notation past \(C(C(\Omega_22,0),0)\).

In this blog post, I will explain the method of converting 2nd system expressions in Taranovsky's C to hydras labelled with 1s and 2s, and how the method relates to the definition of the notation. This method can make it easier to understand the notation easier, and also could help in the analysis of the notation. Expressions of Taranovsky's C will be expressions in the 2nd system unless stated otherwise.

Definition of Taranovsky's C
Here, I will state the definition of Taranovsky's C to avoid ambiguities.

In this definition \(\alpha\),\(\beta\),\(\gamma\),\(\delta\) represent ordinals, and \(n\) represents a nonnegative integer.

Start of definition

\(\alpha\) is \(0\)-built from below from \(<\beta\) iff \(\alpha<\beta\).

\(\alpha\) is \(n+1\)-built from below from \(<\beta\) iff the standard representation of \(\alpha\) does not use ordinals greater than \(\alpha\) except as a subterm of an ordinal \(n\)-built from below from \(<\beta\).

The standard representation of an ordinal in \(n\)th system uses two constants \(0\) and \(\Omega_n\), and a binary function \(C\).

To write an expression in postfix form, reverse the characters of the expression and remove commas and parentheses.

To compare two ordinals expressed in the \(n\)th system, convert them to postfix forms, and comp are the postfix forms lexicographically where \("C"<"0"<"\Omega_n"\).

In the \(n\)th system, \(0\) and \(\Omega_n\) are standard, and \(C(\alpha,\beta)\) is standard iff \(\alpha\) and \(\beta\) are standard, \(\beta\) is \(0\) or \(\Omega_n\) or \(C(\gamma,\delta)\) with \(\alpha\leq\gamma\), and \(\alpha\) is \(n\)-built from below from \(1\) are converted to \(C(\alpha_n,\alpha_{n-1},...\alpha_2,\alpha_1;\beta\). \(\alpha_k\) for \(1\leq k\leq n\) and \(\beta\) are \(C\)-expressions.

Note that the order of \(\alpha_k\) are reversed.

Semicolon Elimination
In this step, every expressions of the form \(C(\alpha_1,\alpha_2,...,\alpha_n;0)\) are converted to \(D(\alpha_1,\alpha_2,...,\alpha_n)\) and every expressions of the form \(C(\alpha_1,\alpha_2,...,\alpha_n;\Omega_2)\) are converted to \(E(\alpha_1,\alpha_2,...,\alpha_n)\).

If the steps were applied to a standard \(C\)-expression, after this step, there will be no more \(C\)s in the expression.

Hydra Writing
First, I will define the word parent.

Let \(f\) be either of \(D\) or \(E\). If an expression contains an expression of the form \(f(\alpha_1,\alpha_2,...,\alpha_n)\), the parent of \(\alpha_k\) for \(1\leq k\leq n\) are \(f\).

In this step, a hydra is generated using the converted expression using the following rules:

Scan the expression one character by character from the left. \(\Omega_2\) is considered to be one character.

If the character is a comma or parentheses, do not add any nodes.

If the character is a \(0\) or a \(D\), write a node labelled with \(1\) as a child of the node that was written when the parent of it was scanned (if any).

If the character is a \(\Omega_2\) or a \(E\), write a node labelled with \(2\) as a child of the node that was written when the parent of it was scanned.

Using these three steps, we can convert any \(C\)-expression to a hydra labelled with only \(1\)s and \(2\)s.

Examples
In this section, I will give several examples of the conversion rules.

Relation to the definition of Taranovsky's C
In this section, I will explain how the conversion rules are related to the definition of Taranovsky's C.

Size Comparison
We can compare hydras by converting them to sequences of number pairs where the 1st number represents how tall a node is and the 2nd number represents the number inside nodes and compare the pair sequences. The conversion from hydras to sequences to number pairs is the same as the conversion in pair sequence system.

This comparison is equivalent to how \(C\)-expressions compare to each other.

Standardness Rule 2
One of the requirements for \(C\)-expression \(C(\alpha,C(\gamma,delta))\) to be standard is \(\alpha\leq\gamma\). The conversion makes \(\alpha\) and \(\gamma\) two children of the same node, with \(\gamma\) being first. This is very similar to one of the characteristics of standard pair sequences. (For example, pair sequence \((0,0)(1,1)(2,1)(2,2)\) is not standard because \((2,1)>(2,2)\).)

Extension to general \(s\)th system
It is possible to extend the conversion method I explained above into general \(s\)th system. To do so, replace all occurrences of \(\Omega_2\) with \(\Omega_s\).

Conclusion
This method of converting \(C\)-expressions to labelled hydras makes \(C\)-expressions a lot easier to read. Further research is needed about how the "n-built from below" requirement for a \(C\)-expression to be standard relates to the hydra, but when that is done we might be able to deduce the "expension rule" of the hydra system and consequently, facilitate the analysis of the notation.