User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-35470197-20190918031607/@comment-11227630-20190918040617

The idea of the algorithm is that it traverses every term with amount of C's less than \(L(\alpha)+k\), from larger to smaller, until a standard term.

The algorithm suits ordinal notation of Taranovsky's style, namely the comparison for postfix forms is lexicographical where the binary function symbol "C" is the least character.

You can take left subterms of a term until you can't, so terms can be written as \(C(\cdots C(C(\gamma,\beta_1),\beta_2)\cdots,\beta_m)\) (where \(\gamma\) is 0 or Ω), which in postfix form is \(\beta_m\cdots\beta_2\beta_1\gamma CC\cdots C\).

Step 1~2: To make the term smaller, if \(\gamma=\Omega\), then change it to \(0\) (generally, change a larger constant character into a smaller one); if \(\gamma=0\) (the least constant character), we should change the corresponding character into "C", but C means a binary function, so we have some more actions, as the algorithm states.

Step 3~5: New term reduced by step 1~2 may be not large enough, so we need to enlarge it to "the largest term with bounded amount of C's below the term before reduced". The idea is simple - it works as adding ΩΩ...Ω in the postfix form.

Step 6: Forming a "immediately smaller" term, we check it standard or not.