User:Wythagoras/Rado's sigma function/Multi-headed TMs

This page is about multi-headed TM's.

Ikosarakt's definition
Ikosarakt defines multi-headed TM's as following:

''Earlier I wrote that \(\Sigma(n)\) be extended to multiple heads, when we just alternate steps of them. I think we can do it more complex if in each step new head picks according to the transition rules. Formally, for n heads we have n rulesets and each rule has following syntax:''

Input: 

Output:   

Why I think Ikosarakt's definition doesn't work
So each head has a own ruleset that refers to each other. Each head has its own halt state, of course. What happens if a head would refer to a head that has been halted?

My own definition
My definition is about the same as Ikosarakt's definition, but it has one ruleset with one halt command for all heads. Each rule has the following syntax:

Input:

Output:

\(\Sigma(1,2,2)\)
1 _ 1 r 1 2 2 _ 1 r halt 2 1 1 l 1 1 \(\Sigma(1,2,2) ≥ 3\) and \(\S(1,2,2) ≥ 4\)

\(\Sigma(1,3,2)\)
1 1 _ 2 r 1 2 1 2 2 1 l 1 1 1 2 _ 1 r 1 1 1 2 1 1 r halt \(\Sigma(1,3,2) ≥ 4\) and \(\S(1,2,2) ≥ 6\)

\(\Sigma(2,2,2)\)
1 1 _ 1 r 1 2 2 1 _ 1 r 2 2 2 1 1 1 r 1 2 1 2 1 1 l 2 1 1 2 _ 1 r halt 2 2 1 1 r 2 2 2 2 _ 1 r 2 1 \(\Sigma(2,2,2) ≥ 6\) and \(\S(2,2,2) ≥ 11\)