Maximum Consecutive Ones Function

The Maximum Consecutive Ones function (named $num(n)$ by Ben-Amram) is a Busy Beaver function which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol Turing machines which leave all their 1s consecutively. Unlike $\Sigma (n)$, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

Champions

Known Values of num(n) function

Domain	$num(n)$	Champion TM	Halting Config
BB(1)	1	1RZ--- (bbch)	$0^{\infty }\;1\;{\textrm {Z>}}\;0^{\infty }$
BB(2)	4	1RB1LB_1LA1LZ (bbch)	$0^{\infty }\;1\;{\textrm {<Z}}\;1^{3}\;0^{\infty }$
BB(3)	6	1RB1LC_1RC1LZ_1LA0LB (bbch) and 3 others	$0^{\infty }\;1\;{\textrm {<Z}}\;1^{5}\;0^{\infty }$
BB(4)	12	1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)	$0^{\infty }\;1^{12}\;{\textrm {Z>}}\;0^{\infty }$
BB(5)	165	1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB (bbch)	$0^{\infty }\;1^{165}\;{\textrm {Z>}}\;0^{\infty }$

Note: All TMs here are listed in TNF-1RB format with the exception that sometimes the halt transition is chosen to be 1LZ when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: 1RB1RZ_0RC1RB_1LC1LA (bbch), 1RB1LC_1LA1RB_1LB1RZ (bbch), 1RB1RA_1LC1RZ_1RA1LB (bbch), 1RB1LC_1RC1RZ_1LA0LB (bbch).