Maximum Consecutive Ones Function

The Maximum Consecutive Ones function (named $num(n)$ by Ben-Amram, Julstrom and Zwick) 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 $\mathrm{\Sigma }(n)$, 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	Source
BB(n,1)	= 0	0RZ (bbch)	$0^{\mathrm{\infty }}\text{Z}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$
BB(1,m) (m >= 2)	= 1	1RZ--- (bbch)	$0^{\mathrm{\infty }}1\text{Z}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$	Ben-Amram, Julstrom and Zwick (1996)
BB(2)	= 4	1RB1LB_1LA1LZ (bbch)	$0^{\mathrm{\infty }}1\text{<mo><</mo>}\text{Z}{1}^3{0}^{\mathrm{\infty }}$	Ben-Amram, Julstrom and Zwick (1996)
BB(3)	= 6	1RB1LC_1RC1LZ_1LA0LB (bbch) and 3 others	$0^{\mathrm{\infty }}1\text{<mo><</mo>}\text{Z}{1}^5{0}^{\mathrm{\infty }}$	Ben-Amram, Julstrom and Zwick (1996)
BB(4)	= 12	1RB0LA_1RC1LB_1LB1RD_1RZ0RA (bbch)	$0^{\mathrm{\infty }}{1}^{12}\text{Z}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$	Ben-Amram, Julstrom and Zwick (1996)
BB(5)	= 165	1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB (bbch)	$0^{\mathrm{\infty }}{1}^{165}\text{Z}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$	Announcement by Andrés Sancho in Feb 2025
BB(6)	≥ ≈ 10↑↑15.60465	1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE (bbch)	$0^{\mathrm{\infty }}\text{Z}\text{<mo>></mo>}01^{\approx 10{\uparrow }^{2}15.60465}{0}^{\mathrm{\infty }}$	Pavel Kropitz in 2022
BB(2,3)	= 9	2RB1RZ1LB_1LA2LB1RB (bbch)	$0^{\mathrm{\infty }}{1}^7\text{Z}\text{<mo>></mo>}{1}^2{0}^{\mathrm{\infty }}$
BB(3,3)	≥ 95524079	1RB2RC1LA_2LA1RB1RZ_2RB2RA1LC (bbch)		nickdrozd in 2026
BB(4,3)*	> 10↑↑↑↑4	1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)	$0^{\mathrm{\infty }}1\text{Z}\text{<mo>></mo>}1(11{)}^{2e_4+8}{0}^{\mathrm{\infty }}$	Pavel Kropitz in 2024
BB(4,4)	$>2{\uparrow }^{15}5$	1RB3LB1LD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_1LZ---1LD1LD (bbch)	$0^{\mathrm{\infty }}\text{<mo><</mo>}\text{Z}{1}^{\sigma }{0}^{\mathrm{\infty }}$	User:Loader3229 in 2026, Based on Pavel Kropitz's BB(3,4) champion.
BB(17,5)	$>f_{{\epsilon }_0}(20)$	Machine code (Not TNF) 4RF---0RF0RD1LQ_1RZ1RB2LC------_0LI4RC2RC3RC2RC_1LE1RD2RD3RD---_3LA1LE2LE3LE1LG_0LG1RF2RF3RF---_0RI0LH4RD3RI2LH_2LA1LH2LH3LH0RC_3RN---0LA0LI2RF_2RO---2LJ3LJ---_2LL---2RK3RK4RK_3RO2RO2LL3LL4LL_3LL---2RM3RM4RM_4LJ1LN2LN3LN1LO_1RB4RP1RK0RM4LN_4LJ1RP2RP3RP0RP_1RB1LQ2LQ3LQ1LQ (bbch)	$0^{\mathrm{\infty }}{1}^{\sigma }\text{Z}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$	User:Loader3229 in 2026

* $e_4=f_2(\frac{2e_3+4}{3},f_1^2(91))$, where $e_3=f_2(\frac{2e_2+7}{3},f_1^2(91))$, where $e_2=f_2(\frac{2e_1+7}{3},f_1^2(91))$, where $e_1=2^{95}\times 3-5$. $f_1(n)=2^{n+4}\times 3-5$ and $f_2(a,b)=f_1^{2\times f_2(a-1,b)+11}(1)$, where$f_2(0,b)=b$.

Note: All TMs here are listed in TNF-1RB or TNF-2RB format with the exception that sometimes the halt transition is chosen to be 0RZ if that is necessary to make all final 1s consecutive or 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).

The num(5) champion was found in 2009 by Joachim Hertel.^{p. 282}

Instruction-Limited Maximum Consecutive Ones

The Instruction-Limited Maximum Consecutive Ones function num_i(n) returns the maximum number of consecutive 1s left on the tape at halt across all n-instruction (unlimited symbols and states) Turing machines which leave all consecutive ones only, no other symbols.

Champions for num_i

n	num_i(n)	Champion	Discovered by	Halting Tape
0	0	--- (bbch)	Trivial	0^inf A> 0^inf
1	1	1RZ (bbch)	Trivial	0^inf 1 Z> 0^inf
2	3	1RB---_1LA--- (bbch)	Trivial	0^inf 1 <A 1 0^inf
3	4	1RB1LB_1LA--- (bbch)		0^inf 1 B> 111 0^inf
4
5	9	2RB---1LB_1LA2LB1RB (bbch)