Introduction

In late July 2020, Scott Aaronson posed the Beeping Busy Beaver problem, which asks: among n-state quasihalting Turing machine programs, which one runs the longest before quasihalting? This function is super-uncomputable, so exact values cannot be obtained even in principle. Nevertheless, I spent much of August and September of 2020 trying to establish nontrivial lower bounds for some early values of n, and I managed to find a few.

This posts summarizes the known results so far, with values for both the shift and sigma functions (the number of steps executed and the number of marks left on the tape, also known as “activity” and “productivity”). Following Pascal Michel’s excellent historical survey of Busy Beaver candidates, the current champion will be listed along with a few other good programs. Also included, for comparison, are the Busy Beaver champions for each category. (The “shift champion” is the program with the greatest shift score; in the event of a tie, the winner is the program with the greater sigma score. The “sigma champion” is the program with the greatest sigma score; in the event of a tie, the winner is the programm with the least shift score.)

All machines are listed in normal form. Following Lafitte and Papazian, state count will be represented with a black circled number, and symbol count with a white circled number. All programs were discovered by me unless otherwise noted. I’m “pretty sure” in all champ claims except for the ❹② case.

❷②

There seem to be exactly four quasihalting ❷② machines. They all quasihalt in six steps, which is how long it takes for the ❷② Busy Beaver champion to halt. None of them have a higher sigma score, so apparently the extra power of quasihalting over halting doesn’t “kick in” for such small programs.

PROGRAM SHIFT SIGMA NOTES
`1RB 1LB 1LA 1RH` 6 4 BB / BBB champ
`1RB 1LB 1LB 1LA` 6 3
`1RB 1LB 0LB 1LA` 6 2
`1RB 0LB 1LB 1LA` 6 2
`1RB 0LB 0LB 1LA` 6 1

❸②

The ❸② shift champion was given in the Aaronson survey.

I think these are all the quasihalting ❸② programs with sigma score > 4.

PROGRAM SHIFT SIGMA NOTES
`1RB 0LB 1LA 0RC 1LC 1LA` 55 6 Shift champ
`1RB 1RC 1LC 1RA 1RA 1LA` 9 6 Sigma champ
`1RB 0LB 1RC 0RC 1LC 1LA` 54 6
`1RB 0LC 1LB 0RC 1LC 1LA` 52 5
`1RB 0LC 0LC 0RC 1LC 1LA` 51 5
`1RB 0LC 1LA 0RC 1RC 1RB` 49 5
`1RB 0LC 0RC 0RC 1LC 1LA` 48 5
`1RB 1RC 1LC 0LB 1RA 1LA` 22 5
`1RB 1RH 1LB 0RC 1LC 1LA` 21 5 BB shift champ
`1RB 1LC 1RC 1RH 1LA 0LB` 11 6 BB sigma champ

❹②

Three of the four most active programs end up ultimately with a sigma score of zero. This is a strange state of affairs.

The programs with sigma scores 25 and 39 were found by Terry and Shawn Ligocki. (Or just Shawn? Maybe it’s like Lennon-McCartney.)

These are just programs with shift score > 1000. Quasihalting ❹② programs are not especially rare, and it’s entirely possible that there are programs with lower shift scores but higher sigma scores.

PROGRAM SHIFT SIGMA NOTES
`1RB 0LC 1LD 0LA 1RC 1RD 1LA 0LD` 66349 0 Shift champ (?)
`1RB 1RC 1LC 1RD 1RA 1LD 0RD 0LB` 2819 69 Sigma champ (?)
`1RB 1RA 0RC 0RB 0RD 1RA 1LD 1LB` 2568 0
`1RB 1RA 0RC 1LA 1LC 1LD 0RB 0RD` 2512 0
`1RB 1RC 1RD 0LC 1LD 0LD 1LB 0RA` 2332 56
`1RB 0LC 1RC 1LD 1RD 0RB 0LB 1LA` 1459 35
`1RB 0LC 1LD 0RC 1RA 0RB 0LD 1LA` 1459 25
`1RB 1LC 1LC 0RD 1LA 0LB 1LD 0RA` 1164 39
`1RB 1LB 1RC 0LD 0RD 0RA 1LD 0LA` 1153 20
`1RB 1LB 1LA 0LC 1RH 1LD 1RD 0RA` 107 13 BB shift champ
`1RB 0RC 1LA 1RA 1RH 1RD 1LD 0LB` 96 13 BB sigma champ

❷③

These appear to be all the quasihalting programs with high shift or sigma scores.

PROGRAM SHIFT SIGMA NOTES
`1RB 2LB 1LA 2LB 2RA 0RA` 59 8 Shift champ
`1RB 2LB 1RA 2LB 2LA 0RA` 43 10 Sigma champ
`1RB 0LB 1RA 1LB 2LA 2RA` 45 3
`1RB 2RA 2LB 2LB 2LA 0LA` 40 5
`1RB 1LA 2RA 2LA 2LB 2RB` 17 8
`1RB 2LB 1RH 2LA 2RB 1LB` 38 9 BB champ