User:RobinCodes/Next Steps

Possible next steps by domain

In this article I want to explore possible next steps in each domain and set forth goals about them.

BB(2,5):

This domain is almost exhausted (especially decider-wise), though far from as much as 3x3. I think the goal here is to work on the holdouts, apply deciders in a distributed-computation manner with high parameters (perhaps break the holdouts list into small pieces, that is, ~10 machines) and encourage people to run deciders with high parameters on them. Another important goal is to translate the 9 proofs and 2 proofs of equivalence into a formal language (Rocq).

List of machines which have not yet been manually inspected.

1RB2RA3LA4LA2RB_2LA3RA---0RA1LA | Chaotic via long. analysis | Notes by mxdys
1RB2RA3LA4LA2RB_2LA3RB---0RA1LA | Chaotic via long. analysis
1RB2LA0RB4LB0LA_1LA3LA1RA4RA--- | Does not halt in 1e13
1RB---3RA2LA2RB_2LB3LA4LB4RA0RA | Does not halt in 1.25e13
1RB3RB1LA2LA3RA_1LB2RA4RB0LA--- | Does not halt in 2e13
1RB2RA4LA1RB4RB_1LB2LA3RA---0LB | Does not halt in 5e13
1RB---4LB0LA4RA_2LB2LA3RA4LB0RB
1RB4RA1LA4RB2LA_2LB3LA1RB2RA---
1RB2RB3LA4LA1LA_2LB3RA---4RA1RB
1RB3RB3LA4LA2RB_2LB3RA---1RA1LA
1RB4RB4RA1LA3LA_1LB2LA3RB2RB---
1RB3LA1LA2RB2RA_2LA4RA3LB1RA---
1RB3RB---4RA2RA_2LA2RA3LB4LB1LB
1RB---3LB4RB0LA_2LB3LA3RB4RA0RA

A few of them were categorized by @dyuan as “quasi-cascading”, but their individual behaviour is still unanalyzed. Top 10 machines most susceptible to block-analysis: (src)

1RB3LA1LA1RA1RA_2LB2RA---4RB1LB
1RB3LA1LA1RA3RA_2LB2RA---4RB1LB
1RB2LB3LA0RA1LB_2LA4RA3RB3LA---
1RB3LB---4LA1RB_2LA4LA4LB3RB1RA
1RB---3LB4RB0LA_2LB3LA3RB4RA0RA
1RB3LA1LA2RB2RA_2LA4RA3LB1RA---
1RB2LA0RB1LB---_1LA3RA1RA4LB2RB
1RB2LA1LA4RA2LA_0LA3RB3LB2RB---
1RB3LA1RA4LA2RA_2LA---1LA0RA3RB
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB

List of unverified proofs for individual machines (9 total proofs).

BB(3,3):

This domain is basically exhausted, and all 4 informal holdouts are what seem like mathematically very hard problems. In my opinion, the goal here is to prove the 3 holdouts solved informally by Legion via Longitudinal Analysis, then to describe the high-level behaviour of the remaining four machines, along with reasons for why we expect / don't expect them to halt, and with proofs of equivalency between machines 153, 758 and 531, 532.

Regardless;

The least analyzed holdout:

1RB1LB2LC_1LA2RB1RB_---0LA2LA (397)

Top 2 machines most susceptible to block-analysis: (src)

1RB0LB0RC_2LC2LA1RA_1RA1LC--- (153)	
1RB2LC1RC_2LC---2RB_2LA0LB0RA (758)

The machines are in the same equivalence class and as such, achieve the same score on the susceptibility metric.

List of unverified proofs for individual machines (3 total proofs).

BB(6):

The goal is pretty straightforward in my opinion. Reduce the holdouts list to as far as we can, until all machines that are left seem way too hard (i.e. seemingly mathematically hard problems in their high-level behaviour), then write those problems up formally for potentially mathematicians to solve. Shorter-term goals include running the new FAR decider in a distributed-computation manner with higher params, and simulating all machines to 1e15. (Currently 239 holdouts)

Simulation

Sim out all machines to 1e13 by no later than April. ACHIEVED!

Simulate all machines out to 1e15 and start exploring beyond.

Deciders

Apply new FAR decider as described in [1] in a distributed computing effort.

Spreadsheet of holdouts: spreadsheet. There are 106 machines with completely unknown behaviour.

List of machines that are tractable to simulate until they very likely halt: Machines at the Edge

List of machines that "should be solvable/easy": More extensive list, forum here

1RB1LE_0RC1RC_0RD0RB_1RE0RF_1LA0LE_---1RB
1RB0LE_1LC1LB_0RD0LC_1RA0RE_1RF1RD_0RA---
1RB---_0LC0RB_1LE0LD_1LA1LC_1LF0RD_1RB1RF
1RB0RE_1RC0LE_1LD1LC_0RA0LD_1RF1RA_1LB---
1RB1RA_0LC0RB_1LF0LD_1LE1LC_0LF---_1LA1LF
1RB1RA_0LC0RB_1LF0LD_1LE1LC_1LB---_1LA1LF
1RB0LE_1LC1LB_0RD0LC_1RA0RE_1RF1RD_0LA---
1RB1RA_0LC0RB_1LF0LD_1LE1LC_0RB---_1LA1LF
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_0LC---
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_0RA---
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_1RA---
1RB1RA_0LC0RB_1LF0LD_1LE1LC_1LC---_1LA0RD
1RB1RA_1LC1LB_0RD0LC_1RA0RE_1RF1RD_0LB---
1RB1RC_0RC---_1RD0RA_1RE1RD_1LF1LE_0RC0LF
1RB1RA_1LC1LB_0LD0LC_0RE0RF_0RF---_1RA1RD
1RB0LE_1LC1LB_0RD0LC_1RA0RE_0RF1RD_0LD---
1RB1RA_0RC0RB_0LD0LE_0LE---_1LF1LC_1LA1LF
1RB0RF_1LC0RD_---1RD_0LE1LD_1LA1LB_1RA1RF
1RB1RA_1RC0RA_1LD0RE_---1LE_0LF1LE_1LB1LC
1RB1RC_1LC0LE_1RF0LD_0RA1RD_1LB1LE_---1RD
1RB1RD_1LC0LE_1RA0LB_1RA1RF_0RA0RC_1RE---
1RB1LB_1LC0RD_0LE1LA_0RF1RA_1LE0LA_---0RB

Top 10 machines which visit the least amount of cells in 1 billion steps: (src) (Full list) List for 1 million steps, some machines may be missing (src)

1RB1RA_0LC0RB_1LF0LD_1LE1LC_0RB---_1LA1LF 66
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_0LC--- 66
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_0RA--- 66
1RB0RE_1RC1RB_1LD1LC_0RA0LD_1RF1RA_1RA--- 66
1RB1RA_0LC0RB_1LF0LD_1LE1LC_1LC---_1LA0RD 67
1RB1RA_1LC1LB_0RD0LC_1RA0RE_1RF1RD_0LB--- 67
1RB0LE_1LC1LB_0RD0LC_1RA0RE_0RF1RD_0LD--- 67
1RB1RA_1LC1LB_0LD0LC_0RE0RF_0RF---_1RA1RD 67
1RB1RC_0RC---_1RD0RA_1RE1RD_1LF1LE_0RC0LF 67
1RB1RA_0RC0RB_0LD0LE_0LE---_1LF1LC_1LA1LF 68

Top 10 machines* most susceptible to block-analysis: (src)

1RB1LE_1LB1LC_1RD0LE_---0RB_1RF1LA_0RA0RD				
1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE
1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB
1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD				
1RB1RD_1LC0RD_1RF1RB_1LE0RA_0LB0LE_---1RC				
1RB1RD_1LC0RD_1RF1RB_1LE0RA_0LB0LE_---0RC				
1RB1LF_1RC1RF_0LD1RE_---1LC_0RB1RE_1LA0LE				
1RB1LA_1LC0RE_1LF1LD_1RE0LC_1LB1RA_---0LA				
1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE				
1RB0LE_1LC1RA_1RA1LD_0LC0LA_0RE1RF_0RC---

*excluding analyzed ones (may be outdated)

List of unverified proofs for individual machines (2 total proofs).