Non-halting Turing machine

(Very rough sketches and outlines)

Partial classification

• Cyclers
• Translated cyclers
• Bouncers (first seen in 2x2)
• Bells
  • Exponential bells (first seen in 3x2)
  • Cubic bells (first seen in 3x2)
  • Quartic bells (first seen in 5x2)
  • etc...
  • Irregular bells
• Counters (first seen in 2x2)
  • Regular
  • Irregular
  • Exponential/tetration...
• Hybrid (counter-counter, bouncer-counter)
• Fractals
  • Cube-like fractals (first seen in 4x2)
  • Christmas tree fractals (first seen in 2x4)
• Spaghetti
  • Asymptotically parabolically-shaped spaghetti e.g. 1RB0LB_1RC1LD_1LA0RC_0LA1LD (bbch)
  • Asymptotically irregular spaghetti e.g. 1RB0RB_1LC1RC_0RA1LD_1RC0LD (bbch). I suspect most irregular spaghetti, including this example, eventually become translated cyclers.
• Cellular automata (first seen in 5x2 and 3x3)

Sometimes phase transitions can occur. For example, 1RB1RC_1LC0RA_0LB0LD_1LA1LD (bbch) starts out as an irregular bell but phase transitions into a translated cycler with period 4222 at step 29754825.

An informal argument, in light of the uncomputability of the Busy Beaver function, suggests that there can never be a complete classification of behaviors of Turing machines for arbitrarily high numbers of states.

Records

Translated cycler preperiod

• 3x2: 101 (proven): 1RB1LB_0RC0LA_1LC0LA (bbch) (period = 24)
• 4x2: 119,120,230,102 (current champion): 1RB1LC_0LA1RD_0RB0LC_1LA0RD (bbch) (period = 966,716)
• 2x4: 293,225,660,896 (current champion): 1RB2LA0RA3LA_1LA1LB3RB1RA (bbch) (period = 483,328)

Translated cycler period

• 3x2: 92 (proven): 1RB0LA_0RC1LA_1LC0RB (bbch) (preperiod = 0)
• 4x2: 212,081,736 (current champion): 1RB0LA_0RC1RD_1LD0RB_1LA1RB (bbch) (preperiod = 5,248,647,886)
• 2x4: 33,209,131 (current champion): 1RB0RA3LB1RB_2LA0LB1RA2RB (bbch) (preperiod = 63,141,841)