Bell

A bell is a non-halting Turing machine whose behavior can be described as a sequence of bouncers of increasing size. There are two main variants of bells, depending on whether the bouncers participate in tape growth or not. A bell whose bouncers grow the tape on each bounce is called an exponential bell or simply bell for short, while a bell whose bouncers do not grow the tape on each bounce is called a cubic bell.

Sometimes a Turing machine displays bell characteristics for some number of steps before phase transitioning to another behavior, such as halting. A famous example is the 5-state busy beaver winner. These may also be called bells, or more precisely, transient bells, to indicate that they only show bell-like features up to a certain point.

Examples

Spacetime diagram of the bell 1RB0LC_1RC1RA_1LA0RB (bbch).

Exponential bell

1RB0LC_1RC1RA_1LA0RB (bbch) is an example of an exponential bell with 3 states and 2 symbols.

Analysis

Transcript: (A0 B0 ((C1 B0)^2k (C0 A1)^2k+1)^{(k = 0 .. n-1)} (C0 B1)²ⁿ (C0 A1)²ⁿ C0 (A0 B1)²ⁿ⁺¹)^{n=a₀, a₁, ...},

where $a_{0}=1$ and $a_{k}=2a_{k-1}+3{\text{ for }}k\geq 1.$

Spacetime diagram of the cubic bell 1RB0LC_1RC1RA_1LA0RB (bbch).

Close-up of the cubic bell 1RB0LC_1RC1RA_1LA0RB (bbch).

Cubic bell

1RB0LC_1LA1RB_0RA1LC (bbch) is an example of a cubic bell with 3 states and 2 symbols.

Analysis

Transcript: ((A0 B1^k B0 A1 C1^k C0 A1 C0)^{(k = n, n-1, .. 1)} A0 B0 A1 C0)^{(n ≥ 0)}.