Skelet 10
1LC0LA_---0LC_0RD1LA_1LB1RE_1RD0RE (bbch), called Skelet #10, was one of Skelet's 43 holdouts and one of the last holdouts in BB(5). It is a double fibonacci counter. It is one of the few TMs to have required an individual proof of non-halting in Coq-BB5.[1]
Behavior
Skelet 10 implements two base fibonacci counters, one on the right tape side and one on the left tape side. The TM halts if these counters become desynchronised.[1] The right side counter can be described by the following rules:[2]
A> 0 10^k 0 --> <D 0^2k 10 B> 10 10^k 0 --> <D 0^2k+1 10
These rules are equivalent to the Zeckendorf increment rules:[2]
Z(n) = 0 10^k 0 w ==> Z(n+1) = 0^2k 10 w Z(n) = 10 10^k 0 w =0> Z(n+1) = 0^2k+1 10 w Where Z(n) represents n in the little-endian (least significant digit is on the left) representation of Zeckendorf notation.
and can be restated as:[2]
Z(n) = 0 w ==> A> Z(n) --> <D Z(n+1) Z(n) = 1 w ==> B> Z(n) --> <D Z(n+1) Where Z(n) = 0 w means that the least-significant bit in Z(n) is 0.
The left side counter encodes base fibonacci differently and is also reversed to big-endian (least significant digit is on the right).[2] The two least-significant symbols are also omitted.[2] These differences can be described using the following definitions:[2]
T(0w) = T(w)00 T(10w) = T(w)1010 T(empty) = empty where w is any valid Zeckendorf expression. --> This encodes the Zeckendorf representations in a way consistent with the left fibonacci counter by encoding 10 as 1010 and 0 as 00. This also reverses the order of the Zeckendorf representation to big-endian. L(n) is defined in the following way: T(Z(n)) = v ab ==> L(n) = v where v is any valid Zeckendorf expression. --> This removes the two least-significant tape symbols.
There are now 3 possible cases for 3 possible values of the two least-significant digits of the left counter:[2]
Z(n) = 00 w --> T(Z(n)) = T(w) 00 00 --> L(n) = T(w) 00 Z(n) = 0 10^k+1 0 w --> T(Z(n)) = T(w) 00 10 1010^k+1 00 --> L(n) = T(w) 00 10^2k+1 10 Z(n) = 10 10^k 0 w --> T(Z(n)) = T(w) 00 1010^k+1 --> L(n) = T(w) 00 10^2k 10
These are affected by incrementing in the following way:[2]
Z(n) = 00 w --> L(n) = T(w) 00 ==> Z(n+1) = 10 w --> L(n+1) = T(w) 10 Z(n) = 0 10^k+1 0 w --> L(n) = T(w) 00 10^2k+1 10 ==> Z(n+1) = 0^2k+2 10 w --> L(n+1) = T(w) 10 10 10^2k+1 Z(n) = 10 10^k 0 w --> L(n) = T(w) 00 10^2k 10 ==> Z(n+1) = 0^2k+1 10 w --> L(n+1) = T(w) 10 10 00^2k
The TM follows these rules on the left:[2]
00 <D --> 10 B> 00 10^k 00 <D --> 10 10 00^k A>
These can be rewritten as:[2]
Z(n) = 00 w ==> L(n) <D --> L(n+1) B> Z(n) ≠ 00 w ==> L(n) <D --> L(n+1) A>