Real Numbers Representing The History of a Turing Machine.
Epistemics: Recreational. This idea may relate to alignment, but mostly it is just cool. I thought of this myself, but I’m positive this is an old and well known.
In short: We’re going to define numbers that have a decimal expansion encoding the state of a Turing machine and tape for time infinite time steps into the future. If the machine halts or goes into a cycle, the expansion is repeating.
Take some finite state Turing machine T on an infinite tape A. We will have the tape be 0 everywhere.
Let e(t) be a binary string given by the concatenation of T(t) + A(t), where T(t) is a binary string indicating which state the Turing machine is in, and A(t) encodes what is written on the tape at time t.
E(t) is the concatenation of e(0) + e(1) + … e(t) and can be thought of as the complete history Turing machine.
Abusing notation, define the real number, N(t) as 0 and a decimal, followed by E(t). That is, the digit in ith decimal place is the ith digit in E(t)
Then E(inf) = the infinitely long string encoding the history of our Turing machine and N(inf) is the number with an infinite decimal expansion.
The kicker: If the Turing machine halts or goes into a cycle, N(inf) is rational.
Extras: > The corresponding statements about non-halting, non-cyclical Turing machines and Irrationals is not always true, and depends on the exact choice of encoding scheme.) >Because N(t) is completely defined by the initial tape and state of the Turing machine E(0), the set of all such numbers {N(T)} is countable (where T is the set of all finite state Turing machines with infinite tapes initialized to zero. > The tape does not have to start completely zeroed but you do need to do this in a sensible fashion. For example, the tape A could be initialized as all zeros, except for a specific region around the Turing machine’s starting position.
Real Numbers Representing The History of a Turing Machine.
Epistemics: Recreational. This idea may relate to alignment, but mostly it is just cool. I thought of this myself, but I’m positive this is an old and well known.
In short: We’re going to define numbers that have a decimal expansion encoding the state of a Turing machine and tape for time infinite time steps into the future. If the machine halts or goes into a cycle, the expansion is repeating.
Take some finite state Turing machine T on an infinite tape A. We will have the tape be 0 everywhere.
Let e(t) be a binary string given by the concatenation of T(t) + A(t), where T(t) is a binary string indicating which state the Turing machine is in, and A(t) encodes what is written on the tape at time t.
E(t) is the concatenation of e(0) + e(1) + … e(t) and can be thought of as the complete history Turing machine.
Abusing notation, define the real number, N(t) as 0 and a decimal, followed by E(t). That is, the digit in ith decimal place is the ith digit in E(t)
Then E(inf) = the infinitely long string encoding the history of our Turing machine and N(inf) is the number with an infinite decimal expansion.
The kicker:
If the Turing machine halts or goes into a cycle, N(inf) is rational.
Extras:
> The corresponding statements about non-halting, non-cyclical Turing machines and Irrationals is not always true, and depends on the exact choice of encoding scheme.)
>Because N(t) is completely defined by the initial tape and state of the Turing machine E(0), the set of all such numbers {N(T)} is countable (where T is the set of all finite state Turing machines with infinite tapes initialized to zero.
> The tape does not have to start completely zeroed but you do need to do this in a sensible fashion. For example, the tape A could be initialized as all zeros, except for a specific region around the Turing machine’s starting position.