User:Vel!/pu/gamma

The gamma function is sort of a question of computability. Let me explain.

Consider a solid string of ones. This means all the ones on the tape are connected by each other. So you can get from any of them to any other, from one to one, without passing through a non-one symbol. So, like, 11111111111111.

Now, consider a functional turing machine. A turing machine is "functional" if, when inputted with a solid string of ones, will always halt. The function of a functional turing machine is the function f, where f(n) is the amount of ones on the tape after inputting with a solid string of n ones.

Now, consider the set Gn. If contains every function of a functional turing machine with 2 colors and n states. γn is the one, out of them, that eventually dominates all others.

Now, is the function, say, γ100, computable? Well, we don't know what it IS, but we could definitely compute it is if we did know. We will be referring to this article for turing machines.

γ1, is, rather trivially, equal to f(n) = n+1, or f0 in the FGH. We are not exactly sure what γ2 is, but searching for it is still plausible.

The function 2n can be formulated in a 3-state machine. 2n can be formulated in a 6-state machine. 2↑↑n can be formulated in an 11-state machine. Generalizing the idea (but not consistently with these records), 2↑kn for k>0 can be done with 6k+2 states. So this notation is at least on par with the FGH.

In fact, with 18 states it is possible to make a TM that achieves expandal growth. So it shouldn't be hard to make ω+2, ω+3, ω2, ω2+1, ω3, ω4,...- possibly even ω2 and beyond. Really, we can just go up and up and up and up and up  and up and up and up  and up and up and up and up and up