Is there a term for/literature about the concept of the first number unreachable by an n-state Turing machine? By “unreachable,” I mean that there is no n-state Turing machine which outputs that number.
Obviously such “Turing-unreachable numbers” are usually going to be much smaller than Busy Beaver numbers (as there simply aren’t enough possible different n-state Turing machines to cover all numbers up to to the insane heights BB(n) reaches towards) , but I would expect them to have some interesting properties (though I have no sense of what those properties might be). Anyone here know of existing literature on this concept?
Thanks! Is there any literature on the generalization of this, properties of “unreachable” numbers in general?Just realized I’m describing the basic concept of computability at this point lol.
Is there a term for/literature about the concept of the first number unreachable by an n-state Turing machine? By “unreachable,” I mean that there is no n-state Turing machine which outputs that number. Obviously such “Turing-unreachable numbers” are usually going to be much smaller than Busy Beaver numbers (as there simply aren’t enough possible different n-state Turing machines to cover all numbers up to to the insane heights BB(n) reaches towards) , but I would expect them to have some interesting properties (though I have no sense of what those properties might be). Anyone here know of existing literature on this concept?
It’s the lazy beaver function: https://googology.fandom.com/wiki/Lazy_beaver_function
Thanks! Is there any literature on the generalization of this, properties of “unreachable” numbers in general?Just realized I’m describing the basic concept of computability at this point lol.