Generalizing Turing Machines: ω1-Computation
While Turing machines are well established as the principal framework for theoretical computation on the natural numbers, no machine model for computation on uncountable domains has achieved any similarly dominant status.
We will give an overview of various methods that have been
explored, including computable analysis, the Blum-Shub-Smale model of
computation on the real numbers, local computability, α-recursion
theory, and certain models of computation over ordinal time (one of which
turns out to be equivalent to
Much of the talk will be expository, describing these different notions of computation, but we will finish with a specific example from ω1-recursive model theory which shows how phenomena that were unknown under Turing computation on ω can arise in the context of ω1-computable model theory.