Ordinal machines and admissible recursion theory

Ordinal machines and admissible recursion theory

We generalize standard Turing machines working in time ω on a tape of length ω to abstract machines with time α and tape length α, for α some limit ordinal. This model of computation determines an associated computability theory: α-computability theory. We compare the new theory to α-recursion theory, which was developed by G. Sacks and his school. For α an admissible ordinal, the basic notions...

An ordinal analysis of admissible set theory using recursion on ordinal notations ∗ Jeremy Avigad

The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible se...

An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations

The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible se...

Ordinal Recursion Theory

Primitive Recursion and State Machines

Methods for specifying Moore type state machines (transducers) abstractly via primitive recursive functions and for definining parallel composition via simultaneous primitive recursion are discussed. The method is mostly of interest as a concise and convenient way of working with the complex state systems found in computer programming and engineering, but a short section indicates connections t...