Computable isomorphisms for certain classes of infinite graphs

We investigate (2,1):1 structures, which consist of a countable set [Formula: see text] together with a function [Formula: see text] such that for every element [Formula: see text] in [Formula: see text], [Formula: see text] maps either exactly one element or exactly two elements of [Formula: see text] to [Formula: see text]. These structures extend the notions of injection structures, 2:1 structures, and (2,0):1 structures studied by Cenzer, Harizanov, and Remmel, all of which can be thought of as infinite directed graphs. We look at various computability-theoretic properties of (2,1):1 structures, most notably that of computable categoricity. We say that a structure [Formula: see text] is computably categorical if there exists a computable isomorphism between any two computable copies of [Formula: see text]. We give a sufficient condition under which a (2,1):1 structure is computably categorical, and present some examples of (2,1):1 structures with different computability-theoretic properties.

UNIFORM PROCEDURES IN UNCOUNTABLE STRUCTURES

This article contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the article extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of ${\Cal R}^n $ contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

COMPUTABILITY AND UNCOUNTABLE LINEAR ORDERS I: COMPUTABLE CATEGORICITY

We study the computable structure theory of linear orders of size $\aleph _1 $ within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.