This paper is a contribution to the following natural problem in complexity theory:
(*) Is there a complexity theory for isomorphism types of recursive countable relational structures? I.e. given a recursive relational structure ℛ over the set N of nonnegative integers, is there a nontrivial lower bound for the time-space complexity of recursive structures isomorphic (resp. recursively isomorphic) to ℛ?
For unary recursive relations R, the answer is trivially negative: either R is finite or coinfinite or 〈N, R〉 is recursively isomorphic to 〈N, {x ϵ N: x is even}〉.
The general problem for relations with arity 2 (or greater) is open.
Related to this problem, a classical result (going back to S. C. Kleene [4], 1955) states that every recursive ordinal is in fact primitive recursive.
In [3] Patrick Dehornoy, using methods relevant to computer science, improves this result, showing that every recursive ordinal can be represented by a recursive total ordering over N which has linear deterministic time complexity relative to the binary representation of integers. As he notices, his proof applies to every recursive total order type α such that the isomorphism type of α is not changed if points are replaced by arbitrary finite nonempty subsets of consecutive points.
In this paper we extend Dehornoy's result to all recursive total orderings over N and get minimal complexity for both time and space simultaneously.