On the computability of cofinal Fra\"iss\'e limits

TL;DR

This paper investigates the computational complexity of constructing cofinal Fra"issé limits for certain classes of finite structures, showing they require higher computational resources than traditional Fra"issé limits.

Contribution

It establishes the exact Turing degrees needed to compute cofinal Fra"issé limits, demonstrating they can require 0''' and sometimes 0'' in contrast to Fra"issé limits which need only 0'.

Findings

Cofinal Fra"issé limits can require the oracle 0''' for construction.

Some ages require the oracle 0'' to compute their cofinal Fra"issé limit.

Fra"issé limits can be constructed from the age using the oracle 0'.

Abstract

For any collection of finite structures closed under isomorphism (i.e., an age) which has the Hereditary Property (HP), the Joint Embedding Property (JEP), and the Cofinal Amalgamation Property (CAP), there is a unique (up to isomorphism) countable structure which is cofinally ultrahomogeneous with the given age. Such a structure is called the cofinal Fra\"iss\'e limit of the age. In this paper, we consider the computational strength needed to construct the cofinal Fra\"iss\'e limit of a computable age. We show that this construction can always be done using the oracle 0''', and that there are ages that require 0''. In contrast, we show that if one assumes the strengthening of (CAP) known as the Amalgamation Property (AP), then the resulting limit, called the Fra\"iss\'e limit, can be constructed from the age using 0'. Our results therefore show that the more general case of cofinal Fra\"iss\'e limits requires greater computational strength than Fra\"iss\'e limits.