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 Fraïssé limit of the age. In this paper, we consider the computational strength needed to construct the cofinal Fraïssé 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 Fraïssé limit, can be constructed from the age using 0'. Our results therefore show that the more general case of cofinal Fraïssé limits requires greater computational strength than Fraïssé limits.

翻译：对于在同构意义下封闭（即一个年龄）且满足遗传性（HP）、联合嵌入性（JEP）与共尾合并性（CAP）的任意有限结构集合，存在一个唯一的（在同构意义下）可数结构，该结构相对于给定年龄是共尾超齐次的。这样的结构被称为该年龄的共尾Fraïssé极限。本文中，我们考虑构造可计算年龄的共尾Fraïssé极限所需的计算强度。我们证明该构造总可以使用谕示0'''完成，并且存在需要0'''的年龄。相比之下，我们证明若假设（CAP）的强化版本——即合并性（AP）——则所得极限（称为Fraïssé极限）可以仅使用0'从年龄构造出来。因此，我们的结果表明，共尾Fraïssé极限这一更一般的情形比Fraïssé极限需要更强的计算能力。