Clean Citation Style 002
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Gao, Ziyuan; Jain, Sanjay; Khoussainov, Bakhadyr; Li, Wei; Melnikov, Alexander; Seidel, Karen; Stephan, Frank Random Subgroups of Rationals. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2019: 25:125:14
This paper introduces and studies a notion of algorithmic randomness for subgroups of rationals. Given a randomly generated additive subgroup \((G,+)\) of rationals, two main questions are addressed: first, what are the modeltheoretic and recursiontheoretic properties of \((G,+)\); second, what learnability properties can one extract from \(G\) and its subclass of finitely generated subgroups? For the first question, it is shown that the theory of \((G,+)\) coincides with that of the additive group of integers and is therefore decidable; furthermore, while the word problem for \(G\) with respect to any generating sequence for \(G\) is not even semidecidable, one can build a generating sequence \(\beta\) such that the word problem for \(G\) with respect to \(\beta\) is corecursively enumerable (assuming that the set of generators of \(G\) is limitrecursive). In regard to the second question, it is proven that there is a generating sequence \(\beta\) for \(G\) such that every nontrivial finitely generated subgroup of \(G\) is recursively enumerable and the class of all such subgroups of \(G\) is behaviourally correctly learnable, that is, every nontrivial finitely generated subgroup can be semantically identified in the limit (again assuming that the set of generators of \(G\) is limitrecursive). On the other hand, the class of nontrivial finitely generated subgroups of \(G\) cannot be syntactically identified in the limit with respect to any generating sequence for \(G\). The present work thus contributes to a recent line of research studying algorithmically random infinite structures and uncovers an interesting connection between the arithmetical complexity of the set of generators of a randomly generated subgroup of rationals and the learnability of its finitely generated subgroups.