This presentation is based on joint research with Nikolay Bazhenov, Ivan Georgiev, Dariusz Kalociński, Luca San Mauro, and Stefan Vatev (in alphabetical order).

Punctual structure theory considers structures whose domain is \(\mathbb{N}\) and all relations and functions from their signature are uniformly primitive recursive.

In [1] it has been proved that every computable structure of each of the following classes has a punctual presentation: equivalence structures, linear orderings, Boolean algebras and some more. On the other hand, among others, computable torsion abelian groups do not always have a punctual presentation. Punctual equivalence structures were also studied in [2].

We have studied a different type of structures which we call injective structures. They are of the form \((\mathbb{N},f)\) where \(f\) is a unary injective function. An injective function can be decomposed into various orbits, each of which is either a cycle, or constitutes an \(\mathbb{N}\)-chain or a \(\mathbb{Z}\)-chain. A function which can be decomposed into cycles only is called a cyclic function.

We have shown that every computable injective structure which is not cyclic is punctually presentable. So is every computable injective structure which has some cycle length occurring infinitely many times.

When none of these cases occurs, the situation is less clear. We have constructed a computable cyclic injective structure without a punctual presentation. We have also considered the following concept. Suppose that \(f\) is a cyclic function and that \(g: \mathbb{N} \to \mathbb{N}\) is an enumeration of all cycle lengths of \(f\) in which each cycle length is repeated as many times as it appears in \(f\). We have proved several charaterizations tying punctual presentability of \((\mathbb{N},f)\) to computational properties of \(g\).

We have also had success in proving some cases of a theorem which states a connection between instrinsic punctuality of an additional relation on a punctual injective structure and definability of that relation using a certain restricted class of quantifier-free formulas of first-order logic, assuming that a certain effectivness condition is satisfied. This is aimed to lead to a more general punctual version of Ash-Nerode Theorem.

Bibliography

1. Iskander Kalimullin, Alexander Melnikov, Keng Meng Ng,Algebraic structures computable without delay,Theoretical Computer Science,vol. 674 (2017), pp. 73–98.
2. Nikolay Bazhenov, Keng Meng Ng, Luca San Mauro, Andrea Sorbi,Primitive recursive equivalence relations and their primitive recursive complexity,Computability,vol. 11 (2022), pp. 187–221.