On permutations with decidable cycles

TL;DR

This paper studies recursive permutations with decidable cycles, characterizing their structure, conjugacy, and normal forms, revealing computational limitations and the non-enumerability of the set of such permutations.

Contribution

It introduces new normal forms for permutations with decidable cycles and characterizes conjugacy and membership in this class using computable isomorphisms and maximality conditions.

Findings

Cycle decidability and finiteness have the maximal degree of the Halting Problem.

Conjugacy in Perm is undecidable and not recursively enumerable.

Perm is not a group and cannot be fully computed or enumerated.

Abstract

Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $\mathrm{Perm}$, the group of all recursive permutations $\mathcal{G}$. Multiple equivalent computable representations of decidable equivalence relations are provided. $\mathcal{G}$-conjugacy in $\mathrm{Perm}$ is characterised by computable isomorphy of cycle equivalence relations. This result parallels the equivalence of cycle type equality and conjugacy in the full symmetric group of the natural numbers. Conditions are presented for a permutation $f \in \mathcal{G}$ to be in $\mathrm{Perm}$ and for a decidable equivalence relation to appear as the cycle relation of a member of $\mathcal{G}$. In particular, two normal forms for the cycle structure of permutations are defined and it is shown that conjugacy to a permutation in the first normal form is equivalent to membership in $\mathrm{Perm}$. $\mathrm{Perm}$ is further characterised as the set of maximal permutations in a family of preordered subsets of automorphism groups of decidable equivalences. Conjugacy to a permutation in the second normal form corresponds to decidable cycles plus decidable cycle finiteness problem. Cycle decidability and cycle finiteness are both shown to have the maximal one-one degree of the Halting Problem. Cycle finiteness is used to prove that conjugacy in $\mathrm{Perm}$ cannot be decided and that it is impossible to compute cycle deciders for products of members of $\mathrm{Perm}$ and finitary permutations. It is also shown that $\mathrm{Perm}$ is not recursively enumerable and that it is not a group.