A propositional inference rule A/B is admissible in an arithmetical
theory T iff for every substitution *f*
of sentences for the propositional
variables if *f*(A) is T-provable, then so is *f*(B).
(The logic of T can then be identified with the admissible rules
of the form true/A.)

Visser showed that the admissible rules of HA are the same as those
of intuitionistic propositional logic **Int**,
hence decidable by Rybakov.
This is a strengthening of propositional
De Jongh theorem.
The propositional logics of HA+MP and HA+ECT_{0} also
coincide with **Int**, however the question about their admissible rules
remains open. For the proofs of the abovementioned results of
Visser and more discussion see

A. Visser. Rules and arithmetics. Department of Philosophy, Utrecht University, Logic Group Preprint Series 186, June 1998.