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+ECT0 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.