H-theories, fragments of HA and PA-normality

For a classical theory T , H(T ) denotes the intuitionistic theory of T -normal (i.e. locally T ) Kripke structures. S. Buss has asked for a characterization of the theories in the range of H and raised the particular question of whether HA is an H-theory. We show that T i ∈ range(H) iff T i = H(T ). As a corollary, no fragment of HA extending iΠ1 belongs to the range of H. A. Visser has already proved that HA is not in the range of H by different methods. We provide more examples of theories not in the range of H. We show PA-normality of once-branching Kripke models of HA + MP , where it is not known whether the same holds if MP is dropped. 1991 Mathematics Subject Classification: 03F55, 03F30.