Comparing Constructive Arithmetical Theories Based on NP-PIND and coNP-PIND

In this note we show that the intuitionistic theory of polynomial induction on ∏1^b+-formulas does not imply the intuitionistic theory I S2¹ of polynomial induction on ∑1^b+-formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polynomial induction. We also investigate the relation between various other intuitionistic first-order theories of bounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories.