A Short Note on Essentially Σ1 Sentences

David Guaspari [G] conjectured that a modal formula is essentially Σ1 (i.e., is is Σ1 under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form 2B. This conjecture was proved first by A. Visser. Then, in [DJP], the authors characterized essentially Σ1 formulas of languages with witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of I∆0 + Supexp and we address a similar problem for I∆0 + Exp.