The ${\mathrm{\Sigma }}_1$-Provability Logic of HA*

For the Heyting Arithmetic HA, HA* is defined as the theory ${A\mid {\sf HA}\vdash A^{\Box}}$,where$A^{\Box}$iscalledtheboxtranslationof$A$. We characterize the ${\mathrm{\Sigma }}_1$-provability logic of HA* as a modal theory ${\sf iH}_\sigma^*$.