The $\Sigma$_1 Provability Logic of HA
  In this paper we introduce a modal theory $H_{\sigma}$, which is sound and complete for arithmetical $\Sigma$_1 substitutions in ${\bf HA}$, in other words, we will show that $H_{\sigma}$ is the $\Sigma$_1-provability logic of ${\bf HA}$. Moreover we will show that $H_{\sigma}$ is decidable. As a by-product of these results, we show that ${\bf HA} + \Box\bot$ has de Jongh property. 
Mohammad Ardeshir and S. Mojtaba Mojtahedi
math.LO