On Decidability Within the Arithmetic of Addition and Divisibility

The arithmetic of natural numbers with addition and divisibility has been shown undecidable as a consequence of the fact that multiplication of natural numbers can be interpreted into this theory, as shown by J. Robinson [14]. The most important decidable subsets of the arithmetic of addition and divisibility are the arithmetic of addition, proved by M. Presburger [13], and the purely existential subset, proved by L. Lipshitz [11]. In this paper we define a new decidable fragment of the form QzQ1x1 ... Qnxnϕ(x,z) where the only variable allowed to occur to the left of the divisibility sign is z. For this form, called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal L}{_{\mid}^{(1)}}$\end{document} in the paper, we show the existence of a quantifier elimination procedure which always leads to formulas of Presburger arithmetic. We generalize the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal L}{_{\mid}^{(1)}}$\end{document} form to ∃ z1,... ∃ zmQ1x1...Qnxnϕ(x,z), where the only variables appearing on the left of divisibility are z1, ..., zm. For this form, called \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\exists{\mathcal L}{_{\mid}^{(*)}}$\end{document}, we show decidability of the positive fragment, namely by reduction to the existential theory of the arithmetic with addition and divisibility. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal L}{_{\mid}^{(1)}}$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\exists{\mathcal L}{_{\mid}^{(*)}}$\end{document} fragments were inspired by a real application in the field o