A Solvable Class of Quadratic Diophantine Equations with Applications to Verification of Infinite-State Systems

A κ-system consists of κ quadratic Diophantine equations over nonnegative integer variables s1, ..., sm, t1, ..., tn of the form: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{array}{*{20}c} {\sum\limits_{1 \leqslant j \leqslant l} {B_{1j} \left( {t_1 , \ldots ,t_n } \right)A_{1j} \left( {s_1 , \ldots ,s_m } \right) = C_1 \left( {s_1 , \ldots ,s_m } \right)} } \\ \vdots \\ {\sum\limits_{1 \leqslant j \leqslant l} {B_{kj} \left( {t_1 , \ldots ,t_n } \right)A_{kj} \left( {s_1 , \ldots ,s_m } \right) = C_k \left( {s_1 , \ldots ,s_m } \right)} } \\ \end{array} $$\end{document} where l, n, m are positive integers, the B’s are nonnegative linear polynomials over t1, ..., tn (i.e., they are of the form b0 + b1t1 + ... + bntn, where each bi is a nonnegative integer), and the A’s and C’s are nonnegative linear polynomials over s1, ..., sm. We show that it is decidable to determine, given any 2-system, whether it has a solution in s1, ..., sm, t1, ..., tn, and give applications of this result to some interesting problems in verification of infinite-state systems. The general problem is undecidable; in fact, there is a fixed k > 2 for which the k-system problem is undecidable. However, certain special cases are decidable and these, too, have applications to verification.