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

Abstract

A κ-system consists of κ quadratic Diophantine equations over nonnegative integer variables s ₁, ..., s _m, t ₁, ..., t _n of the form:

$$ \begin{array}{*{20}c} {\sum\limits_{1 \leqslant j \leqslant l} {B_{1j} \left( {t_1 , \ldots ,t_n } \right)A_{1j} \left( {s_1 , \ldots ,s_m } \right) = C_1 \left( {s_1 , \ldots ,s_m } \right)} } \\ \vdots \\ {\sum\limits_{1 \leqslant j \leqslant l} {B_{kj} \left( {t_1 , \ldots ,t_n } \right)A_{kj} \left( {s_1 , \ldots ,s_m } \right) = C_k \left( {s_1 , \ldots ,s_m } \right)} } \\ \end{array} $$

where l, n, m are positive integers, the B’s are nonnegative linear polynomials over t ₁, ..., t _n (i.e., they are of the form b ₀ + b ₁ t ₁ + ... + b _n t _n, where each b _i is a nonnegative integer), and the A’s and C’s are nonnegative linear polynomials over s ₁, ..., s _m. We show that it is decidable to determine, given any 2-system, whether it has a solution in s ₁, ..., s _m, t ₁, ..., t _n, 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.

The research of Oscar H. Ibarra has been supported in part by NSF Grants IIS-0101134 and CCR02-08595.