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

A k-system consists of k quadratic Diophantine equations over nonnegative integer variables s1, ..., sm, t1, ..., tn of the form: Σ 1≤j≤l B1j(t1, ..., tn)A1j(s1, ..., sm) = C1(s1, ..., sm) Σ 1≤j≤l Bkj(t1, ..., tn)Akj(s1, ..., sm) = Ck(s1, ..., sm) 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.

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