Linear Equations with Ordered Data

TL;DR

This paper extends the concept of linear equations to ordered data vectors, revealing complex computational relationships with vector addition systems and identifying conditions for polynomial-time solvability.

Contribution

It introduces a new generalization of linear equations for ordered data vectors and analyzes their computational complexity, contrasting with unordered data cases.

Findings

Nonnegative-integer solvability is equivalent to the reachability problem in vector addition systems.

The complexity of solving these equations is exponential in the general case.

Dropping the nonnegative-integer restriction allows polynomial-time solutions.

Abstract

Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an exponential blowup) with the reachability problem for (plain) vector addition systems. This high complexity is surprising, and contrasts with NP-completeness for unordered data vectors. Also surprisingly, we achieve polynomial time complexity of the solvability problem when the nonnegative-integer restriction on solutions is dropped.