Linear Combinations of Unordered Data Vectors

TL;DR

This paper investigates the expressibility of data vectors as finite sums of base vectors, providing complexity bounds and reductions for reversible cases, with applications to counter machines and Petri nets.

Contribution

It introduces a bounded witness approach for data vector expressibility, characterizes reversibility over groups, and establishes complexity results for related decision problems.

Findings

Positive instances can be witnessed in a bounded domain subset.

Expressibility over groups reduces to subgroup membership checks.

Reversibility simplifies the problem to subgroup membership, with NP and P complexity results.

Abstract

Data vectors generalise finite multisets: they are finitely supported functions into a commutative monoid. We study the question if a given data vector can be expressed as a finite sum of others, only assuming that 1) the domain is countable and 2) the given set of base vectors is finite up to permutations of the domain. Based on a succinct representation of the involved permutations as integer linear constraints, we derive that positive instances can be witnessed in a bounded subset of the domain. For data vectors over a group we moreover study when a data vector is reversible, that is, if its inverse is expressible using only nonnegative coefficients. We show that if all base vectors are reversible then the expressibility problem reduces to checking membership in finitely generated subgroups. Moreover, checking reversibility also reduces to such membership tests. These questions naturally appear in the analysis of counter machines extended with unordered data: namely, for data vectors over $(\mathbb{Z}^d,+)$ expressibility directly corresponds to checking state equations for Coloured Petri nets where tokens can only be tested for equality. We derive that in this case, expressibility is in NP, and in P for reversible instances. These upper bounds are tight: they match the lower bounds for standard integer vectors (over singleton domains).