Reachability in Two-Dimensional Unary Vector Addition Systems with States is NL-Complete

TL;DR

This paper proves that reachability witnesses in two-dimensional unary vector addition systems with states always have pseudo-polynomial length, enabling a more space-efficient verification algorithm when vectors are given in unary.

Contribution

It establishes the existence of pseudo-polynomial length witnesses in unary VASS, resolving an open question and improving the complexity of reachability verification.

Findings

Existence of pseudo-polynomial length witnesses in unary VASS

Improved logarithmic space algorithm for reachability verification in unary case

Resolution of an open problem from Blondin et al. 2015

Abstract

Blondin et al. showed at LICS 2015 that two-dimensional vector addition systems with states have reachability witnesses of length exponential in the number of states and polynomial in the norm of vectors. The resulting guess-and-verify algorithm is optimal (PSPACE), but only if the input vectors are given in binary. We answer positively the main question left open by their work, namely establish that reachability witnesses of pseudo-polynomial length always exist. Hence, when the input vectors are given in unary, the improved guess-and-verify algorithm requires only logarithmic space.