Separability in Büchi Vass and Singly Non-Linear Systems of Inequalities

TL;DR

The paper resolves the complexity of regular separability for Büchi VASS languages, proving $EXPSPACE$-completeness and $PSPACE$-completeness in fixed dimension. It introduces singly non-linear systems of inequalities (SNLS) to model the non-linear constraints that arise in inseparability witnesses and shows that SNLS feasibility is tractable: if a solution exists, there is one with exponentially bounded numerators/denominators, via quantifier elimination and root separation. A Rackoff-like bound is then developed for SNLS within the Büchi VASS setting, yielding doubly-exponential witness lengths; combined with Demri’s selective unboundedness, this leads to an $EXPSPACE$ decision procedure, with improved bounds in fixed dimension. The results provide tight complexity bounds and a practical framework for certificates-based separability in infinite-state systems, linking non-linear inequality analysis with automata-theoretic verification methods.

Abstract

The omega-regular separability problem for Büchi VASS coverability languages has recently been shown to be decidable, but with an EXPSPACE lower and a non-primitive recursive upper bound -- the exact complexity remained open. We close this gap and show that the problem is EXPSPACE-complete. A careful analysis of our complexity bounds additionally yields a PSPACE procedure in the case of fixed dimension >= 1, which matches a pre-established lower bound of PSPACE for one dimensional Büchi VASS. Our algorithm is a non-deterministic search for a witness whose size, as we show, can be suitably bounded. Part of the procedure is to decide the existence of runs in VASS that satisfy certain non-linear properties. Therefore, a key technical ingredient is to analyze a class of systems of inequalities where one variable may occur in non-linear (polynomial) expressions. These so-called singly non-linear systems (SNLS) take the form A(x).y >= b(x), where A(x) and b(x) are a matrix resp. a vector whose entries are polynomials in x, and y ranges over vectors in the rationals. Our main contribution on SNLS is an exponential upper bound on the size of rational solutions to singly non-linear systems. The proof consists of three steps. First, we give a tailor-made quantifier elimination to characterize all real solutions to x. Second, using the root separation theorem about the distance of real roots of polynomials, we show that if a rational solution exists, then there is one with at most polynomially many bits. Third, we insert the solution for x into the SNLS, making it linear and allowing us to invoke standard solution bounds from convex geometry. Finally, we combine the results about SNLS with several techniques from the area of VASS to devise an EXPSPACE decision procedure for omega-regular separability of Büchi VASS.