Algebraic Closure of Matrix Sets Recognized by 1-VASS

TL;DR

This paper investigates the algebraic closure of matrix sets recognized by one-counter automata, providing a decidable procedure for certain classes and proving undecidability for more complex language classes.

Contribution

It introduces a method to compute the Zariski closure for matrix sets recognized by one-counter languages and establishes undecidability results for indexed languages.

Findings

Decidable procedure for one-counter language-recognized matrix sets

Undecidability results for matrix sets recognized by indexed languages

Novel adaptation of Simon's factorization forests for infinite matrix monoids

Abstract

It is known how to compute the Zariski closure of a finitely generated monoid of matrices and, more generally, of a set of matrices specified by a regular language. This result was recently used to give a procedure to compute all polynomial invariants of a given affine program. Decidability of the more general problem of computing all polynomial invariants of affine programs with recursive procedure calls remains open. Mathematically speaking, the core challenge is to compute the Zariski closure of a set of matrices defined by a context-free language. In this paper, we approach the problem from two sides: Towards decidability, we give a procedure to compute the Zariski closure of sets of matrices given by one-counter languages (that is, languages accepted by one-dimensional vector addition systems with states and zero tests), a proper subclass of context-free languages. On the other side, we show that the problem becomes undecidable for indexed languages, a natural extension of context-free languages corresponding to nested pushdown automata. One of our main technical tools is a novel adaptation of Simon's factorization forests to infinite monoids of matrices.