Algebraic Closure of Matrix Sets Recognized by 1-VASS

2025-07-12

AI Summary

This work addresses the decidability of computing the Zariski closure of matrix sets generated by one-dimensional vector addition systems with states (1-VASS), thereby establishing the boundary of decidability for algebraic closures of matrix sets corresponding to context-free languages. We present the first effective algorithm for computing the Zariski closure of matrix sets recognized by counter languages, and prove undecidability for indexed languages—corresponding to nested-stack automata—thus precisely characterizing the decidability frontier. Our method integrates 1-VASS semantics, matrix semigroup theory, and Simon’s factorization forest technique to develop novel analytical tools for infinite matrix monoids. The main result is a decidable procedure for computing the Zariski closure of matrix sets generated by 1-VASS. This yields the first nontrivial decidable subclass for automatic derivation of polynomial invariants in affine programs featuring recursive procedure calls.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Compute Zariski closure for context-free matrix sets
• Decide polynomial invariants for recursive affine programs
• Handle algebraic closure for indexed language matrices

Innovation

• Compute Zariski closure for one-counter languages
• Undecidability shown for indexed languages
• Adapt Simon's factorization forests for matrices