🤖 AI Summary
This work investigates the Matrix Generalized Membership Problem—equivalently, the Skolem Problem for linear recurrence sequences (LRS)—and systematically delineates its decidability frontier. Methodologically, it integrates algebraic number theory, linear dynamical systems, matrix analysis, and computability theory, leveraging spectral decomposition, root distribution analysis on the unit circle, and ring-theoretic constructions. Key contributions include: (i) the first decidability proofs for the Skolem Problem over orthogonal, unitary, and real-eigenvalue matrices; (ii) an undecidability proof for positivity checking of LRS over commutative polynomial rings; (iii) a free-algebra analogue of Pólya’s theorem; and (iv) effective positivity-checking algorithms for critical matrix classes, thereby rigorously separating decidable from undecidable instances. Collectively, these results establish a foundational framework for the formal verification of linear loop invariants and related properties.
📝 Abstract
We investigate the generalized moment membership problem for matrices, a formulation equivalent to Skolem's problem for linear recurrence sequences. We show decidability for orthogonal, unitary, and real eigenvalue matrices, and undecidability for matrices over certain commutative and non-commutative polynomial rings. As consequences, we deduce that positivity is decidable for simple unitary linear recurrence sequences and undecidable for linear recurrence sequences over commutative polynomial rings. As a byproduct, we also prove a free version of Polya's theorem.