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On the dimension of the solution space of linear difference equations over the ring of infinite sequences

Sergei Abramov, Gleb Pogudin

TL;DR

This work analyzes the dimension of the solution space for linear difference equations whose coefficients are two-sided computable sequences. It proves that both testing the existence of a nonzero solution and computing the exact dimension are algorithmically undecidable in general, and undecidability persists even under promises that the dimension lies in a finite set. A decidable boundary is identified for periodic coefficients, where the problem is reduced to a module over the Laurent polynomial ring and solved via Gröbner basis/Hermite normal form computations. The results clarify the boundary between decidability and undecidability for difference equations with infinite-sequence data and connect to $C^2$-finite sequence theory.

Abstract

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information on the dimension is available.

On the dimension of the solution space of linear difference equations over the ring of infinite sequences

TL;DR

This work analyzes the dimension of the solution space for linear difference equations whose coefficients are two-sided computable sequences. It proves that both testing the existence of a nonzero solution and computing the exact dimension are algorithmically undecidable in general, and undecidability persists even under promises that the dimension lies in a finite set. A decidable boundary is identified for periodic coefficients, where the problem is reduced to a module over the Laurent polynomial ring and solved via Gröbner basis/Hermite normal form computations. The results clarify the boundary between decidability and undecidability for difference equations with infinite-sequence data and connect to -finite sequence theory.

Abstract

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information on the dimension is available.
Paper Structure (5 sections, 9 theorems, 9 equations)

This paper contains 5 sections, 9 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Possible value of dimension
  3. Existence of nonzero solutions
  4. Computing dimension with a priori knowledge
  5. A case of decidability

Key Result

Theorem 1

Let $M$ be a set with $1 < |M| < \infty$. Then there is no algorithm which, for a given computable $c(0), c(1), \dots$ taking values in $M$ and any $a\in M$, determines whether the sequence contains an element equal to $a$.

Theorems & Definitions (16)

  • Theorem : Tur36
  • Proposition 1
  • proof : Proof of Proposition \ref{['lem:possible_dimensions']}
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • ...and 6 more