Computing with locally effective matrices

Abstract

In this work, we start from the naive notion of integer infinite matrix (i.e., the functions of the set ℤ^{ℕ × ℕ} = {f: ℕ × ℕ → ℤ} ). Then, several undecidability results are established, leading to a convenient data structure for effective machine computations. We call this data structure a locally effective matrix. We study when (and how) the standard matrix calculus (Ker and CoKer computations) can be extended to the infinite case. We find again several undecidability barriers. When these limitations are overcome, we describe effective procedures for computing in the locally effective case. Finally, the role played by these data structures in the development of real symbolic computation systems for algebraic topology (based on the effective homology notion) is illustrated.

Keywords:

• Data structure
• Locally effective matrix
• Algebraic topology

Acknowledgements

The authors were partially supported by MCyT and FEDER, project TIC2002-01626.