Eike Neumann - Decision problems for linear recurrences involving arbitrary real numbers

lmcs:6880 - Logical Methods in Computer Science, August 10, 2021, Volume 17, Issue 3 - https://doi.org/10.46298/lmcs-17(3:16)2021
Decision problems for linear recurrences involving arbitrary real numbersArticle

Authors: Eike Neumann

    We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.

    Volume: Volume 17, Issue 3
    Published on: August 10, 2021
    Accepted on: June 16, 2021
    Submitted on: November 4, 2020
    Keywords: Computer Science - Logic in Computer Science


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