Algorithmic Cost in "Exact Real Computation"

Abstract

Turing completeness of a programming language or system characterizes its expressive power; and the strong Church-Turing hypo-/thesis refines such from qualitative to polynomial-time equivalence. Exact Real Computation (ERC) is a novel numerical programming language paradigm: for the imperative processing of continuous data as entities appearing as exact, i.e. devoid of rounding errors [doi:10.1007/978-3-662-44199-2_107]. ERC has been designed [doi:10.46298/lmcs-20(2:17)2024] as convenient and practical alternative, namely proven qualitatively equivalent, to the Turing machines originally underlying Computable Analysis [doi:10.1007/978-3-642-56999-9,doi:10.1007/978-1-4684-6802-1]. The present work quantitatively strengthens this qualitative Turing-completeness: We assign bit-costs to ERC's operational primitives (including partial/multivalued tests) in such a way that any real function incurring polynomial cost becomes Turing-computable in polynomial time, and vice versa. Runtime measurements on implementations in the iRRAM C++ library confirm our theoretical performance predictions.