Invitation to Real Complexity Theory: Algorithmic Foundations to Reliable Numerics with Bit-Costs

TL;DR

This paper introduces Real Complexity Theory, a resource-based foundation for rigorous numerical computations over continuous data, providing sound semantics, compositionality, and links to complexity classes.

Contribution

It proposes a novel theoretical framework for reliable numerical algorithms grounded in complexity theory, bridging the gap between theoretical computer science and numerical engineering.

Findings

Provides a formal semantics for continuous computations

Ensures closure under composition of algorithms

Relates numerical complexity to classical complexity classes

Abstract

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and efficiency is demonstrated empirically. We advertise REAL COMPLEXITY THEORY: a resource-oriented foundation to rigorous computations over continuous universes such as real numbers, vectors, sequences, continuous functions, and Euclidean subsets: in the bit-model by approximation up to given absolute error. It offers sound semantics (e.g. of comparisons/tests), closure under composition, realistic runtime predictions, and proofs of algorithmic optimality by relating to known classes like NP, #P, PSPACE.