Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2603.29427

Beyond Bits: An Introduction to Computation over the Reals

Tillmann Miltzow
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a lightweight and accessible approach to computation over the real numbers, with the aim of clarifying both the underlying concepts and their relevance in modern research. The material is intended for a broad audience, including instructors who wish to incorporate real computation into algorithms courses, their students, and PhD students encountering the subject for the first time. Rather than striving for completeness, we focus on a carefully selected set of results that can be presented and proved in a classroom setting. This allows us to highlight core techniques and recurring ideas while maintaining an approachable exposition. In some places, the presentation is intentionally informal, prioritizing intuition and practical understanding over full technical precision. We position our exposition relative to existing literature, including Matousek's lecture notes on ER-completeness and the recent compendium of ER-complete problems by Schaefer, Cardinal, and Miltzow. While these works provide deep and comprehensive perspectives, our goal is to offer an accessible entry point with proofs and examples suitable for teaching. Our approach follows modern formulations of real computation that emphasize binary input, real-valued witnesses, and restricted use of constants, aligning more closely with contemporary complexity theory, while acknowledging the foundational contributions of the Blum--Shub--Smale model.

Explore related subjects

cs.CCcs.CGcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tillmann Miltzow. 2026-03-31. Beyond Bits: An Introduction to Computation over the Reals. https://arxiv.org/abs/2603.29427

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗