Memoryless computation: new results, constructions, and extensions

Maximilien Gadouleau; Soren Riis

Home/Archive/arXiv

arXiv · 1111.6026

Memoryless computation: new results, constructions, and extensions

Maximilien Gadouleau·Soren Riis

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we are interested in memoryless computation, a modern paradigm to compute functions which generalises the famous XOR swap algorithm to exchange the contents of two variables without using a buffer. This uses a combinatorial framework for procedural programming languages, where programs are only allowed to update one variable at a time. We first consider programs which do not have any memory. We prove that any function of $n$ variables can be computed this way in only $4n-3$ variable updates. We then derive the exact number of instructions required to compute any manipulation of variables. This shows that combining variables, instead of simply moving them around, not only allows for memoryless programs, but also yields shorter programs. Second, we show that allowing programs to use memory is also incorporated in the memoryless computation framework. We then quantify the gains obtained by using memory: this leads to shorter programs and allows us to use only binary instructions, which is not sufficient in general when no memory is used.

Explore related subjects

cs.CC math.CO

Keep this discovery

BibTeX RIS

Maximilien Gadouleau, Soren Riis. 2011-11-25. Memoryless computation: new results, constructions, and extensions. https://arxiv.org/abs/1111.6026

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

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

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↗