Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1811.01347

Some Results on the Circuit Complexity of Bounded Width Circuits and Nondeterministic Circuits

Hiroki Morizumi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that there is a Boolean function $f$ which cannot be computed by any nondeterministic circuit of size $O(n)$ and width $o(n)$. To the best of our knowledge, this is the first result on the lower bound of (nonuniform) bounded width circuits computing an explicit Boolean function, even for deterministic circuits. Actually, we prove a more generalized lower bound. Our proof outline for the lower bound also provides a satisfiability algorithm for nondeterministic bounded width circuits. In the second part of this paper, we consider the power of nondeterministic circuits. We prove that there is a Boolean function $f$ such that the nondeterministic $U_2$-circuit complexity of $f$ is at most $2n + o(n)$ and the deterministic $U_2$-circuit complexity of $f$ is $3n - o(n)$. This is the first separation on the power of deterministic and nondeterministic circuits for general circuits. In the third part of this paper, we show a relation between deterministic bounded width circuits and nondeterministic bounded width circuits. As the main consequence, we prove that $\mathsf{L/quasipoly} \supseteq \mathsf{NL/poly}$. As a corollary, we obtain that $\mathsf{L/quasipoly} \supset \mathsf{NL}$. To the best of our knowledge, this is the first result on $\mathsf{L}$ with large (more precisely, superpolynomial size) advice.

Explore related subjects

cs.CC

Keep this discovery

BibTeXRIS

Hiroki Morizumi. 2018-11-04. Some Results on the Circuit Complexity of Bounded Width Circuits and Nondeterministic Circuits. https://arxiv.org/abs/1811.01347

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↗