arXiv subjects

Hiroki Morizumi

Publications and source records attributed to Hiroki Morizumi.

6 recordsLinked to original sources

arXiv2023

An Approach to Circuit Lower Bounds via Bounded Width Circuits

In this paper, we investigate an approach to circuit lower bounds via bounded width circuits. The approach consists of two steps: (i) We convert circuits to (deterministic or nondeterministic) bounded width circuits. (ii) We prove lower bounds for the bounded width circuits. For the second step, we prove that there is an explicit Boolean function $f$ as follows: If a nondeterministic circuit of size $s$ and width $w$ computes $f$, then $w = Ω(\frac{n^4}{4^{\frac{s}{n}}s^3}) - \frac{\log_2 s}{2}$. For the first step, we give some observations, which include a relation between conversions to bounded width circuits and a standard pebble game.

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arXiv2023

Computation with Large Advice

In this paper, we consider a new direction of computation, which we call computation with large advice. We mainly consider constant space computation with large advice in Turing machines, and prove the following facts: (i) The class of decision problems solvable by a constant space Turing machine with polynomial-size advice includes nonuniform-{\sf NC}$^1$, (ii) The class of decision problems solvable by a constant space Turing machine with quasipolynomial-size advice equals nonuniform-{\sf polyL}. The facts mean constant space computation with large advice has unexpected computational power. On the other hand, we mention bounded time computation with large advice, and attempt to propose a concept of ``algorithms with large advice''. In the proposal, advice is precomputed data for a problem and a fixed instance size, and we expect efficient algorithms by large or huge advice.

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arXiv2019

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

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.

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arXiv2016

Zero-Suppressed Computation: A New Computation Inspired by ZDDs

Zero-suppressed binary decision diagrams (ZDDs) are a data structure representing Boolean functions, and one of the most successful variants of binary decision diagrams (BDDs). On the other hand, BDDs are also called branching programs in computational complexity theory, and have been studied as a computation model. In this paper, we consider ZDDs from the viewpoint of computational complexity theory. Our main proposal of this paper is that we regard the basic idea of ZDDs as a new computation, which we call zero-suppressed computation. We consider the zero-suppressed version of two classical computation models, decision trees and branching programs, and show some results. Although this paper is mainly written from the viewpoint of computational complexity theory, the concept of zero-suppressed computation can be widely applied to various areas.

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arXiv2015

Lower Bounds for the Size of Nondeterministic Circuits

Nondeterministic circuits are a nondeterministic computation model in circuit complexity theory. In this paper, we prove a $3(n-1)$ lower bound for the size of nondeterministic $U_2$-circuits computing the parity function. It is known that the minimum size of (deterministic) $U_2$-circuits computing the parity function exactly equals $3(n-1)$. Thus, our result means that nondeterministic computation is useless to compute the parity function by $U_2$-circuits and cannot reduce the size from $3(n-1)$. To the best of our knowledge, this is the first nontrivial lower bound for the size of nondeterministic circuits (including formulas, constant depth circuits, and so on) with unlimited nondeterminism for an explicit Boolean function. We also discuss an approach to proving lower bounds for the size of deterministic circuits via lower bounds for the size of nondeterministic restricted circuits.

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arXiv2008

A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas

In this note, we consider the minimum number of NOT operators in a Boolean formula representing a Boolean function. In circuit complexity theory, the minimum number of NOT gates in a Boolean circuit computing a Boolean function $f$ is called the inversion complexity of $f$. In 1958, Markov determined the inversion complexity of every Boolean function and particularly proved that $\lceil \log_2(n+1) \rceil$ NOT gates are sufficient to compute any Boolean function on $n$ variables. As far as we know, no result is known for inversion complexity in Boolean formulas, i.e., the minimum number of NOT operators in a Boolean formula representing a Boolean function. The aim of this note is showing that we can determine the inversion complexity of every Boolean function in Boolean formulas by arguments based on the study of circuit complexity.

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