The graph of multiplication is equivalent to counting

doi:10.1016/0020-0190(92)90180-4

Information Processing Letters

Volume 41, Issue 4, 18 March 1992, Pages 199-201

https://doi.org/10.1016/0020-0190(92)90180-4 Get rights and content

Abstract

Counting is AC⁰-reducible to the graph of multiplication. Hence the graph of multiplication is equivalent under AC⁰ reductions to majority and to the function form of multiplication.

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Cited by (6)

A lower bound for integer multiplication on randomized ordered read-once branching programs
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We prove an exponential lower bound $2^{Ω(n/logn)}$ on the size of any randomized ordered read-once branching program computing integer multiplication. Our proof depends on proving a new lower bound on Yao’s randomized one-way communication complexity of certain Boolean functions. It generalizes to some other models of randomized branching programs. In contrast, we prove that testing integer multiplication, contrary even to a nondeterministic situation, can be computed by randomized ordered read-once branching program in polynomial size. It is also known that computing the latter problem with deterministic read-once branching programs is as hard as factoring integers.
A bounded arithmetic AID for Frege systems
2000, Annals of Pure and Applied Logic
In this paper we introduce a system $AID$ (alogtime inductive definitions) of bounded arithmetic. The main feature of $AID$ is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems $F$ in Buss (Ann. Pure Appl. Logic 52 (1991) 3–29). We show that $AID$ proves the soundness of $F$ , and conversely any $Σ_{0}^{b}$ -theorem in $AID$ yields boolean sentences of which $F$ has polysize proofs. Further we define $Σ_{1}^{b}$ -faithful interpretations between $AID+Σ_{0}^{b}$ -CA and a quantified theory $QALV$ of an equational system $ALV$ in Clote (Ann. Math. Art. Intell. 6 (1992) 57–106). Hence $ALV$ also proves the soundness of $F$ .
Chapter 17 Computation models and function algebras
1999, Studies in Logic and the Foundations of Mathematics
Efficient data structures for Boolean functions
1994, Discrete Mathematics
The hardware of computers, e.g. circuits, sequential circuits or VLSI chips, realizes Boolean functions. The design of efficient hardware is a fundamental issue in computer design. Because of the large cost for the physical construction of a new chip, the logical synthesis and the verification as well as the generation of test patterns have to be performed before the chip is built. For these purposes data structures for Boolean functions supporting operations like the evaluation on a given input, the satisfiability test, the synthesis of a representation for f = g ⊗ h for some binary operator ⊗ from representations for g and h are necessary tools. The corresponding state of the art data structure is the ordered binary decision diagram (OBDD). Efficient algorithms for the operations on OBDDs and the expressive power of OBDDs of polynomial size are discussed. A generalized data structure called graph-driven binary decision diagram is presented. The new data structure allows for many important functions a representation of polynomial size, even for functions whose representations by OBDDs have exponential size. Efficient algorithms for the operations on graph driven BDDs are outlined.
A lower bound for integer multiplication with read-once branching programs
1999, SIAM Journal on Computing
VTC <sup>0</sup>: A second-order theory for TC <sup>0</sup>
2004, Proceedings - Symposium on Logic in Computer Science

^∗: Supported in part by NSF Grant DMS-8902480.

View full text

The graph of multiplication is equivalent to counting

Abstract

On uniformity within NC1

J. Comput. System Sci.

Bounded Arithmetic

Revision of Ph.D. Thesis

The Boolean formula value problem is in ALOGTIME

Proc. 19th Ann. ACM Symp. on Theory of Computing

Constant depth reducibility

SIAM J. Comput.

On uniformity within NC¹