Complex Algebras of Arithmetic

An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper, we investigate the algebraic structure of complex algebras of natural numbers, and make some observations regarding the complexity of various theories of such algebras

A Logical Characterization of Constant-Depth Circuits over the Reals

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202

Simplified Floating-Point Units for High Dynamic Range Image and Video Systems

The upcoming arrival of high dynamic range image and video applications to consumer electronics will force the utilization of floating-point numbers on them. This paper shows that introducing a slight modification on classical floating-point number systems, the implementation of those circuits can be highly improved. For a 16-bit numbers, by using the proposed format, the area and power consumption of a floating-point adder is reduced up to 70% whereas those parameters are maintained for the case of a multiplier.This work was supported in part by the Ministry of Education and Science of Spain and Junta of Andalucía under contracts TIN2013-42253-P and TIC-1692, respectively, and Universidad de Málaga.Campus de Excelencia Internacional Andalucía Tech

Multiple Product Modulo Arbitrary Numbers

AbstractLetnbinary numbers of lengthnbe given. The Boolean function “Multiple Product”MPnasks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems,SIAM J. Discrete Math.7, 285–292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits of depth 2. In this paper, we investigate the complexity of computingMPnmodulomby depth-2 threshold circuits. It turns out that for all but a few integersm, exponential size is required. In particular, it is shown that form∈{2, 4, 8}, polynomial-size circuits exist, form∈{3, 6, 12, 24}, the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allowmto grow withn

On the class of graphs with strong mixing properties

We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph