Journal of Applied Mathematics

Volume 2015 (2015), Article ID 827572, 7 pages

http://dx.doi.org/10.1155/2015/827572

## Function Synthesis Algorithm of RTD-Based Universal Threshold Logic Gate

^{1}Hangzhou Institute of Service Engineering, Hangzhou Normal University, Hangzhou 311121, China^{2}Department of Information Science & Electronic Engineering, Zhejiang University, Hangzhou 310027, China

Received 19 March 2015; Revised 13 May 2015; Accepted 31 May 2015

Academic Editor: Georgios Sirakoulis

Copyright © 2015 Maoqun Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The resonant tunneling device (RTD) has attracted much attention because of its unique negative differential resistance characteristic and its functional versatility and is more suitable for implementing the threshold logic gate. The universal logic gate has become an important unit circuit of digital circuit design because of its powerful logic function, while the threshold logic gate is a suitable unit to design the universal logic gate, but the function synthesis algorithm for the -variable logical function implemented by the RTD-based universal logic gate (UTLG) is relatively deficient. In this paper, three-variable threshold functions are divided into four categories; based on the Reed-Muller expansion, two categories of these are analyzed, and a new decomposition algorithm of the three-variable nonthreshold functions is proposed. The proposed algorithm is simple and the decomposition results can be obtained by looking up the decomposition table. Then, based on the Reed-Muller algebraic system, the arbitrary -variable function can be decomposed into three-variable functions, and a function synthesis algorithm for the -variable logical function implemented by UTLG and XOR2 is proposed, which is a simple programmable implementation.

#### 1. Introduction

With the improvement in integrated circuit integration, the complementary metal oxide semiconductor (CMOS) technology is gradually approaching its physical limitations. The resonant tunneling device (RTD) has better performance and features, such as negative differential resistance characteristic, self-latching, high speed, and functional versatility [1, 2]. The universal logic gate, which has a powerful logic function, has become an important unit to implement -variable logical functions [3], and the RTD is more suitable for implementing the universal logic gate because of its negative differential resistance characteristic [4–6]. So, the RTD will probably become the main electronic device in the next generation of integrated circuits [7, 8].

Though the circuit of an -variable logical function implemented by the universal logic gate will be simpler, a different universal logic gate requires its corresponding synthesis algorithm to implement a function. Some function synthesis algorithms have been proposed in the literature [9–12], but these algorithms are not suitable for implementing an arbitrary -variable function by the RTD-based universal threshold logic gate (UTLG) [13]. And the algorithm [14] which can implement a three-variable nonthreshold function by UTLGs is relatively complicated, and the implemented circuit structure is also complicated.

In this paper, based on the Reed-Muller expansion, the three-variable nonthreshold functions are classified. Two categories of these are analyzed, and a new decomposition algorithm of the three-variable nonthreshold functions is proposed. Then a function synthesis algorithm which can implement an arbitrary -variable logical function by UTLGs is proposed. The proposed function synthesis algorithm provides a new scheme for designing integrated circuits by RTD devices.

#### 2. Background

##### 2.1. Threshold Logic

A threshold logic gate is defined as a logic gate with binary input variables and a single binary output. Its internal parameters are as follows: binary input variables, , a set of integer weights, , and a threshold and an output , such that its input-output relationship can be expressed as [15]

Formula (1) can also be presented as . If a logic function can be implemented with a single threshold logic gate, the function is called a threshold function; otherwise, it is called a nonthreshold function [15].

##### 2.2. Spectral Technique

Spectral technique is a mathematical transformation method. It can convert binary data from the Boolean domain into the spectral domain by matrix transformations, and the information will not be lost [16]. In the spectral domain , for an -variable logical function , its input and output have kinds of states, and the truth vector is

The spectral-coefficient vector is given bywhere is a Rademacher-Walsh matrix.

As for a three-variable function, the spectral-coefficient vector is , where is a zero-order spectral-coefficient, and , , and are one-order, two-order, and three-order spectral coefficients, respectively.

##### 2.3. Reed-Muller Expansion

A Reed-Muller expansion is a standard expansion in the AND/XOR algebraic system. A given function can be expressed as the XOR of basic entry; its coefficient is called the RM expansion coefficient [17]. Given an -variable function , its RM expansion coefficient vector (0-polarity; in this paper we only use 0-polarity expansion coefficient) iswhere , , is the Kronecker product, and is the truth vector of the function in the Boolean domain. As for a three-variable function , its RM expansion is

#### 3. Decomposition Algorithm of Three-Variable Nonthreshold Functions

In this section, the three-variable nonthreshold functions are classified, and a new decomposition algorithm of the three-variable nonthreshold functions is proposed.

##### 3.1. Determine the Three-Variable Threshold Function

The zero-order and one-order spectral-coefficient can determine whether a function is a threshold function [16]. The spectral-coefficient classification table of all the three-variable threshold functions is given in Table 1, and some conclusions as follows [16].