Research Article Function Synthesis Algorithm of RTD-Based Universal Threshold Logic Gate

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1 Applied Mathematics Volume 5, Article ID 8757, 7 pages Research Article Function Synthesis Algorithm o RTD-Based Universal Threshold Logic Gate Maoqun Yao, Kai Yang, Congyuan Xu, and Jizhong Shen Hangzhou Institute o Service Engineering, Hangzhou Normal University, Hangzhou 3, China Department o Inormation Science & Electronic Engineering, Zhejiang University, Hangzhou 37, China Correspondence should be addressed to Maoqun Yao; yaomaoqun@63.com Received 9 March 5; Revised 3 May 5; Accepted 3 May 5 Academic Editor: Georgios Sirakoulis Copyright 5 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. The resonant tunneling device (RTD) has attracted much attention because o its unique negative dierential resistance characteristic and its unctional versatility and is more suitable or implementing the threshold logic gate. The universal logic gate has become an important unit circuit o digital circuit design because o its powerul logic unction, while the threshold logic gate is a suitable unit to design the universal logic gate, but the unction synthesis algorithm or the n-variable logical unction implemented by the RTD-based universal logic gate (UTLG) is relatively deicient. In this paper, three-variable threshold unctions are divided into our categories; based on the Reed-Muller expansion, two categories o these are analyzed, and a new decomposition algorithm o the three-variable nonthreshold unctions 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 n-variable unction can be decomposed into three-variable unctions, and a unction synthesis algorithm or the n-variable logical unction implemented by UTLG and XOR is proposed, which is a simple programmable implementation.. 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 perormance and eatures, such as negative dierential resistance characteristic, sel-latching, high speed, and unctional versatility [, ]. The universal logic gate, which has a powerul logic unction, has become an important unit to implement n-variable logical unctions [3], and the RTD is more suitable or implementing the universal logic gate because o its negative dierential resistance characteristic [4 6]. So, the RTD will probably become the main electronic device in the next generation o integrated circuits [7, 8]. Though the circuit o an n-variable logical unction implemented by the universal logic gate will be simpler, a dierent universal logic gate requires its corresponding synthesis algorithm to implement a unction. Some unction synthesis algorithms have been proposed in the literature [9 ], but these algorithms are not suitable or implementing an arbitrary n-variable unction by the RTD-based universal threshold logic gate (UTLG) [3]. And the algorithm [4] which can implement a three-variable nonthreshold unction 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 unctions are classiied. Two categories o these are analyzed, and a new decomposition algorithm o the three-variable nonthreshold unctions is proposed. Then a unction synthesis algorithm which can implement an arbitrary n-variable logical unctionby UTLGs is proposed. The proposed unction synthesis algorithm provides a new scheme or designing integrated circuits by RTD devices.. Background.. Threshold Logic. A threshold logic gate is deined as a logic gate with n binary input variables and a single binary output. Its internal parameters are as ollows: n binary input

2 Applied Mathematics variables, {x i } (i =,,...,n), a set o integer weights, {w i } (i=,,...,n), and a threshold T and an output,such that its input-output relationship can be expressed as [5] {, i w i x i T = { i= {, otherwise. Formula () can also be presented as = w x + +w n x n T. I a logic unction can be implemented with a single threshold logic gate, the unction is called a threshold unction; otherwise, it is called a nonthreshold unction [5]... Spectral Technique. Spectral technique is a mathematical transormation method. It can convert binary data rom the Boolean domain into the spectral domain by matrix transormations, and the inormation will not be lost [6]. In the spectral domain {, +}, orann-variable logical unction (x,...,x n ), its input and output have n kinds o states, and the truth vector is Y=((+,+,...,+) () (+,+,..., ) (,,..., )) T. The spectral-coeicient vector R is given by n R=(r r r n r r,...,n ) T =T n Y, (3) where T n is a n nrademacher-walsh matrix. As or a three-variable unction, the spectral-coeicient vector is R = (r r 3 r r 3 r r 3 r r 3 ) T,wherer is a zero-order spectral-coeicient, and r i, r ij,andr ijk are one-order, two-order, and three-order spectral coeicients, respectively..3. Reed-Muller Expansion. A Reed-Muller expansion is a standard expansion in the AND/XOR algebraic system. A given unction can be expressed as the XOR o basic entry; its coeicient is called the RM expansion coeicient [7]. Given an n-variable unction (x,...,x n ), its RM expansion coeicient vector B (-polarity; in this paper we only use - polarity expansion coeicient) is B=[T] n F, (4) where [T] n = [T] [T] [T], [T] = ( ), is the n Kronecker product, and F is the truth vector o the unction in the Boolean domain. As or a three-variable unction (x,x,x 3 ), its RM expansion is (x,x,x 3 )=b b x 3 b x b 3 x x 3 b 4 x (5) b 5 x x 3 b 6 x x b 7 x x x Decomposition Algorithm o Three-Variable Nonthreshold Functions In this section, the three-variable nonthreshold unctions are classiied, and a new decomposition algorithm o the threevariable nonthreshold unctions is proposed. () Table : Spectral-coeicient r i classiication o all the threevariable threshold unctions. n r i a i 8 n= Determine the Three-Variable Threshold Function. The zero-order and one-order spectral-coeicient can determine whether a unction is a threshold unction [6]. The spectralcoeicient classiication table o all the three-variable threshold unctions is given in Table, and some conclusions as ollows [6]. () The zero-order and one-order spectral-coeicients o thegivenunctionarecalculatedandarrangedinnumerically descending magnitude order, and this is the r i o the unction; i the r i appear in Table, the unction is a threshold unction.thethresholdvalueoathresholdunctionand weights can be obtained by a i in Table. Therelationship between the w i and listed a i value is w i = a i i=,, 3. (6) The sign o w i is the same as r i.thethresholdt can be calculated as T= 3 ( a i + ). (7) i= () I the unction is not a threshold unction, the derived number r i will not appear in Table. 3.. New Decomposition Algorithm o Three-Variable Nonthreshold Functions. According to the properties o the spectral coeicient, the three-variable nonthreshold unction can be decomposed into three-variable threshold unctions by a series o conversions [4], but or some three-variable nonthreshold unctions, this algorithm [4] is relatively complex. So, we will propose a simpler algorithm which can decompose those three-variable nonthreshold unctions into three-variable threshold unctions. Forallthethree-variableunctions,thereare4threshold unctions and 5 nonthreshold unctions [6]. First,according to Table, the 4 threshold unctions are divided into three categories [6]. () Among the absolute values o all the zero-order and one-order spectral-coeicients, there is one number equal to 8 and three numbers equal to. ()Amongtheabsolutevaluesoallthezero-andoneorder spectral-coeicients, there is one number equal to 6 and three numbers equal to. (3) Among the absolute values o all the zero-order and one-order spectral-coeicients, there is one number equal to and three numbers equal to 4.

3 Applied Mathematics 3 Then, the 5 nonthreshold unctions can be divided into our categories. () Among the absolute values o all the spectral coeicients, the maximum number is rom the two-order spectral coeicient. () Among the absolute values o all the spectral coeicients, the maximum number is rom the three-order spectral coeicient. (3) Among the absolute values o all the zero-order and one-order spectral-coeicients, there is one number equal to 4. (4) Among the absolute values o all the zero-order and one-order spectral-coeicients, there are two numbers equal to 4. The numbers o these our categories o three-variable nonthreshold unctions are 54, 8, 33, and 47, respectively. I the unction belongs to the second and third categories o the three-variable nonthreshold unctions, the algorithm [4] which decomposes the unction will be relatively complex. A nonthreshold unction is decomposed into the threshold unction by the conversion o (x) x i and x i x i x j,sowe propose a new decomposition algorithm o the three-variable nonthreshold unctions based on the Reed-Muller algebraic system, and the decomposition process is as ollows. () The RM expansion coeicient vectors o all the three-variable threshold unctions are calculated as a reerence table. () The RM expansion coeicient vector o the given unction is calculated. (3) Search the reerence table and ind two threshold unctions in which the XOR result o their RM expansion coeicient vectors just equals the calculated RM expansion coeicient vector in step. (4) The three-variable nonthreshold unction can be expressed as the XOR o the ound threshold unctions o step 3. Ater applying the process to the second and third categories o the three-variable nonthreshold unctions, except two special unctions, = x x x 3 and = x x x 3, the others can be presented by the XOR o two threshold unctions; that is, (x,x,x 3 )= (x,x,x 3 ) (x,x,x 3 ),where and are threshold unctions. The decomposition results o the second and third categories o the three-variable nonthreshold unctions are shown in Tables and 3. Inthetables,eachvectorothecolumns is the output o the three-variable unction or an 8-input combination rom to. Taking the irst column o the irst vector ( )as an example in Table,it expresses the three-variable unction = m i (i = 3, 4), where m i is the minterm o the unction, and can be expressed as =,where = m i (i = 3, 4, 5, 6, 7) and = m i (i = 5, 6, 7), and they are the three-variable thresholdunctionsdenotedbythesecondandthirdcolumns o the irst row in Table. 4. The Synthesis Algorithm o n-variable Function Based on UTLG The literature [3] proposed an RTD-based universal logic gate (UTLG) which can implement an arbitrary threevariable threshold unction with a single UTLG. Figure shows the schematic and symbol o UTLG; its input-output relationship can be expressed as = {, c +c +c 3 c 4 c 5 c 6 {, otherwise. { The irst and ourth categories o the three-variable nonthreshold unctions can be implemented by 3 UTLGs, and the second and third categories o the three-variable nonthreshold unctions can be implemented by 7 UTLGs [3]. In our proposed decomposition algorithm o three-variable nonthreshold unctions we introduced a bivariate XOR unction, which cannot be implemented by a single UTLG, so according to the structure o MOBILE circuit [8], we design anrtd-basedbivariatexorgate(xor).figure shows the schematicandsymboloxor.thus,thesecondandthird categories o the three-variable nonthreshold unctions can be implemented by UTLGs and XOR, and it simpliies the circuit. Figure 3 showsthesimulationresultsoxor circuit by HSPICE; the parameters o RTD and HFET are the same as UTLG circuit [3], and as observed rom Figure 3,the proposed XOR has the correct logic unctionality. Currently, there is no algorithm which can implement arbitrary n-variable unctions by the UTLG.Thereore,we proposed an n-variable unction synthesis algorithm based on UTLG. First, we analyze the RM expansion o a our-variable unction (x,x,x 3,x 4 ) [7]; =b b x 4 b x 3 b 3 x 3 x 4 b 4 x b 5 x x 4 b 6 x x 3 b 7 x x 3 x 4 b 8 x b 9 x x 4 b x x 3 b x x 3 x 4 b x x b 3 x x x 4 b 4 x x x 3 b 5 x x x 3 x 4. By observation, the last eight terms o the RM expansion are equal to the irst eight terms o the RM expansion by multiplying x, respectively, and the operations o terms in the RM expansion unction are XOR, so the our-variable unction can be expressed as = x, the variables o are x, x 3 and x 4, and the RM expansion coeicient vector o is equal to the irst eight elements o the RM expansion coeicient vector o.thevariableso are also x, x 3,and x 4, and the RM expansion coeicient vector o is equal to the last eight elements o the RM expansion coeicient vector o. For an n-variableunction,basedonthereed-muller algebraic system [7],itcanalsobeexpressedas= x, so we propose the synthesis algorithm o n-variable unctions based on UTLG; its process is as ollows. (8) () Calculate the RM expansion coeicient vector B o the given n-variable unction. () In the irst decomposition, = x,the subunction is the n variable o the unction except x, and its RM expansion coeicient vector B is equal to the irst n/elementsob;thesubunction

4 4 Applied Mathematics Table : The second category o the three-variable nonthreshold unctions. = m i = m i = m i = m i = m i = m i Table 3: The third category o the three-variable nonthreshold unctions. = m i = m i = m i = m i = m i = m i is also the n variable o the unction except x, and its RM expansion coeicient vector B is equal to the last n/elementsob. (3) In the second decomposition, = x, = 3 x 4,thesubunctions,, 3,and 4 are the n variables o unctions except x and x, and their RM expansion coeicient vectors are equal to the irst (n )/elementsob,thelast(n )/ elements o B,theirst(n )/elementsob,and the last (n )/elementsob,respectively. (4) Repeat step 3, until the subunctions are the threevariable unctions. (5) Judging all the three-variable subunctions, i the unction is a threshold unction, it can directly be implemented by UTLG; i the unction belongs to the irst and ourth categories o the three-variable nonthreshold unctions, it can be implemented by 3 UTLGs [4]; i the unction belongs to the second and third categories o the three-variable nonthreshold unctions, looking up the decomposition results rom Tables and 3, it can be implemented by UTLGs and XOR; i the unction is =x x x 3 or =x x x 3, it can be implemented by UTLG and XORs. (6) The x i+ (i+)l o each decomposition equation ij = (i+)k x i+ (i+)l can be regarded as the bivariate unction, which can be implemented by UTLG. And the bivariate XOR operation o the decomposition equations can be implemented by XOR. Example. Implement the our-variable unction: =x x 3 x 4 + x x 3 x 4 + x x x 3 + x x x 4 +x x x 4 + x x x 3 x 4. The RM expansion coeicient vector B o is B = [ ], the can be decomposed as = x, is a three-variable unction, and its RM expansion coeicient vector B is equal to the irst 8 elements o B;thatis,B = [ ], =x x 3 +x x 4 + x 3 x 4 + x x 3 x 4. is a three-variable unction, and its RM expansion coeicient vector B is equal to the last 8 elements o B;thatis,B = [ ], =x x 4 +x 3 x 4. (9)

5 Applied Mathematics 5 CLK A A A A C 3 C C A A A.5A C C C 3 C 4 C 6 C 5 C 4 C 5 C 6 (a) (b) Figure : The universal threshold logic gate (UTLG), (a) schematic (b) symbol. CLK A A x x A.5A x XOR x x x (a) (b) Figure : The bivariate XOR gate (XOR), (a) schematic (b) symbol. CLK x x Voltage (V) Time (ns) Figure 3: Transient waveorms o the proposed XOR. belongs to the second category o the three-variable nonthreshold unctions; looking up Table, =, = x 3 +x 4, = x x 3 x 4. belongs to the ourth category o the three-variable nonthreshold unctions, and it can be decomposed as [4] = +, =x x 4, =x 3x 4.Then can be implemented by 6 UTLGs and XORs. Figure 4 shows the UTLG and XOR implementation o this unction. Example. Implement the ive-variable unction: =x x x 3 x 4 x 5 +x x x 3 x 4 x 5 + x x 3 x 4 x 5 + x x x 3 x 4 +x x x 3 x 4 + x x 3 x 4 x 5 + x x 3 x 4 x 5 + x x x 3. () The RM expansion coeicient vector B o is B = [ ], canbedecomposedas = x, is a our-variable unction, its RM expansion coeicient vector B is equal to the irst 6 elements o B,that is, B = [ ], is a our-variable unction, and its RM expansion coeicient vector B is equal to the last 6 elements o B, thatis, B =[ ]. Continue to decompose. = x, is a three-variable unction, its RM expansion coeicient vector B 3 is equal to the irst 8 elements o B,thatis,B 3 = [ ], is also a three-variable unction, and its RM expansion coeicient vector B 4 is equal to the last 8 elements o B,thatis,B 4 =[ ]. Continue to decompose. = 3 x 4, 3 is a three-variable unction, its RM expansion coeicient vector B 5 is equal to the irst 8 elements o B,thatis,

6 6 Applied Mathematics x x 3 x 4 XOR x XOR Figure 4: UTLG and XOR implementation o Example. x 3 x 4 x 5 x 3 x 4 XOR XOR x XOR Figure 5: UTLG and XOR implementation o Example. B 5 = [ ], 4 is a three-variable unction, and its RM expansion coeicient vector B 6 is equal to the last 8elementsoB,thatis,B 6 = [ ].,, 3,and 4 are the three-variable threshold unctions, and can be implemented by 7 UTLGs and 3 XORs. Figure 5 shows the UTLG and XOR implementation o this unction. The proposed synthesis algorithm o the n-variable unction based on UTLG can decompose the arbitrary nvariable unction into three-variable unctions, and all the three-variable nonthreshold unctions are divided into our categories, and two categories o them can be implemented by UTLG and XOR. Thus, arbitrary n-variable unctions canbeimplementedbyutlgandxor,andithegiven

7 Applied Mathematics 7 unction is decomposed into the threshold unctions, the circuit structure will be simple. Up to now, there is no algorithm which can implement arbitrary n-variable unction by UTLG. Reerence [3] just proposed the algorithm which can implement arbitrary three-variable unction. So, we provided new logic units UTLG and XOR and a new algorithm to implement RTDbased arbitrary n-variable unction. 5. Conclusion In this paper, the 5 three-variable nonthreshold unctions are divided into our categories, and a new decomposition algorithm o the three-variable nonthreshold unctions is proposed. I the unction belongs to the second and third categoriesothethree-variablenonthresholdunctions,itcan be implemented by UTLGs and XOR, and it simpliies the implemented circuit structure. Based on the Reed-Muller algebraic system, the arbitrary n-variable unction can be decomposed into three-variable unctions, and the unction synthesis algorithm or the n-variable unctionwhich can be implemented by UTLG and XOR is proposed; that is, the arbitrary n-variable unction can be implemented by UTLG andxor,andithegivenunctionisdecomposedinto three-variable threshold unctions, the circuit structure will be much simpler. The proposed logic units and algorithm present a new method to implement the arbitrary n-variable unction by RTD-based devices. Conlict o Interests The authors declare that there is no conlict o interests regarding the publication o this paper. Acknowledgments This work was supported by the National Natural Science Foundation o China under Grant nos. 674 and and Zhejiang Provincial Natural Science Foundation o China under Grant nos. LY3F and LY5F. Reerences [] N. Muramatsu, H. Okazaki, and T. 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