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Race-Haeard and SkipHaeard in Multivalued Combinational CircuitsXunwei Wu Xiexiong Chen Ji ho ng Shen

Dept. of Electronic Engineering, Hangzhou University, Hangzhou, China

AbstractThis paper discussesracehazards based on AND /ORexpression of functions in multivalued combinational cir-

cuits, and proposes techniques for eliminating race-haeardsby algebraic and K-map's means. Furthermore, this pa-per analym the skiphazard, another inherent hazards inmultivalued circuits, and points out that it is a n o d esponse for multivalued circuits. They can be restrained byusing the input signals with fast transition or a small loadcapacitor.

1 IntroductionIn m n t years, the march on multivalued logic hasattracted attention of logic designera because it can increasethe information density of circuite, and thus leading to the

reduction of th e number of connections and pins, and thesaving of on-chip area of IC.llal However, it also leads tothe complexity in designing and analyzing circuits alongwith the increaae of the number of signal levels. Taking theternary logic circuits as example, there exist three levels(0, , 2) for signals in circuits. Therefore, when analyzingternary circuits, we must consider the response o six tran-sitions of input signalsbetween hree voltage levels (0 s ,1 s 2, 0 2) as well as the mponse of circuits*to hreevalues of input signals.

For ternary Combinational circuits, raceh iwards mayoccur when an input signal I transits between two adja-cent levels (0, 1) or (1, 2). Thus the task for analyzingrace-hazard doubled. Besides, when input 2 transits between two non adjacent levels (0, 2), it must pass throughthe state of z=1, and the circuit will respond to the tran-sient input (t =1) so that the transient spurious outputmay occur. Therefore, in ternary and higher-& logiccir-

cuits there exists another new ha zard -skip haza rd. Nowwe know it is a complex work to analyze the hazards ofternary and higher-radix combinational circuits, and it cannot be avoided in the design of multivalued circuits. In thispaper the race-hsard and skiphazard in ternary circuitsam mainly diJ3cuw3d.2 R a ce -h a rd of ternary combinational circuits

According to the practical case of ternary combma-tional circuits, the analysis of their race-hazards may bebased on the following assumptions:(1) The combinational circuitsam based upon the two-level AND/OR or OR/AND realization in Post algebra.(2) Each variable in the expression of functions appearsin literal operation form.(3) Consider only the hazards due to single input

changes, and don't consider the hazards due to multipleinput changes.Based on the above assumptions, let us consider the cir-cuitscorresponding o two-level AND/OR form first. Whendiacuasing the hazards under input c changes, we can de-

note a function i ( ~ ,, 2 . . .) as follows:J(z )=10-'to 11*'2't 2 e%' t d , (1)

where j ( z ) , m , a l , ~ , d an be regarded as the functions ofthe remaining input variables y, 3,...

In ternary logic the poesible adjacent-level changes forvariable z are 0 1 and 1=2. When the variable t t a k ea ked value 0, 1, 2, the function f will have a definiteoutput: f(O)=ootd, /(l)=al+d , j ( 2 )=o2+d. However, when

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z changes between 0, 1 or 1, 2 (x: 0 1 or 1e ), thecircuits designed by using Eq.(1) may produce the spuriousoutput, i. e. the hazards may occur. The condition tocause hazards can be listed as follows:

(1) The input signal z has a transition 0*1or 1s .(2) The signal z b transmitted to the output by sev-

eral opened paths. Wring Eq.(l)as an example, t can betransmitted to the OR gate at the output through threepaths (~9 , '114c1, 02z1). Only if #a, the C o m p n d h gi-th path is opened, and a# exists.

(3)The propagation delay time of gates for each pa this different 80 that the changes of a,%+, o l Y , a Z V ausedby the transitio n of t are not simultaneous , .e. the raceexists. For example, when z has a transition o +I, it ispossible tha t a o Y has changed from do to 0, but clY hasnot changed from 0 to al yet, then the OR operation ofthese two terms instantaneously produce a 0output. Whenz has the changes 1+ and 1e , the similar cases mayalso happen.

(4) When t has the 0 1change, if d

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0 during the process of the change 00 -+ (11, i.e. at the dis-junction of the groupings 00 and al . Therefore the value ofthe function undergoes a transition: 00 +d -+(d) -+ o1+d. Ifd is less than a0 a "1, a transient hazard with smaller valued may appear during the change of the function. The two'redundant terms added are denoted in Fig.l(b). They cor-respond to a o o l F and a10257 respectively. Their existenceremoves the disjunction of groupingsa0 and a1, also groupings a1 and e. bviously, it is more perceimble to handlethe racehazard on K map.

Example 1. Consider the combination circuit shownin Fig.fl(a). Its output be expresseda

/(tl ) =1-51 $1 .zf +$lo *122+5%222+1 *y. (12)Its corresponding K-map is given in Fig.fl(b), where th egates with label 2 denote binary gab. Let's consider them ehazards caused by the changes of t n the circuit first.According to Eq.(12), we have

aO=O, a l = f + ' y l , a2=$t'+yr d = l - V .Hence we get

IIC

(b)

Figure 2 (a) Circuit of example 1(b) K-map exprsssion of function for example 1(c) Hazard-free circuit for example 1

a0 'a1=0,

It can be seen ha t for removing the race-hazards for th e 1=E 2 change of Z, he added redundant term isA = 1.0gp.Wand there is no racehazard for the 0*1change of 2.

Now we consider the ra cehaza rds caused by the changeof y, from Q.(12), we haveG = Y , a ; = * t * , o ~ = 1 t Z 2 , = 1 . ' z 1 .

Then we obtainaI,-a:=O,

a ~ . o ~ = l - l t l ,LY =6+a{aiW=1 Y + 4G1 - V =b l .

The above equations show that the redundant terms arenot needed because there exist no race-hazards at al l whiley undergoes the 0*1 and 1*2 transitions.

Therefore, only the redundant term A = 1 . oy" .%?is needed to eliminate the race-hazards when z,y changeamong adjacent levels, and hence the hazard-free expres-sion of the output function for the circu it shown in F ig.%(a)is

j ( 2 , g ) =1. '2' +I21 . 1 +I C2 .%O+ Y . $ P + l * y + l - y * T F *

Ita corresponding circuit realization ie shown in Fig.fl(c),where the circuit in th e dotted lines is the additional part.The above procesa to detect and eliminate hazards hasbeen e x p d n Fig.fl(b) by the dotted linea, where'thegroupingY .%p and the grouping 1 . 4 ' are disjunctivecir-cles, and hence the racehazard will occur when z undergoes1 2 transition with I =0 . Thus a redundant grouping

1 -'1p-aE6 a requ redto eliminate the hazard. Since he dis-junctive parts in groupings 2*.2.2$ and 1 '21 arecovered bythe grouping I .y l here no hazard occur. Similarlyl thedisjunctive par ts in groupings 121 . 1 and 1.y re coveredby the grouping I-Y, nd hence there also nohazard occur.Therefore,only the redundant grouping 1 .%O !F is neededto add.

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Example 2. Design a hazard-free T gate.T gate isa kind of ternary universal-logic-gate. It may

beexpressedasT=do020+ 112' +d&?.

From the above discussion we can derive the function of ahazard-free T gate.T=do0zot l't' +&Y+d o d l V + d 1 & P .

The above equation can be written as Is]T = d o O ~ ~ + d l ( ' ~ ' d ~ P +& v t &%~=doozo +dl (do+q d 2+w)+d2Y. (13)

It is obvious that the circuit realization corresponding tothe above equation is hazard-free, but it has not a two-levelAND/OR onstruction. It may be pointed that a amall loadcapacitor also can be used for restraining the race-hazards,however, the input signals with fas ter transition s areuselessfor it.

3 Skiphazard in ternary combinational circuitsWe have restricted our discusclionso far to race-hazards

caused by the changes of input signal z between adjacentlevels, i.e. z has a transition 0= 1 or 1 s 2. Now weconsider the changes of z between non adjacent levels (02 2). Since in ternary circuits taking voltage or currenta6 signal, the voltage or current corresponding o the iogicvalue 1 always takes the middle value, it must pass throught=1 while c has a transition 0s . Thus the change o f t 0=2can be divided into sequential change 0e nd 1*2.However, he time keeping 0 I 1 i ery short (dependingon the rise time and fall time of the signal 2). Let us stillanalyze Eq.(l). When z has a transition o -t 2, I(z) mayundergo three transient values a, ol + d , d in the process ofits change from 90+d to 02 + d . If d is less than a0 -02, dis the racehazard among these three value, and we havementioned the method in the previous section to removeit. However, al +d is a normal response of the circuits toz :1, and it cannot be eliminated. This spurious transientoutput j(1) =o1 +d is temed skiphazard, and it is theinherent hazard for multjvalued circuits. If t changes from0 to 2,we can discuss the skiphaza rds, which m a y occur,for various values of l(o),j(i),j(2)as follow:

(1) If ](I) = j ( o ) or {(I) =j(~),hen no skiphazardsoccur since this transient outpu t /(I) is the same as he pre-vious or later output I(0)or f(2), and thus no hazarda canbe observed. For example, f ( o ) , j ( l ) , j ( ~ )re (OOO), (Ool),(211), (220), (221) nd (222).(002), (011)l(02211(W,110)l(111)l W ) , W , 2%

(2) If the output j (1 ) is set between f(0) and I@),ohazards can be seen also since the middleoutput must a ppear while the output transits between I(o) and f(2). Forexample, 1(0),1(1),j(2) are (012) nd (210).

(3) If /(1) >"9,(2))li.e. I(1)>f(0)+m,Peak-shaped skiphazard will occurwhen / (0) ,I ( l ) , j (2) are (OlO),(020), (021),(120) nd (121),as shown in Fig.3(a).

(4) If f(1)-cm w ( o ) ~ m ) ) ,.e. I(1) uo+% +d (peak-shaped skiphaza rd oc c m ) andUI +d

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Since the transient /(I) produced due to skiphazardsis a normal response of the circuit, it cannot be removedby adding a redundant term to the output for eliminatingracehazards as mentioned in the previous section. For re-straining skiphazards, we may use the input signals withshort rise time and fall time. Moreover, if the outputis connected with a small load capacitor, it can also re-strain skiphaz ards. Figure 4(a) shows the &OS circuitfor threshold operation '21 based on the theory of clippingvoltage switchesP1 The output of the ci rc dt is2 (high level)only if ~ 1 .SPICE simulation of this circuit has beenperformed using 3pm technique. The results axe shown inFig.4(b) and 4(c). In FigA(b), curve@ is the input sig-nal with transition time (between OV and 4V) Ons, andcurv e@ is the t d e n t output &anrteristic with a 2OOWload reaktance, where two spurious p u b re ust the skiphazards of the circuit. Curve0 hows the transient ont-put characteriiticwith a 2Wkn load resistance and a 0.7PFP

t

0 50 100 150dns )

I

I , Q0 2o r(ns) 'O

load capacitor. It can be seen that the magnitudes of twospurious pulses are reduced obviously, and are less thanIV. t shows that a small load capacitor can restrain theskiphazards, which may be the input capacitor C,, of thenext MOS gate. Figure 4(c) gives the simulation result ofthe circuit by using the input signal with faster ransitions.Curve@ is the input signal with the transition t i e (be-tween OV and 4V) 4ns, nd curve@is the transient outputcharacteristic with a 200kn oad resistance. It may be seenthat the magnitudes of two spurious pulses are obviouslysmaller than that of curve @ in Fig%(b), and hence it isshown that skiphazards can be also &rained by using theinput signals with faster transitions.4 Conchu4ons

Firstly, this paper discusses he racehazards of ternarycombinational circuits based on two-level AND/OR xpres-sions due to certain variable change among adjacent levels.We analyze their occurring process and conditions, an themethod to remove race-hazards y both algebraic and K-map means. They totally correspond to the counterpartsin binary circuits,and have several advantages such as easyto operate, easy to understand, etc.firther, this paper discussesanother inherent hazardsfor multivalued circuits-akiphazarda when an inpu t vari-able changes between non adjacent levels, and proposes themethodsto detect the Bkiphazards by algebraic and K-map

means. Since the skiphazards are he normal response ofthe circuib, they cannot be removed by using the techniquefor removing race-hazards. For restraining skiphazards,several methods such as using the input signalswith fastertransitions and connecting a small load capacitor at theoutput areproposed in this paper.Although our discussion in this paper is restricted tohazards in ternary circuits, it can be extended to multi-valued circuits with a higher radix. Of course, the case ofracehazards and skiphazards aremore complicated.%IC-

ing R=4 as an example,for eachvariable three racehazards(O S 1 , 1 * 2 ,2 =3) and threeskiphazards (O P 2 , 1 ~ 3 ,0 S)are required to consider. Therefore, the analysis forthe hazardsof the multivalued circuits ismore cumbersome.

Figure 4 (a) Ternary &OS circuit for realizing 121(b) Jnput signalwith slower transition 0 2(c) Input signal with faster transition 0 e

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AcknowledgmentsThe project has been supported by the National Nat-

ural Science Foundation of the Peoples Republic of China.References[l] Hurst, S.L.,Multiplevalued logic - ta status and itafuture, IEEE %nu. on Computer, C-33, 160-1179, 1984.[2]Butler, S.W., Butler, J. T., Profiles of topics and oth-ers of the International Symposiumon multiple valuedlogic for 1971-1991, roecding01 t n d lEEE In:. S p p . on

multiple uolued logic, Sendai, 372-379, 992.[3] Higuchi,T.,.Kameyama,M., Static-hazard-free T-gatefor ternary memory element and it application to ter-

nary counters, IEEE Item. on Computer,016,1212-1221,1977.[4] Wu, X., Zhao, X., Degign of ternary &OS circuits

based on the theory of clipping voltage switches, nter-national Journal 01Medronieu, 75, 91-102, 993..

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