Circuits and Systems, 2013, 4, 466-471
Published Online November 2013 (http://www.scirp.org/journal/cs)
http://dx.doi.org/10.4236/cs.2013.47061
Open Access CS
Two Analytical Methods for Detection and Elimination of
the Static Hazard in Combinational Logic Circuits
Mihai Grigore Timis, Alexandru Valachi, Alexandru Barleanu, Andrei Stan
Automatic Control and Computer Engineering Faculty, Technical University Gh.Asachi, Iasi, Romania
Received August 18, 2013; revised September 18, 2013; accepted September 26, 2013
Copyright © 2013 Mihai Grigore Timis et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, the authors continue the researches described in [1], that cons ists in a comparative study o f two methods
to eliminate the static hazard from logical functions, by using the form of Product of Sums (POS), static hazard “0”. In
the first method, it used the consensus theorem to determine the cover term that is equal with the product of the two
residual implicants, and in the second method it resolved a Boolean equation system. The authors observed that in the
second method the digital hazard can be earlier detected. If the Boolean equation system is incompatible (doesn’t have
solutions), the considered logical function doesn’t have the static 1 hazard regarding the coupled variable. Using the lo-
gical computations, this method permits to determine the needed transitions to eliminate the digital hazard.
Keywords: Combinational Circuits; Static Hazard; Logic Design; Boolean Functions
1. Introduction
Under certain conditions, on the output of the logical
signals may occur unwanted transitions. These transition s
are known as glitches. The logic glitch is a kind of un-
wanted noise presenting inthe output signal that can ini-
tiate an uncontrollable process. In the next level there is
an input signal [2].
We can distinguish three types of noise that is intro-
duced in CLC (Combinational Logic Circuits), called ha-
zards (Static, Dynamic and Function Hazards).
In the following we consider only the stat ic hazard pro-
blem in combinational logic systems, called static hazard
“0”.
 Static 1 hazard, also called SOP (Sum of Products)
hazard—a glitch that occurs in otherwise steady-state
1 output signal from SOP logic;
 Static 0 hazard, also called POS (Product of Sums)
hazard—a glitch that occurs in otherwise steady-state
0 output signal from POS logic.
Static Hazards in Two-Level Combinational Logic
Circuits (Consensus Method [2]).
We will initially define:
 Coupled variable; a variable input is complemented
within a term of function and un complemented in an-
other term of the same function.
 Coupled term; one of two terms containing only one
coupled variable.
 Residue; the part of a coupled term that remains after
removing the coupled variable.
 Hazard cover (or consensus term).
The RPI (Redundant Prime Implicant) required to eli-
minate the static hazards:
AND the residues of coupled p-term to ob tain the SOP
hazard cover,
OR the residues of coupled s-term to obtain the POS
hazard cover.
POS example: any logic function can be described as:


01ii
yexex 
where



01211
11211
,,,,0,,,
,,,,1,,,
nni i
nni i
eyxxx xx
eyxxx xx
 
 




0
0
(1)
Sometimes, the same function can be describes as:


ii
yaxbxc (2)
Using the (1) form, we can say:
0
1
eac
ebc

 (2.1)

M. G. TIMIS ET AL. 467
Using the algorithm described in [3], if , the
expression from (2), (2.1) doesn’t present static hazard in
relation with the
cab
i
x
input, and if a = b = 0, then results
c = 0.
The condition to have static hazard in relation with the
i
x
input, is when a = b = 0 and c = 1.
The consensus method [4] consists of determination of
coupled terms, then by removing the coupled variables
we obtain residual values.
That meaning the (2) equation can be written like:




ii
yaxbxcab 

(3)
It can be observed that the expression of the function is
multiplied by the sum of residual values, the new expres-
sion presents static hazard in relation with the i
x
input.
We proposed as example the 4 inputs logic function:



43210
0
,,,,
1,3,6,7,9,11,14,17,19,20,25,27,28
0,5,10,16,2 3,2 4,29,31
yyxxxxx
R
R




(4)
Using the Quine-McCluskey minimization method we
obtain the equation from (5) and also the residual values
determined by 0
x
input:


432141020
4210
yxxxxxxxxx
xxxx
 


414210
2
242410412124
axxxxxx
bx
abxxxxxxxxxxxxx
 

 
1
(5)
The expression of no static hazard in relation with 0
x
input:




432141020
42 10421
yxxxxxxxxx
xx xxxxx
 

(6)
2. Method of Resolving of Boolean Equations
[5]
In this paragraph we apply the consensus method [5] and
the method of solving some specific Boolean equations.
If


ii
yaxbx c, by resolving the next system
equations it can be determined the vectors input values
which presents static hazard.
0
0
1
a
b
c



(7)
If the (7) system has no solution, the function doesn’t
presents static hazard in relation with i
x
.
Therefore, the expression of the function becomes:




414210
2
4321
0
0
1
axxxxxx
bx
cx x x x




(8)
Therefore, 20x

imposes the reduction of the sys-
tem to: 41
0xx

 or 41 1xx
So, the solution is:
4
3
2
1
1
0
0
x
x
x
x




(8.1)
So, the function will hav e hazard at commutation


43210
100001000116 17
110 00110 012425
xxxxx


So, in the POS relation will be added the multiplied
prime implicant 42
x
1
xx.


The function will have the same expression like in (7).
3. Static Hazards in Two-Level
Combinational Logic Circuits
We will consider two analytical methods to detect and
eliminate this type of hazard:
(A) Consensus method [1]
We will initially define:
 Coupled variable; a variable input is complemented
within a term of function and un complemented in an-
other term of the same function.
 Coupled term; one of two terms containing only one
coupled variable.
 Residue; the part of a coupled term that remains after
removing the coupled variable.
 Hazard cover (or consensus term).
The RPI (Redundant Prime Implicant) required to
eliminate the static hazards:
 AND the residues of coupled p-term to obtain the
SOP hazard cove r,
 OR the residues of coupled s-term to obtain the POS
hazard cover.
Example 1. Lets consider the logic function




2101 2,3,5,7fxxx R.
a) SOP example: will be determined the prime impli-
cants using Veitch-Karnaugh or Quine-McCluskey me-
thods, as:



21
10
20
2,3
3,7
5,7
Axx
Bxx
Cxx




(9)
One of the minimal equations is:
Open Access CS

M. G. TIMIS ET AL.
468
21 20
y
AC xxxx  (10)
where we have :
 coupled variable: 2
x
 coupled terms: 2120
,
x
xx x
 residues: ,
10
x
x
 consensus term: 10
x
x 
Therefore, the logic expression that has no static haz-
ard in relation to 2
x
variable is:
21 2010
y
xx xxxx (11)
b) POS example: will be determined the prime impli-
cants using Veitch-Karnaugh or Quine-Mc Cluskey me-
thods, as:



21
10
20
0, 1
0,4
4,6
ax x
bxx
cx x



(12)
One of the minimal equations is:




21 20 10
yxxxx xx

(13)
where we have :
 coupled variable: 2
x
 coupled terms: 202
,1
x
xx x
 residues: ,
10
x
x
 consensus term: 10
x
x
The equation (13) shows no static 0 hazard.
Example 2. Let’s consider the function of four varia-
bles .
 
3210 10,1,2,5,6,7,8,9,10,14yfxxxxR
SOP hazard: will be determined the prime implicants
using Quine-McCluskey method, as:








310
310
320
321
310
21
20
10
1, 5
2,6
5,7
6,7
10,14
0, 1,8,9
0,2,8, 10
2,6, 10,1 4
Axxx
Bxxx
Cxxx
Dxxx
Exxx
Fxx
Gxx
Hxx








(14)
Applying the Patrick method [6], going from prime
implicants table will be determined all SOP solutions .
Let’s consider the logical variables attached to the
prime implicants as follows: 0, if 0, the A
prime implicant is present in the logical function expres-
sion, otherwise 0 (A prime implicant is not pre-
sent in the logical function expression), etc.
i
ppA1p
0p
Therefore, considering the correspondence 1
pB

,
,3,4, 5, 62
pCpDpEpFpG

,
, in the table illustrated in Table 1 is shown the
7
pH
Table 1. The SOP coverage table.
dec. equiv.0 125 6 7 8 9 1014
pi
p0 11
p1 1 1
p2 1 1
p3 1 1
p4 1 1
p5 11 1 1
p6 11 1 1
p7 1 1 1 1
Patrick coverage:
It writes the coverage equation:



 
 

5605167
02 13723
565467 47
1
pp pp ppp
pp ppp pp
pppppp pp



 
(15)
Simplifications are made by using the laws of Boolean
algebra: the redundance law, the iden tity law and the dis-
tributive law.

 







5167 137
47 20 23
5713674
203
5714346
203
57 145 3456
203
1
or
1
or
1
or
1
pppp ppp
pp pp pp
ppppp pp
ppp
ppppppp
ppp
ppppp pppp
ppp
 








(16)
A version of the optimal solution corresponds to
572
ppp

 triplet, i.e.


3210
21 10320
,,,yfxxxx FHC
xx xxxxx


 (17)
The cost of this function in SOP implementation is:





2110320 310CyCx xCxxCxx x

 
(It was considered the variables ,
ii
x
x, available at
input).
It can verify that any other coverage has a higher cost.
For example, the coverage which cor-
responds to 5703
pp p p
21 10310321
yFHAD
x
xxxxxxxxx



 (18)
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M. G. TIMIS ET AL. 469
has the cost .

114Cy
4. The Static Hazard Elimination
(B) The consensus method
We apply the same method as in [7], only that it has a
strong computing nature. Any logic function can be
written as:
01ii
y
ex ex (11),
where

01211
,,,,0,,,
nni i
eyxxx xx
 

0
0
,

11211
,,,,1,,,
nni i
eyxxx xx
 
.
Obviously, if ii
y
ax bx c  
ebcee
(12), then
, .
0
eac1
If we add the term to relation (11), the function
presents no hazard towards
01
i
x
.
In terms of the consensus method, the term that covers
the static 1 hazard is

01
eeac bccab  (19),
therefore for the form (11) will be , and for the
form (12), . 01
ee
ab
Considering the second example, we will have:
hazard in relation to the input 0
x
:
21 10320
y
FHCxxxxxxx  (20),
where

012112
132213221
01123 221
321 21
,
,
exxxxx
exxxxxxxx
eex xxxxx
x
xx xxFD


  

(21)
By adding
F
D term to relation (14), it obtains:

2110 320 321
yFHC FD
x
xxxxxxxxx
 

(22)
hazard in relation to the input 1
x
:

02320230
10 32 320032
0123003220
exxxxxxx
e x xx xxxx xx
eex xxxxxxxG


  
(23)
Therefore, the expression of the function becomes:
21 10320
32120
yFHCDG
x
xxxxxx
xxx xx



(24)
hazard in relation to the input 2
x
:
010101
1303110
exxxxxx
exxxxxx
0



Therefore,




01103 0311 0
310310 10
10 310
eex xxxxx xx
xxxxxx xx
xxxxxH A



 
(25)
The expression of the function becomes:
21 10320
321 20 310
320321 10
yFHCDGA
xx xxxxx
x
xx xxxxx
x
xxxxxxx t







(26)
hazard in relation to the input 3
x
:
0202110
exxxxxxt

,
where
21 1020310
txxxxxxxxx


1
et


012 02110
eexxxxxxtt t


(27)
so that remains the same expression (20), which has no
hazards in relation to 3
x
.
From the relation (20), it sees that the expression of
the function without SOP hazards contains all prime im-
plicants without B and E.
(C) The method of solving of some Boolean equations
[8]
A logic function can be written as:
ii
y
ax bx c

 (28)
where



12110
121 10
,,,,0,,,
,,,,1,,,.
nni i
nni i
ac fxxxxx
bc fxxxxx
 
 
,





According to a theorem from [8], a logic function ex-
pressed as SOP, presents a static hazard in the situation
1
ii
xx

, a situation deducted by solving the following
system of logical equations:
1
1
0
a
b
c



(29)
We return to the same function, (14):
21 10320
y
FHCxx xxxxx

 .
hazard in relation to the input 0
x
:


10 32021
y
xx xxxxx

  (30)
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M. G. TIMIS ET AL.
470
The function will present SOP hazard, if
1
32
21
1
1
0
x
xx
xx



(31)
Therefore, 3, , , which imposes a
hazard at commutation
0x21x11x
3210
0110 0111,
x
xxx

which imposes the adding of the prime implicant
to function.

6,7D

The function becomes
2110 320 321
yFHCD
x
xxxxxxxxx


(32)
hazard in relation to the input 1
x
:
2
2
032
0
320
3
1
0
1
or 0
0
ax
x
bx xx
x
cxxx
x 






(33)
Therefore, we will have hazards in the following
cases:


3210
00000010 0,2
1000 10108,10
xxxx


The previous commutations are equivalent to the
implicant .

0,2,8, 10G
The function becomes
21 10320
321 20
yFHCDG
x
xxxxxx
xxx xx



(34)
hazard in relation to the input 2
x
:
10
3
30 31
100 1
1
0
1
and 0,1
axx
x
bxxxx
cxxxx




(35)
We will have the solution:
3
1
0
0
0
1
x
x
x



(35.1)
The corresponding commutation is:

3210
00010101 1,5
xxxx
.
Therefore, the term is added to the func-
tion.

1, 5A
And therefore:
y
FHCDGA

 (36)
hazard in relation to the input 3
x
:
21 10320
32120 310
yxxxx xxx
x
xx xxxxx


 (37)
202110
21 1020
1
0
0
axx xxxx
b
cxxxx xx




 
(38)
Because one of the terms is zero, we have no
hazards in relation to that variable.

,ab

5. Conclusions
The contribution of the authors consists in that by
analysis of two methods of detection/elimination of the
static hazard, insisting of the POS method for the logic
function which wasn’t analyzed in [1].
The boolean equation [2,3], presents some advantages
instead the consensus methods, the most important to
determine the transactions which causes static hazard.
It concludes that the classical method of the 70s, the
method of solving some specific Boolean equations [4],
presents some advantages compared to consensus
method [5], which has a strong heuristic nature.
In the first method it used the consensus theorem to
determine the cover term that is equal with the product of
the two residual implicants [6], and in the second method
it resolved a Boolean equation system [7]. The authors
observed that in the second method the digital hazard can
be earlier detected. If the Boolean equation system is
incompatible (doesn’t have solutions), the considered lo-
gical function doesn’t have the static 1 hazard regarding
the coupled variable. Using the logical comp utations , this
method permits to determine the needed transitions to
eliminate the digital hazard.
From the both methods, we can observe that static 1
hazard can be removed by adding the prime implicants
step by step.
The same method with the same conclusions was ap-
plied to the static 0 hazard (POS), using the duality theo-
rem [8,9].
The authors observed that in the second method the
digital hazard can be earlier detected. If the Boolean equa-
tion system is incompatible (doesn’t have solutions), the
considered logical function doesn’t have the static 1 haz-
ard regarding the coupled variable. Using the logical com-
Open Access CS

M. G. TIMIS ET AL.
Open Access CS
471
putations, this method permits to determine the needed
transitions to eliminate the digital hazard.
REFERENCES
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