.I

\

REPORT) R-354 . h Y . 1967 -.-)

~ C O O R D I ~ ~ D ~ T E D SCIENCE LABORATORY, -.

rn w k3 rn 0 A PROCEDURE FOR a RANKING DIAGNOSTIC

TEST INPUTS CI

( A C C E S S I O N N U M B E R )

/UNIVERSITY OF ILLINOIS - URBANA, ILLINOIS

https://ntrs.nasa.gov/search.jsp?R=19670023989 2019-03-02T22:08:20+00:00Z

T h i s work was supported by the Jo in t Se rv ices E l e c t r o n i c s

Program (U.S. Army, U. S. Navy, and U.S. A i r Force) under

Con t rac t D A - 2 8 043 AMC 000(73(E).

Reproduction i n whole o r i n p a r t is permi t ted for any purpose of t h e

United S t a t e s Government.

D i s t r i b u t i o n of t h i s r e p o r t is unl imited. Q u a l i f i e d r e q u e s t e r s may

o b t a i n copies of t h i s r e p o r t from DDC.

I 1 1 I

8 1 E 1 1 I 1 8 1 8 I t 1 I

a

iii

ACKNOW LEDGMEN T

The author is indebted to Professor Gernot Metze whose suggestions,

comments, and cr i t ic isms helped greatly during the research work and the

writing of this thesis.

The author is also grateful to Ralph Marlett and R o s s Schneider for

their helpful discussions and to Edward Davidson for reading through the man-

uscript and making suggestions.

The. support extended by the National Aeronautical and Space Admin-

is tration is appreciated.

iv

TABLE O F CON.TENTS

Page

1. INTRODUCTION. . . . . . . . . . . . . . . . . . . 1

2. PACKAGE LEVEL DIAGNOSIS. . . . . . . . . . . . . 4

2. 1 Prime Implicant Diagnostic P rocedure . . . . . . 6 2. 2 Probability Weighting Procedure . . . . . . . . . 9 2. 3 Assumed Number of Outputs. . . . . . . . . . . 11 2.4 Example with Probabilistic Weights . . . . . . . . 15 2.5 Comparison. . . . . . . . . . . . . . . . . . . 23

3. PROBABILISTIC WEIGHT FORMULA. . . . . . . . . . 29

3. 1 Weight Formula Derivation . . . . . . . . . . . 3.2 Multi-Output Example . . . . . . . . . . . . . 29

34

4. FUTURE WORK AND CONCLUSIONS. . . . . . . . . . 43

4. 1 4.2 Future Research Areas . . . . . . . . . . . . 4. 3 C onclu s ion s . . . . . . . . . . . . . . . . .

Sequential Machine Extension . . . . . . . . . .

LIST O F REFERENCES. . . . . . . . . . . . . . . . .

43 46 48

49

APPENDIX. . . . . . . . . . . . . . . . . . . . . . . 50

1. IN TR ODU C TION

Today, digital computers a r e being built larger and with greater

complexity.

flights, tele-phone systems, and plant control which require highly reliable

operation.

needed to detect and locate the faulty component shoula be minimized.

Such a faulty component will be called the fault.

cate and detect faults in computers a r e needed to aid maintenance personnel

with maintenance of computers.

They a r e being used in applications connected with space

When a component in the machine becomes faulty, the time

Rapid procedures to lo-

Let a machine be defined as the logical circuit of a computer.

This thesis will be mainly concerned with combinational circuits. A ma-

chine which has a fault will give a s e t of outputs different f rom that given

by the same machine with no faults, fo r some se t of inputs.

a machine can be detected and specified by examining the outputs of the ma-

chine and by comparing them with the different output sets that a machine

with the different possible faults produces.

ing to the individual faults have been developed [l, 3, 4, 5, 61. The pur-

pose of these procedures has been to find the specific fault in as few tests

as possible and as quickly as possible.

many of these procedures, and that a r e also made in this thesis, are:

1) The machine will have a t most one fault; 2) the possible faults that can

occur in a machine a r e a logic element stuck at "1" or "0, I ' and lines be-

Most faults in

Several procedures f o r diagnos-

The assumptions that a r e made by

1

2

1 I 5 5 1 8 t 8 c e I 1 f

tween the logic elements opened; and 3) the fault is well behaved, i. e. , i t is

not intermittent.

Many computers today are constructed in small packages.

which fault has occurred is not as important as knowing which package con-

tains the fault; for when a package has a fault, the whole package is re-

placed.

ducing redundant tests and thereby for selecting a near optimal s e t of tests

for diagnosing a machine.

ison of a machine containing a fault, to a fault-free machine.

for a machine with a single line output but does not consider the additional

Knowing

Chang [2] has taken this into consideration in his procedure for re-

H i s procedure is based on a pass or fail compar-

It is useful

information that a multi-output machine provides.

defined as an output with two output symbols, "1" and "0"; in this study each

output symbol will be referred to as an output.

problem of selecting a near-optimal s e t of tests for diagnosing a multi-

output machine with the goal of finding only the package wherein the fault

lies.

that can be applied to a multi-output machine.

A single line output is

This study considers the

This procedure therefore removes redundant tes ts f rom a s e t of tests

One criterion for this near-optimal selection of tes ts has been to

find a procedure that can be applied to large combinational logic circuits.

OptimalI procedures have been developed for small combinational circuits

[ l , 31, but if these procedures a re applied to combinational circuits of rea-

sonable size, the computations and the bookkeeping involved would be so

voluminous that they would render the procedures impractical.

The method developed by this study can be applied to large combin-

ational logic circuits since i t examines only the package separation that each

possible additional test gives when applied together with previously selected

tests. The test that gives the greatest separation of packages i s chosen and

added to the l i s t of selected tests.

always chosen, a t least a near-optimal se t i s selected.

not examine the package separations of several combinations of tests applied

together which optimum procedures must do, and therefore the bookkeeping

is greatly reduced, making this test procedure more practical.

Although an optimal s e t of tests is not

This method does

In Chapter 2 various procedures for package diagnosis a r e discus-

sed and a procedure f o r choosing a near-optimal s e t of tests is developed.

Chapter 3 contains the development of a probabilistic formula to calculate

probabilistic weights which a r e used f o r selecting the tests to be applied

for diagnosis.

one stage of an adder i s also found in this chapter. In Chapter 4 a possible

extension of the procedure to sequential machines is given along with other

possible a reas of research.

An example of finding a se t of tests with this procedure fo r

4

1. IC 8 3 8 8

t 8 8 8 I 8 I 8 8 8 1 I

2. PACKAGE LEVEL DIAGNOSIS

In diagnosing failures of a machine, several procedures have been

A test developed for selecting the se t of tests that i s applied to a machine.

i s defined as a possible input to a machine to determine the output of the

machine.

chine. One main goal of computer diagnosis i s to minimize the number of

tests fo r complete diagnosis of amachine s o as to shorten the time for di-

agnosis and reduce program space for a stored program data processor.

Most test selection procedures have been developed on a fault level diagno-

s i s basis [ I , 3, 4, 5, 61.

faulty circuit until the specific fault i s known or i t is known that no fault

exists in the circuit.

The output then is compared with the output of the fault-free ma-

Fault level diagnosis i s defined as diagnosing a

The trend fo r machine construction today i s to build several logic

circuits in the same package f o r the reduction of size and the greater ease

in replacing components. When a fault occurs, the whole package i s re-

placed. Fo r diagnosis of such a circuit olily the package containing the.'

fault need be found, and not the specific fault.

a machine should require fewer tests since less information is needed.

agnosing a circuit until i t is known which package contains the fault, o r un-

til i t is known that the machine contains no faults, will be called diagnosis

to the package level.

Therefore diagnosis of such

Di-

Few procedures have been developed fo r selecting tests for diag-

5

J 0 I 8 8 8

8 B 8 8 1

8 8 8 8 I I

nosis to the package level which determines the faulty package but not an

individual fault.

tes ts for diagnosing a machine to the package level.

useful only if the machine essentially has just two outputs, namely, "1"

and "0.

give m o r e information than just a "1" o r a "0" for each input test. Util-

izing this extra information in selecting a sequence of tes ts improves the

procedure for removing redundant tests.

Chang [2] derived a procedure for removing redundant

His procedure is

However, many machines have more than two outputs and hence

In diagnosing to the package level, a very general procedure that

requi res only a knowledge of the truth table of each package would be de-

sirable.

tions of the packages, diagnose the combinational circuit to the package

level.

each package on its inputs would be needed.

mined, then all possible inputs fo r each package a r e needed to make su re

that the package is fault-free. With dependency of the output on i ts inputs

known, different sets of tests that would indicate whether a package has a

fault would be needed, and f r o m the interconnection of packages a diagnos-

tic procedure would then be developed to diagnose to the package level.

This procedure would then, f r o m the knowledge of the interconnec-

F o r such a procedure, ways to detect dependency of the output of

If no dependency can be deter-

On the other hand, if the circuit of each package is known then i t

is known immediately which faults can occur and what s e t of inputs will

give a cor rec t output for the package even if a fault is in the package and

II I 8 - 8 1 8 t t 8 1 8 8 8 8 8 8 t 1 1

what se t will detect the fault. The problem of separating packages is much

less complex and the diagnostic procedure for the separation can be started

a t once.

known.

For these reasons i t is assumed that the logic in each package is

2. 1 Prime Lmplicant Diagnostic Procedure

Before considering the probability approach, i t will be instructive

to examine an optima1,prQcedure.

th Let p. represent the i package and l e t the possible faults within

1

th i i i the i package be f l , f2 , . . . , fk. Although diagnosis to the package lev-

e l does not require the identification of the separate faults within each pack-

age, each fault must be l isted to insure that a specific fault f . in package i

has been separated f rom the faults of the k

i J

package, where all other faults th

in package i may have been separated from those of package k.

fault must be separated by at least one test, in a s e t of tests, f rom all

faults in the other packages in order to diagnose to the package level.

Thus each

This

leads to a method for diagnosis with multiple outputs which is somewhat an-

alogous to the pr ime implicant problem for logic design,

test to a combinational circuit, different outputs will be given f o r the differ-

ent faults, depending upon the fault.

After applying a

Note that a fault simply changes the

fault-free machine to a different machine with different outputs.

these different outputs the following table (cf. figure 2) can be formed:

F r o m

6

7

f12

p2 i f22

Down the s

00 lo 11 01 00

01 00 01 10 00

ie of the table the different possible tests a r e lis-3d. Across

the top of the table a r e listed all possible pa i r s of faults, such that the pair

of faults is not f rom the same package. A t will be placed in the t m’

i k J n m J n fi./f? table position i f test t

words, if a machine with fault f . produces a different output f r o m a ma-

separates fault f . f rom fault f , or in other

i J

i n J

chine with fault 8. When a machine with fault f . produces the same out-

put a s a machine with fault ? , then the table position t

blank.

fi/k is left n . m’ J n

An example will clarify this procedure. Let a combinational c i r -

and p with two possible faults each. Let the 1’ P2, 3 cuit have packages p

five tests have four possible outputs f o r the various faults as shown in fig-

u r e 1. t l +2 t3 t4 +5

[f? I o 0 11 00 01 00

[ f13 I o 1 00 01 10 11

I

FR-1267

F i g u r e 1

8

8 8

The fault test pattern table is s,,own in figure 2.

fll/ff f l l / fZ fl’/f? f $ f 2 f21/ f$ f 2 / f , 2 f & f ? f2‘/f,3 f,Z/f? f;/f; f ,2/f? f;/f;

FR-1266

F i g u r e 2

Consider the pa i r s of possible faults, i. e. fi/F, as elements of a

be consider-ed a s a se t of these pa i r s and contain ele-

.

J ‘ 4 set. Then le t test t

ment f./? i€ and only if a plus sign is in the t

Then test t will be a pr ime implicant if and only i f any test t S r

t 3 t implies that t = t .

m i k J 9 m J q

f i / f entry of the table.

such that

Thus if a test is not a pr ime implicant i t is a r - S r s .

proper subset of a pr ime implicant test, while prime implicants a r e proper

subsets of none of the other tests.

plus sign in a row of t whenever there is a plus sign in the row t

thermore, since t is not a proper subset of t

plicant.

plicant tests according to this new definition of a pr ime implicant.

In the example, t 1 2 C t since there is a

Fur - 2 l’

o r t i t is a pr ime im- 3’ t4’ 5’ 2

The f i r s t step in the diagnostic procedure is to find the pr ime im-

9

The remaining step is the same as a step in the p r ime implicant

problem for logic design which is that of finding a minimum cover of the

fault pairs. A minimum cover is the minimum number of pr ime impli-

cants whose union will contain all elements fi/fk where i # k. In the ex- J 9

ample, t and t or t and t a r e minimum covers. Thus one concludes

that tes ts t

age level.

2 5 3 5

and either tes t t or t will diagnose the circuits to the pack- 5 2 3

This procedure is an optimum procedure for combinational c i r -

cuits with multiple outputs since a minimum number of tests can be found

by this minimum cover. However, one serious drawback in using this pro-

cedure for diagnosis is the vast amount of bookkeeping necessary to find a

s e t of tes ts t h a t will cover all faults. As an example, suppose that a

combinational circuit has five modules with ten faults in each module;

then the size of the fault pattern a r r a y similar to figure 2 will be the num-

ber of tests by 1000.

pr ime implicants, and finding the minimum cover woul8 be very t ime con-

suming.

Formulating this fault pattern table, finding the

2. 2 Probability Weighting Procedure

A more practical procedure is the following which o rde r s the pos-

sible tes ts by a probability procedure and finds a near-optimal s e t of di-

agnostic tests. For this procedure the tes ts and the outputs of the ma-

10

I I 1 I 1 8 8 1)

1

8 8 8 8 I I

~

chines with different faults a r e listed in tables a s in figure 1.

i of the tables represent the possible faults that cam occur. In the f . , t en-

J k

t ry is listed the output that the machine with fault f . would give if test t

were applied.

The columns

i

J k

Similar to Chang's method for selecting tes ts [2], a weight for each

test i s given f o r the purpose of selecting the best tes t to be applied next to

the machine for diagnosis.

set, of tests i s obtained by this approach while not requiring the exhaustive

tr ials of all possible test combinations. This procedure weights the possi-

ble tests that can be applied to the machine according to probability.

culating' thel pohahilitcjr it i's assunied.that?:ebc;h &utpawfor ever$ test i's equally

likely and that there a r e enough outputs available so that a hypothetical

test could completely diagnose the circuit.

ity that a hypothetical test will diagnose the circuit after the test being

weighted has been applied.

applied to the machine.

the tests a r e applied again to the machine to see how much more they will

separate the remaining packages in conjunction with the test already select-

ed. The probability that another hypothetical €est will separate all remain-

ing packages not separated by the tes t selected and the test being weighted

taken together, i s the new weight. A second test with the highest weight i s

added to the selected test set, and the process i s continued.

At least a near-optimal set , if not an optimal

In cal-

The weight is then the probabil-

The test with the highest weight is chosen and

Some separation of packages is obtained. Next,

Each time a

.

11

new weight is found f o r the tests not selected,

that a hypothetical test wil l separate the remaining packages not separated

by the test being weighted and the selected tests taken together.

packages have been separated the procedure is terminated, and the selected

test se t will'diagnose the machine to the package level.

This weight is the probability

After all

2. 3 Assumed Number of Outputs

To find the weight of each test, i t i s necessary to see how many pack-

ages would still need to be separated if the test to be weighted were appfied

i , a fault f . in a ma- J

af ter previously selected tests. If for a given tes t t m i

Q- J chine gives a different output from fault 8, then f . i s separated from fault

fk Of course the interest is not in separating faults but packages, and so

if all faults of a package have been separated f r o m all other faults in the

other packages, then this package has been separated.

i s important to determine whether the machine is fault-free, then the fault-

f ree machine can be considered as a fault of a fictitious package denoted by

f whose outputs for the tests are the same as the machine's truth table.

This will assure that the fault-free machine is separated from all other

packages with faults. 1

4'

Note also that if i t

0 0

In calculating the weights, i t i s necessary to determine how many

possible outputs need be assumed.

that a hypothetical test wi l l separate all remaining packages not separated

The weight of a test is the probability

I 1 8 I 8 8 8 t 8 8 8 I 8 8 8 1 8 I I

12

by the tests already selected and-the tes t being weighted taken together. A

snffjiient number of outputs needs to be assumed for this hypothetical t es t

in order for the probability not to be zero. The number of assumed outputs

may then be more o r l e s s than the number of actual outputs of the machine.

The number assumed is taken as the maximum number of packages that

have not been separated af ter applying the tests selected for diagnosis

along with any one of the tes ts being weighted.

assumption that each output for every fault in the next applied test is equally

likely to occur. Finding the probability is then simply a process of count-

ing the number of ways that outputs can occur, such that the remaining un-

diagnosed circuit will be diagnosed to the package level.

divided by the total number of possible combinations of outputs to give

what will be called the probabilistic weight, denoted W

of diagnosis the probabilities a r e the weights, and i t has been decided that

a weight of zero is not to be given to any tes t since this would eliminate a

comparison of all tes ts weighted zero. Therefore by using the maximum

number of packages left to be separated as the assumed number of outputs,

none of the weights will be zero.

The probability is under the

This number is

In this procedure P'

Let the probabilities of the various tes ts order the tes ts and call

this the test order assuming a number of outputs. This tes t ordering is

not necessary to choose the test with the highest weight, but i t has been

introduced to help explain some properties of this procedure and to make

8 I 8 1 8 8 8 8 8 t 8 P 8 8 I I 8 1 1

13

comparisons.

designated 1 and will be the test to be applied next for diagnosis.

The tes t with the highest probabilistic weight will have order

Assuming more outputs than the machine actually has, is equiva-

lent to putting the outputs of several tes ts together so that the machine ap-

pears to have more outputs. F o r example, i f only two outputs are possible

then four possible combinations of outputs, i. e., 2 can be obtained by put- 2

ting the outputs of two different tests together. If the number of outputs as-

sumed is equal to the machine's actual number of outputs ra ised to a power

r , then the weight is also the probability of diagnosing the machine with r

tests. . Of course the assumption of all outputs having equal probability for

all faults is the same.

Assuming more outputs may rank: one tes t higher than another,

whereas for fewer assumed outputs the tests may be ranked equally. An ex-

ample of this is seen in figure 3.

Both weights are equal, assuming two outputs; however, assuming

three outputs gives tes t t a higher weight. FOT test t and with three out- 2 2

puts, different combinations of two outputs among the three faults in pack-

age p2, with the third output for fault f2, can skparate the packages, where-

a s with two outputs all three of the faults in package @ had to have the same

output.

1

2

There is less than half the number of combinations for two outputs

among two faults than among three faults, and so with three assumed out-

puts, test t has the higher weight. 2

14

0.125 Assuming 2 outputs 0.296 Assuming 3 outputs wp { z::::

(Wp = Probabilistic weight 1 FR- 1269

F i g u r e 3

There a r e cases where assuming more outputs will switch the or -

der of tes ts that are close together in a test set, but usually the tes t order

will remain the same.

er total faults o r fewer packages left to separate.

possible outputs will increase all the probabilistic weights.

tes ts will not change orders , however, since a fewer number of faults o r a

fewer number of packages will still have m o r e combinations of outputs that

will separate them.

Tests that a r e weighted higher will often leave few-

Increasing the number of

Usually the

2. 4 Example with Probabilistic Weights

h order to make the procedure clearer , consider the diagnosis of

the following example that Chang used in his paper [2].

only two outputs, either 1 or 0; however, the multi-output probability

method is also applicable to this case since a single line output is a triv-

ia l mu1 ti- output.

His example has

The se t of tests that Chang used a r e shown in figure 4,

The test patterns can be written differently to see more clearly

which packages still need to be separated after the f i r s t test has been ap-

plied if the f i r s t test i s t. fo r i = 1, . . . , 8. This is done in figure 5(a) and 1

(b)

In figure 5(b) i t i s seen that t has a "0" in each package. The 1

1 3 1' f2 ' test t does give some information for diagnosis since i t separates f

1 5 1 and f f rom the other faults, and hence the probabilistic weight should not

be zero. An assumption of at least five outputs is required in order that

the probability of the remaining packages being separated by a hypothetical

test is not zero after applying test t Let these five outputs be called a, 1'

b, c, d, and e. A test pattern that will completely separate the packages

that have output "0" in test t must have the same output for all faults in

the package, and this output must not occur in m y of the other packages.

After referring to figure 5(b) one can see that one such possible output

1

pattern that would separate these packages is:

i 6

f l l 1 0 0 1 0 0 0 0

p1 { f ? I 0 0 0 1 0 0 1 1

I

Chang's Weight = 24 30 25 30 35 29 29 26

Chang's Ordering of Tests = e 2 7 2 1 4 4 6

Probabilist ic Weight = 0.0029 0.0177 0.0275 0.0472 0.4261 0.0531 0.0118 0.0059 x 10.'

Probabi li t y Ordering = 8 of Tests

5 4 3 1 2 6 7

FR- 1270

Figure 4

i i

+ l + 2 +3 t4 t5 t 6 t 7 t 8

p3 { :;

FR-1272

F i g u r e 5a

'1 +2 t 3 +4 t5 ' 6 t7 t 8

0 0 0 0

0 0 0 0 r ' i gu re 3b

FR-1273

I 1 8 I I 8 1 t I 1 I 1

8 8 1 I 1 1

e

.

I

I ,

1 0 LU

FR-1271

There a r e 51 ways for five outputs to completely separate the packages.

so there are 5 , ways for five outputs to occurcisince there a r e nine faults.

Therefore the probability of completely separating these remaining faults

9 is 51/51 . also separate the faults f

Al-

9

The next tes t to separate all packages af ter applying tes t t mus t

There are 5(4)(3) ways for five out-

1 5 , and f 1 3

1' f2 1' 3

puts to separate these faults out of a total 5

3 probability for completely diagnosing these faults ii 6 0 / 5 . possible test patterns. The

No dependency exists between the event of separating the 1 ' s in

figure 5(a) and the event of separating the 0 ' s in figure-5(b); and so the

probabilistic weight W P

two independent events.

is simply the product of the probabilities of these

The independence results f rom the assumption

I 1 I I 8 8 1 I 8

I 1 1 8 1 I 8 I 1

e

19

that every output is equally likely for every fault.

F o r tes t t the probabilistic weight is calculated: 1

9 W = (5:/5 ) ( 6 0 / 5 ~ ) = 7200/512 = 0. 0029 X l o m 2 P

By similar procedures the other probabilistic weights a r e

found and recorded in figure 4 along with the tes t order.

Test t has the highest weight and is therefore chosen as the tes t 5

to apply f i r s t in diagnosis. After applying tes t t the remaining test pat- 5

9

terns and packages to be separated a r e shown in figure 6. Figure 6 could

be split up again as figure 4 was in figure 5(a) and (b); however, this is

not necessary to calculate the probabilities. Since there a r e a t most

three packages left to be separated f rom each other, the maximum num-

ber of outputs needed is equal to o r l e s s than three.

separate packages p

Test t does not 1

and p completely, and- thus a t l eas t ;three out- 3’ P4’ 5

puts need to be assumed for calculating the weights.

Let u s choose t a s an example f o r calculating the probability.

and t have been applied to the circui t it can be seen that 5 6

6

After tes ts t

1 1 2 2 1 1 2 2 3 package p with faults f and f Z and package p with faults f and f need

to be separated. With three outputs there a r e 18 ways that these packages

can be separated. Since there a re four faults, the probability of separat-

4 2 1 ing p1 f r o m p is 18/3 . The probability of separating p with fault f

2 2 3

3 2 f r o m p with fault f is 2/3 under the same conditions. It is noted that

package p is completely separated, and so the probability of separating 5

20

Wp = 0.0049 0.0195 0.0073 0.0293 0.0439 0.0146 0.0012

Test Ordering by wp

3 5 2 1 4 7

Chong's Ordering 5 2 5 2 1 4 7 FR- 1274

Figure 6

2i

1 0 0 1 0 0

0 1 0 0 1 0

= 0.0313 0.5000 0.0156 0.0625 0.0625 0.0625

5 Test Ordering by wp

1 6

Chang's Ordering 5 1 6

2

4

2 2

2 3

FR-1275

Figure 7

I I I I I 1 1 I 8 1 B 1 1 8 I B 8 I I

22

4 i t is 1. 0 while the probability of separating p Multi-

plying these independent probabilities together will give the probabilistic

weight of the test.

f rom p 3 4 is 24/3

w = (~/3~)(2/3)(24/3~)(1. 0) = 0.0438 P

Other probabilistic weights a r e recorded in figure 6.

Since t has the highest weight, i t is ayplied and produces the test 6

patterns as shown in figure 7.

arated.

a r e recorded in figure 7.

3 weight, the only remaining packages to be separated a r e p with fault f l

and p with fault f

patterns a r e shown in figure 8.

Package p is omitted since i t has been sep- 5

By the same procedure the probabilistic weights a r e calculated and

After applying test t which has the highest 2

3 4

4 1' All other packages have been separated, and the test

W p = 0.500 0.500 0.500 0.500 0.500

Test Orders 1 1 1 1 1 FR- 1276

Figur 'e 8

2 3

All probabilistic weights a r e the same as the weight of test t2 and

with the same number of assumed outputs. Therefore there is no further

separation obtained by applying any of the remaining tests which indicates

that faults f and f a r e indistinguishable. The diagnosis process then is 3 4 1 1

2' terminated with the application of tests t 5, t6s and t

2. 5 Comparison

It is instructive to compare the s imilar i t ies and differences be-

tween Chang's ordering of the tests and the probabilistic ordering.

time the tests were weighted in this example, the tes t with the highest

weight by Chang's procedure also bad the highest probabilistic weight.

Therefore the test selected to be addednext to the diagnostic s e t of tests

was the same by both procedures.

separated there must be a t least three tests f o r separating all the packages,

and if one examines all other possible combinations of three tes ts , he will

s ee that tests t

both procedures the best possible tests were selected,

Each

Since there a r e five packages to be

and t a r e the best combination for diagnosis. In 5' t6' 2

After ordering the tests in f igu re 4 i t i s noted that the grobabilis-

tic weights ordered test t as second while Chang's procedure ordered 6

test t a s second and test t as fourth. By examining these two tes ts one 2 6

sees that tes t t does not separate any of the packages from the rest . Ev- 2

e r y package after applying tes t t has a zero for at l eas t one fault, and i f 2

24

this test were selected, a minimum of three additional tes ts would be r e -

quired to separate all packages, (Since two of these faults in different

packages have identical test signatures, i. e . , the same outputs for all

tes ts , they a r e indistinguishable; however, this is assumed not known until

the end of the diagnosis.

tes ts would be required. ) On the other hand, tes t t separates package

and pl.

If they could be separated, a t l eas t three m o r e

6

completely f r o m packages p There remain four packages to p5 4

be diagnosed that have 0 ' s recorded under tes t t

have 1 ' s recorded.

completely diagnosed with just two remaining tes ts i f tes t t

t

and three packages that 6

Thus there is a possibility that the circuit could be

is used. (Tes t 6

has been ignored for this comparison. ) 5

For this specific example with test t omitted, the same additional 5

tests a r e required whether tes t t

indicates the reason for the difference in ordering.

slightly more the splitting of the remaining faults to be diagnosed into two

equal pa r t s while the probabilistic weight simply selects the tes t with the

highest probability of complete diagnosis with a certain number of additional

tests.

into equal separations is the test with the highest probabilistic weight as

was seen in the selection of the best tests by both methods in Chang's ex-

ample.

is chosen or tes t t 2 6' but the comparison

Chang's method weights

Of course in many cases the tes t which spli ts the remaining faul ts

Exarnining a select set of test patterns will show m o r e clearly

25

some differences of ordering tests between the probabilistic weights and

Chang's weights. This s e t will also indicate one weakness of both proced-

ures.

Consider the set of tests and their weights a s shown in figure 9.

1' t6' Two se ts of tests will completely separate all faults; either tests t

2 , tg, t4, and t t and t or tests t

and t of the first s e t highest while the probability u r e weights tests t

It can be seen that Chang's proced- 5' 7' 8

1' t6' 8

weighting procedure weights tests t and t of the second se t highest. No- 2 5

tice that every test in the f i r s t set of tests except one has the same number

of 0 ' s and 1 's ' while every test in the second se t separates one package

each. Chang's procedure weights tests t t o r t highest since the 0 ' s

and 1's a r e divided more evenly in them. The probability weighting pro-

1' 6' 8

cedure weights tests t and t 2 5 highest since there a r e more possible com-

binations with five outputs that would separate the remaining four packages

than there a r e combinations which would separate the five packages. After

applying either test t

separate the remaining packages with two tests together, whereas a mini-

mum of three tests is requiredfor complete diagnosis after test t t o r

t

or t 2 5 there a r e output patterns that would completely

1' 6'

Admittedly fo r this example, two tests that would separate the remain- 8'

ing packages after applying t o r t would have the highest weight of the 2 5

tes ts listed and would be chosen f i r s t by both methods; however, this is not

always the case for se t s of packages and tests. The probability weighting

I I I I I U 1 D

1 D I 1 1 I 1 1 I n

e

26

Ch ang's Weight 29 27 2 0 20 27 29 28 29

Chang's Test Order 1 5 7 7 5 1 4 1

Wp Test Order 5 1 3 3 1 5 5 5

wp = 0.0059 0.0430 0.0111 0.0111 0.0430 0.0059 0.0059 0.0059

FR- 1277

F i g u r e 9

27

procedure chooses the test that a t l eas t rnakes a shorter dLagnosis possible.

If eight outputs a r e assumed in the probability weighting procedure,

i t tarns oirt that tes+ + has a higher probabilistic

order switches, indicating that with an additional

there exist more combinations that will separate

&l weight than t Thus the 2'

3 three tes ts (or 2 outputs)

the remaining packages

for tes t t than for test t By assuming three

ordering and the probabilistic ordering a r e the

However, this approach ignores the possibility

This points out one of the reasons for

sible number of assumed outputs that will st i l l

1 2' additional tests, Chang's

2' same for tes ts t and t

of diagnosis with fewer tests.

choosing the smallest pos-

give all tes ts a weight other

1

than zero.

correspond more closely to the minimum number of tests that will diagnose

the circuit.

outputs is to make the computations of the weights easier.

By choosing the smallest number of outputs, the probabilities

Another reason for choosing the smallest possible number of

Neither Chang's procedure nor the probability weighting proced-

u r e always chooses the optimum test sequence for all cases. This can be

seen if the only tes ts available for diagnosing the packages in figure 9 are

and t Chang's procedure wouldgick tes t t f i r s t and 5' 1 t es t s t l , t2$ t3, t4?

then the other four tests.

a r e sufficient to diagnose the circuit to the package level.

1, t2? t6, t7, and t 8' i f the only available tes ts are tests t

ility weighting procedure would pick test t

Test t is not needed since the other four tes ts 1

In the same way,

then the probab-

f i r s t and then the other four 2

28

i a

tests.

age level since the other f o u r tests a r e necessary and separate the packages

Test t in this case contributes nothing to the diagnosis to the pack- 2

themselves.

As was pointed out in the introduction, optimality was traded for

the process of choosing the next best test without looking a t all possible

combinations of tests. A glance at the two cases above yields the optimum

I I 8 I B I 1 I

se t of tests because of the simplicity of this problem and because the eye

can see all possible combinations of tests. But of course in la rger com-

binational logic circuits, considering all possible combinations of tests is

an enormous task.

I I

1 1 1 1

3 . PROBABILISTIC WEIGHT FORMULA

3 . 1 Weight Formula Derivation

Calculating the probabilities can be a fairly tedious job i f done by . hand. In order to facilitate the matter, the following i terative formula has

been developed to permit the probabilities to be calculated by computer.

The following recursive formula calculates the number of possible ways

of assigning NT possible outputs to N packages.

Define :

NX (I1, 12, . . . , I N ) = 1

and

NX (I1, 12, . . ., Ik, 0, 0 , . . ., 0 ) =

F R - 1 3 0 3

T = NT - N - I1 - I2 - ... - I k t k t 1

NT = Total number of outputs assumed

N = Number of packages to be separated

th = Number of faults that are in the k t 1 package

N F k t 1

Each I . represents the number of outputs that appear i n the

j package, for j = 1, 2, . . . , k t 1.

J th

NX (I1, 12, . . . , Ik, 0, . . . , 0) i s the number of ways that T t N - (k t 1 )

outputs will separate all remaining packages indexed greater than k, with

29 I

th I. outputs in the j package for j = 1, . . . , k. J

30

The number of possible combinations of NT outputs that will com-

pletely separate all packages is given by NX(0, 0, . . . , 0). The probabilis-

t ic weight is given by:

NX(0, 0, . ... 0) . w = P N T ~ ~ ~

9

(the total number of faults). where N F T = F N F k k= 1

In order to verify that this formula calculates the probabilistic

weight as desired, consider the individual, terms. The maximum number

of outputs T that may appear in a package and sti l l leave a&sufficient number

of outputs to cover the remaining packages is the total number of assumed

outputs NT less the number of outputs assigned already, and less a t l eas t

one output for each of the remaining packages. Therefore

- (N - ( k t l ) ). The minimum number required is - k T = NT - I - Iz - 7 . - 1

1. The t e rm

is the number of ways that outputs of the total number of outputs NT

less the previously used outputs can be selected. These I outputs are k t 1

then applied to the faults of package k+l.

k t l The number of ways that these \tl outputs can be assigned to N F

I I I I I I 1 I I I I I. I I I 1 I I 1

31

faults is given by:

- 1 NFkt 1

J k t l

('kt1 - Jk+l ' ') p (-1)

= 1 J k t 1

19 X2' x3, X4$ e.. 9 x I' To verify this, consider I outputs labeled x

N F There are (I) ways that these outputs can be arranged where N F is the

number of faults in this package. But we seek only those arrangements

that have used all I outputs; we do not want to include those combinations

with fewer than I outputs. There a r e (:)(I - l)NF combinations that have

NF (I - 1) outputs that have been included in the number (I) , and so we must

N F subtract (;)(I - l)NF from (I) . Next we must consider how many com-

binations of (I - 2) we have so far. The t e rm (I)NF has (;)(I - 2)NF too

(:> many. To see how many of these have already been subtracted in the

(I - l)NF te rm consider the matrix of outputs:

x x 2 3 4

X ... I X

I x - x3 x4 ... x 1

I X x4 *' "

x - x1 2

1 X x x

2 3 4 X ...

The dashes indicate the output missing in the listing of the (1-1) combina-

tions.

'I I I I I I I 1 1 I I I I I 1 I I I I

32

Notice that there are ways to choose any two outputs omitted

and that there a r e rows of the matr ix that can have the same remaining

outputs with one more output omitted. Fo r example, the f i r s t two rows

3' x4' omitting x in the second row and x in the first row have outputs x 1 2

Since all other outputs have an x., deleted for 3 L j I I, all . * * y xl' J

other rows with one more output deleted will have either x o r x 1 2 o r both.

Since a row of the a r r ay represents one of the te rms of (I - 1) outputs,

the number of combinations of (I - 2)NF has been subtracted twice, once too

many, and therefore ( i) (I - 2)NF must be added to (I)NF - ( i)(I-l)NF.

Consider now the number of (I - k) outputs. These combinations

also must be removed f rom the total number of combinations since only I

output combinations are wanted. There are (:)(:)(I - k)NF combinations

in the (i) (I)NF term; (;)(:)(I - k)NF combinations in the

t e rm that have been subtracted and (;) (L ) (I-k)NF combinations in the

(;)(I - 2)NF t e rm that have been added. The validity of this l a s t number

can be demonstrated by the array:

8 I I I 1 I 1 I

3 X 4 X ... I X

I X x4 . * * - x -

2

2 x 3

X ... I X

I X x4 . . * x - - 1

I x - x - ... x 1 3

x1 x2 x3 x4 * . - -

There are (:)ways to choose the k outputs to be deleted. Of

these k outputs there are (;)ways to choose two that have already been de-

leted f rom the rows of the array. Hence there a r e (:)(:)(I - kJNK com-

binations that have k outputs deleted in the t e rm (;)(I - 2 l N ~ 4 By a simi- . - ( )(:)(I - k)NF com- term has included N F m lar argument, the (L) (I - m)

N F binations of k outputs. Therefore the coefficient of (-;)(I - k)

t e r m forms a binomial distribution.

for each

Summing these up to see how m a n y

t imes (;)(I - k)NF need be added o r subtracted indicates that just one

t e r m is needed since the final term of the binomial coefficient is 1 and

k= 0

Tc+ 1 Thus the t e rm in (2) gives the number of combinations of

specific outputs where each of these outputs is used a t least once. The

C I I I 1 1 8 8 1)

I I 8 1 I 1 1 f 1 I

34

product of these two t e rms (1) and (2) gives the total number of allowed com-

binations that can occur with outputs.

The te rm NX(I1, 12, ..* I I 0, ..., 0) ( 3 ) k' k t l '

gives the number of combinations that the packages indexed greater than

possible outputs. By multiplying Ik t 1 k t l have with NT - I1 - I2 - ... -

i t is c lear that k t l ' ( 3 ) by (1) and (2) and summing the possible values of I

NX(Il, 12, . . . , Ik, 0, 0 , . . . , 0) is found and gives the total number of

combinations that will separate completely the packages indexed greater

than k with I outputs in the f i r s t package, I

and so forth up to J, outputs in the k

outputs in the second package, 1 2 th

package.

After NX(0, 0, . . . , 0 ) is calculated, the probabilistic weight is

given by :

NX(0, 0, . . . , 0) NFT w =

P NT

NFT The t e r m NT i s the total number of ways that NT outputs can

be assigned to the total number of faults, NFT.

A program that calculates the probabilistic weight for up to three

packages is included in the Appendix.

3. 2 Multi-Output Example

The example selected is a single stage of a paral le l adder com-

posed of two half adders and one "OR" gate. The circuit is shown in figure

1 1c I c I 1 8 1 8 8 8 I E E I 1 f I 1

35

10.

The faults and tes t inputs were simulated on the simulation portion

of Seshu's Sequential Analyzer [ 6 ] , and the dictionary of possible machine

fai lures is shown in figure 11.

between packages were included arbi t rar i ly in one of the packages. The

fault R1 to CZ OPEN was included in the package H2 while faults C1 to C

Some of the possible faults that could occur

OPEN and C2 to C OPEN were included in package C3. The fault-free ma-

chine called the good machine was considered as a separate package to as-

sure that all faults would be separated f rom i t for complete diagnosis.

The tes t fault pattern in figure 12 l i s t s the various faults that cor-

respond to the dictionary of machine f a i l u r e s . in ;figure 11. The outputs are

shown in the entries of figure 12.

been recorded as well as the test order.

Under the tables the probabilities have

The figures f rom figure 12 to

f igure 17 show the complete diagnosis of the full adder stage to the package

level.

The full adder stage was completely diagnosed to the package level

with tes t inputs 010, 011, 101, 001, 000, and 100. T w o possible tests were

not required, and in three cases the best choice was arbi t rary between two

tests.

the same package since the locations of these faults were close to each

other in the physical circuit. Since these indistinguishable faults were in

the same package, no weight was given to separating them, and therefore

no useless search fo r a tes t that would separate them was made.

It can be noted that a few indistinguishable faults were included in

36

8 1

And % -

Or Not

F i g u r e 10

3 ‘7

DICTIONARY OF MACHINE FAILURES FOR LOGIC CIRCUIT CKTl

MACH NOS FAILURE MACH NOS FAILURE

1 GOOD MACHINE 2 A T O C 1 OPEN

3

5

7

9

11

13

15

17

19

21

23

25

27

29

LC

c1

LC

01

N1

R1

B

c 2

B

0 2

c 2

C

02

R2

TO C1 OPEN

OUTPUT 1

TO 01 OPEN

OUTPUT 1

T O R 1 OPEN

OUTPUT 1

TOCZ OPEN

OUTPUT 1

TO 02 OPEN

OUTPUT 1

TO C OPEN

OUTPU T 1

T O R 2 OPEN

OUTPUT 1

INPUT ORDER A B LC

OUTPUTORDER R2 C

F i g u r e 11

4

6

8

10

12

14

16

18

20

22

24

26

28

c 1

A

01

01

R1

R1

c 2

R1

02

c 1

C

N2

R 2

OUTPUT 0

T O 0 1 OPEN

OUTPUT 0

T O R 1 OPEN

OUTPUT 0

T O C 2 OPEN

OUTPUT 0

TOO2 OPEN

OUTPUT 0

T O C OPEN

OUTPUT 0

T O R 2 OPEN

OUTPUT 0

- 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 26 27 28 29

-

-

38

Test ( Inputs 1 000 001 010 011 100 101 110 111

00 01 10 11 10 01 01 11 00 10 10 01 01 01 11 11 00 10 10 0 : 10 10 01 01 01 0 1 11 11 01 01 11 11 00 10 10 01 00 01 10 11 00 00 10 10 10 01 01 11 00 00 10 10 00 01 10 11 10 10 01 01 10 01 01 11 10 10 01 01 10 01 01 11 00 10 10 01 10 11 01 01 00 00 10 10 00 01 10 11 10 10 01 01 10 11 01 01

00 00 00 01 00 0 0 00 10 00 10 00 10

10 01 01 01 10 01 10 10 10 01 01 01 00 10 01 10 00 01 00 00 01 10 10 01 10 10 11 10 10 01 00 00 01 10 10 11

10 01 01 0 1 01 01 01 11 10 01 10 11 01 01 01 01 00 01 01 11 10 01 01 01 00 01 01 01 10 11 01 11 10 01 11 11 10 11 01 11 00 01 01 01 10 11 11 11

00 10 10 01 10 00 01 lo 00 10 10 00 10 01 00 11 00 10 10 00 10 00 00 10 01 11 11 01 11 01 01 11

00 10 10 01 10 01 01 11

3.1812 2.5365 2.8161 2.6090 2.5365 0.2347 2.6090 0.617' x 10-1

8 4 1 2 4 7 2 6 FR- 1279

F i g u r e 12

I 1 I I 8 1 E 1 1 I I I 1 8 8 8 t I I

39

Test (In Package 8 Fault No.

2 H l [ i 3

8 11 12

15

H2 1 26 :! 27 29

-

-

M { 1- H1{ 5

01 1 100 101 110 000 00 00 00 00 00 00 00 00

00 00 00 10 00 10 10

00 00 00

00

--_

---

---

001 01 10 10 10 00 00 10 00

01 10 00 10 10 10 10

lo 10 10

lo ---

111 10 01 10 00 10 00 10 00

01 10 00 10 10 10 10

lo 10 10

lo

--

--

---

01 01 10 01 01 01 11 01

01 01 01 11 01 1 1 11

00 01 00

01

---

---

-

01 11 01 10 01 10 01 10

01 10 01 01 11 01 11

01 00 00

0 1

---

---

11 11 01 11 11 11 01 11

11 11 11 11 11 11 11

lo 11 10

11 ---

11 01 O! 01 10 10 01 10

01 10 01 01 11 01 11

01 00 00

01

11

01

01 01 01

01 01

--

--

--

_-

01

11

01

11

-- 01

01

--- 11

01

11

11 - - .

01

01

10 10 10

00 0 1

I c 3 { 2 5

10 10 10

lo 01

3.047

--

-

4

01 01 11

01 01

---

01 01 0 1

01 01

---

11 11 01

01 01

---

10 10 10

lo 01

Hl[ l”0 13 - I

j:, ~

1.0042

7

0.0477

4

1.082

1

0.1485

3

1.0821

1

0.0094

6 x10-f

FR- 1280 Figure 13

I I I 8 1 8 I E I D 6 1 8

Package 8 Fau I t No.

3 4 6 11

15 18 21 27 -

22 - 1

2 26 29

-

H 2{ 16

Hl[ 1;

13

WP

TO.

Test ( InDuts) 000 001 100 101 110 111

00 01 10 0 1 0 1 11 00 10 10 0 1 11 11 10 10 10 11 11 11

-------------------

00 00 10 01 0 1 11 00 00 00 01 10 11 00 00 00 01 10 11

.-----------__------- 00 10 10 0 1 10 11

10 10 10 01 01 11

10 10 10 01 01 11

10 10 10 11 01 01

00 10 10 0 1 0 1 01

0 1 0 1 01 0 1 0 1 0 1

0.1159 0.1545 0.5866 4.1062 0.0410 1.7598

5 4 3 1 6 2

.-------------------.

x 10-3

40

F R - 1 2 8 l

Figure 14

R

Package 8 Fault No.

Test (Inputs) 000 001 100 110 111

11 l1 I 10 01 11 10 00 10

11 11

00 10 10 01

10 10 10 01 10 10 01

------- ------- H 1 { 11

H 1 { 2

H 2 ( 2 6

H1 [I H2 { 16 --

H1 { ,'o

00 00 10 01 11 00 00 00 10 11 00 00 00 10 11

00 10 10 10 11

10 10 lo 01 11

10 10 10 01 11

00 lo 10 01 01

01 01 01 01 01

----------- --- -

__--------- ----

Wp 0.4877 8.779 3.9018 1.9509 0.4877

TO. 4 1 2 3 4

x 10-2

F R - 1282

Figure 15

8 8

000 30 00

00

00

10 10

10 10

00

1.2500

- - - -

_ _ -

- - - --

1

package 8 Fault No. 100

01 00

10

- - - -

10

10 10

10 10

10

0.0625

3

- -- -

- - - -

H 1 { 11 -_

Mb WP

TO.

Ii2 {

_ _ _ _ _ _ 10 01

1.00 1.00

1 1

H1 {Po

H2 { 14

WP

T.O.

Test ( I n p u t s )

1 10 11 10

01

----

01

01 01

01 01

01

0.0625

3

111 11 11

11

01

1 1 11

--

1 1 11

---- 0 1

0.250C

1 FR- 1283

F i g u r e 16

q 0.25

FR-1284 F i g u r e 17

43

4. FUTURE WORK AND CONCLUSIONS

4. 1 Sequential Machine Extension

The probability weighting procedure could be extended to sequen-

tial machines.

following tables :

A possible extension would involve forming and using the

(Entries consist of an output plus the next state the machine would go to.)

F i g u r e 18

. . .

. . . FR - 1285

It is assumed that there a r e k states and s packages with a vary-

ing number of faults in each package. A row of the table represents a fault

I 0 1 I 8 1 8 1 8 I 8 I 8 8 8 t 8 1 1

in a given package with i ts outputs f o r various test inputs.

represents a test input with the machine in a specific state.

the tes t tq

machine in state q.

bookkeeping since a fault sequential machine may not only have a different

output but also a different next state when compared with the fault-free ma-

chine.

test as well a s the next state that the machine with the fault would go to.

In applying the probability weighting procedure i t is assumed that an initial

state of the machine i s given or that the machine can be put into a given

state.

tion the packages and shorten the diagnostic process.

Each column

F o r example,

i s the test input t applied to the sequential machine with the m m

The states of the machine must be accounted fo r in the

Therefore each table position entry has the output for the applied

If the state that the machine goes to is observable i t will also parti-

If state i is the initial state then the subarray shown in figure 19

i s formed by all columns of tests with superscript i and all rows.

When this a r r ay is used, the probabilistic weights a r e found and

the f i r s t test to be applied is selected.

the a r ray , f rom which to find the next test, i s obtained by f i r s t examining

the next state that each fault would put the machine in.

plying the f i r s t test puts the machine in state h, then the portion of the f

row under column t

i fo r row f The tests of the new ar ray drop their superscripts. After the

a r ray has been completed i t contains the condition of the machine after the

After the f i r s t test has been applied,

i If fault f . after ap-

3 i j

h h 1’ * . * ’ n

t of figure 18 would be placed in the new a r ray

j’

45

8 8 8 8 I 1 8 1 1

p1 I p2 1

8 I 8 B 1

FR - 1286

Figure 19

secopd test has been applied, This a r r ay thus shows the outputs that the

sequential machine would give and i ts next state for any given fault after

applying the first tes t and any second test.

then calculated and a next test selected.

the machine is diagnosed.

The probabilistic weights a r e

This process is continued until

4. 2 Future Research Areas

Applying probabilistic weights to sequential machines will require,

in addition, a procedure for determining the next test to be applied when

the present possible tests do not give any additional diagnostic information,

but other tests in other states do,

the diagnostic process so that all packages a r e separated that can be, but

indistinguishable faults in different packages do not cause useless tests to

be inserted in the sequence of diagnostic tests.

l em is to provide the detailed cri teria fo r the extension of this probability

weighting method to sequential machines.

A criterion is needed for terminating

A possible research prob-

The probability weighting procedure could b e altered in two ways

that may improve the procedure fo r combinational machines.

ity would be to purposely weight some of the tests zero in order to shorten

the time for calculating the weights.

each time is to select the best test that would next be applied to the machine.

The order of the tests i s unnecessary especially f o r those tests which give

the least information.

zero would not prevent selecting the best test to be applied.

weight of zero to some tests could be done by assuming fewer outputs for

the hypothetical test so that this test cannot separate the remaining pack-

ages.

four outputs a r e assumed possible for the hypothetical test, then the probab-

One possibil-

The purpose for weighting the tests

Weighting the tests that give the leas t information

Giving the

For example, if five packages remain af ter applying a tes t and only

i l ist ic weight is zero. Other tests that reduce the number of packages left

to be diagnosed sufficiently would sti l l have non- zero probabilistic weights

which would provide the means f o r selecting the best test. Under this new

consideration a procedure would have to be developed for determining how

many outputs should be assumed for the next test.

puts may jeopardize the selection of the best tes t while too many assumed

outputs would not take the greatest advantage of the shortened procedure.

Too few assumed out-

Another variation of the probability weighting procedure to consider

is the calculation of the probabilistic weights based on two or three tests

taken a t a time in order to select the best two or three next tes ts for diag-

nosis.

s e t of diagnostic tests but would require more bookkeeping and computa-

tions.

This approach would increase the chances of achieving an optimal

F o r certain problems this may be desirable.

A further a r e a of research would be to examine m o r e closely diag-

nosis to the package level with only a truth table for each package.

sumptions are necessary or useful in forming such a general diagnostic

procedure ?

What as-

A multiple output optimum procedure that would diagnose to the

package level is another research problem,

knight's [l] might be considered.

A procedure similar to Bou-

What types of elements should be included in packages that would

lend themselves better to diagnosis to the package level? Since package lev-

I I I I 8 8 0 I 8 1 I I 8 0 1 1 I I I

, 48 i

e l diagnosis is possible, could this be extended to a computer block diagno- ~

s i s ? If so, what a r e the advantages? The disadvantages? These questions

suggest further a reas of research. I

4. 3 Conclusions

This investigation has produced an instructive, although not prac-

tical, optimum procedure fo r diagnosing multi-output machines to the pack-

age level somewhat analogous to the pr ime implicant problem in logic de-

sign. The investigation a l s o has produced a procedure f o r selecting a se t

of tests sequentially to diagnose multi-output machines to the package level,

thereby eliminating redundant tests.

select the optimum set, i t does select a near-optimal s e t for combinational

machines. The probability weighting procedure has the advantage that the

s e t of selected tests is found without exhaustive t r ia l combinations of tests

which other optimal procedures require.

used f o r diagnosis one at a time. After applying previously selected tests,

the test that has the largest probabilistic weight is chosen and added to the

set. The probabilistic weight is the probability that a hypothetical test

would completely separate all remaining unseparated packages after the sg-

lected tests and the test being weighted had been applied to the machine.

Since diagnosis is accomplished to the package level, fewer tests a r e usual-

l y required than would be fo r diagnosing to the fault level.

Although the procedure may not always

The tes t s a r e added to the se t

LIST O F REFERENCES

49

I 8 8 I I 8 I I

1. Bouknight, W. J . , "On the Generation of Diagnostic Tes t Procedures , Coordinated Science Laboratory Report 292, University of Illi- nois, May 1966.

2. Chang, H. Y., "An Algorithm for Selecting an Optimum Set of Diag- nostic Tests, I ' IEEE Transactions on Electronic Computers, Volume EC-14, Number 5, pp. 706-71'0, October 1965.

3. Howarter, D. R . , "The Selection of Failure Location Tests by Pa th Sensitizing Techniques, I ' Coordinated Science Laboratory Report 316, University of I l l inois , August 1966.

4. Seshu, S. and Freeman, D. N . , "The Diagnosis of Asynchronous Se- qsmtia2Switching Systems, I ' IRE Transactions on Electronic m u t e r & , Volume E C - 1 1 , Number 4, pp. 459-465, August 1962.

5. Seshu, S. , "On an Improved Diagnosis Program, Coordinated Science Laboratory Report 207, University of Illinois, May 1964.

6. Seshu, S . , "The Logic Organizer and Diagnosis P rograms , Coordi- nated Science Laboratory Report 226, University of Illinois, July 1964.

APPENDIX

The program SEPPROB is a FORTRAN program for finding the

probability of separating up to three packages.

MNP is the maximum number of outputs, CKN is the number of tests whose

probabilities a r e to be calculated, NM is the number of packages, and

th N F ( i ) , the number of faults in the i package.

The inputs a r e a s follows:

program sepprob

dimension

'nf(31,jj ( 3 ) , 4 16,16,16)

c o m n nf, jj, nx, mno, U, nm, mindx

C mno=nwdmumnumber of outputs. ckn=nmber of tes ts

read 1 , rum, ckn

1 format ( l i2 , l f3 .1)

ck.jd'.(2l

2 read 3, m,nf(l ),nf(2),nf(3)

3 format (4i2)

ck=ck+l.@

do 4 k=l,mn

4 33 (k)=a'

18h module ,2( 1 i2,l h, ), 1 i2,/, lx, 1 bhprobability = ,l f7.5 1 if (ck. eq. ckn) 1 g,2

19 end

C t o cdculate nx( i1,i2,g,j~f) from stan over j of nx( il ,i2, j,$)

subroutine nxtv(nfk)

dimension

W31,JJ (3),d6,16,16)

c o m n nf, jj, nx, m, 11, nm, mindx

call nindxm(maxi,nfk)

if(nfk.eq.la’) 56J3

53 maxj=mindx+nm-U--l

do 54 nn=l,maxi

imodcg

do 55 mm=l,nn

I I I I 8 8 I D 1 I B 1 I 1 1 I I I I

mmlo=mn-l

55 imodc=imOdc+(-1 ) ' 'mmlo'nbinomc(nn,mmlo) '(nn-nmiLo)"nfk

D t 1 8 8 8 I I 8 1 I 1 I 1 I 1 I 1 1

79

81

w

82

83

84

173

1 74

176

fm-m

binoslc=fnn/rr

nrrlo=nrr-l

do 176 k=l ,nrrlo,l

dl a - k

s~b2-m-k

subm=sub 1 /sub2

binom=binmc'stibm

nbinomc=binomc

'i I I I 8

55

t

"I I

go to 178

177 nbinom=i

178 end

end

X

e .

e .

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Secu n t y Classification

DOCUMENT CONTROL DATA - R 8, D 1 (Secur i ty c l a s s i f i c a t i o n of t i t l e , body of abs trar t and indexing annotation m u s t be entered when the overall report is r J a a s ; t i e d )

2a. R E P O R T 5 E C U R I T Y C L A 551 F I c A T I ON . O R I G I N A T I N G A C T I V I T Y (Corporate author) U n i v e r s i t y of I l l i n o i s UncJassif i ed

Urbana, I l l i n o i s 61801 i Coor d ina t ed S c ience Labor a t or y Zb. G R O U P

I 1 . R E P O R T T I T L E

A PROCEDURE FOR RAHKING DIAGNOSTIC TEST INPUTS

4 . D E S C R I P T I V E N O T E S (ripe ot report and. inc los ive d a t e s )

5 . A U T H O R ( S ) {First name, middle initial, l a s t name) 1 POWELL, The0

,. R E P O R T D A T E

May, 1967

DA 28 043 AMC 00073(E) 2 0 0 14 5 0 1B 3 1F

?.e. C O N T R A C T OR G R A N T N O .

b. P R O J E C T N O .

C .

d.

10. D l S T R l 8 U T l O N S T A T E M E N T

I

\ 7 6 . NO. O F R E F S 7a. T O T A L NO. O F P A G E S t

55 6 9a. O R I G I N A T O R ' S R E P O R T N U M B K R ( S )

t

R-354 I

96. O T H E R R E P O R T NO(S1 (Any other numbers that may be a s s i g n e d th is report)

4

D i s t r i b u t i o n o f t h i s r e p o r t is un l imi t ed I

1 1 . S U P P L E M E N T ~ R Y N O T E S 12. S P O N S O R I N G M I L I T A R Y A C T I V I T Y

Jo i n t Se rv ices E lec tr on i c s Program t h r u U. S. Army E l e c t r o n i c s Command F t Monmouth, N e w J e r s e y 07703

13. A 8 S T R A C T

S/N 01 01 -807-681 1 Securitv Classification

-12-

Fig. 3

Fig. 4 . . -