Test generation

1
Test generation

• Gate-level methods
• Functional testing universal test sets
• Structural test generation
• Path activation conception
• Algorithms D, Podem, Fan
• Test generation for multiple faults
• Test generation for sequential circuits
• Random test generation
• Genetic algorithms for test generation
• High-level and hierarchical methods
• Test generation for digital systems
• Test generation for microprocessors

2
Test generation

• Universal test sets
• 1. Exhaustive test (trivial test)
• 2. Pseudo-exhaustive test
• Properties of exhaustive tests
• 1. Advantages (concerning the stuck at fault
  model)
• - test pattern generation is not needed
• - fault simulation is not needed
• - no need for a fault model
• - redundancy problem is eliminated
• - single and multiple stuck-at fault coverage is
  100
• - easily generated on-line by hardware
• 2. Shortcomings
• - long test length (2n patterns are needed, n -
  is the number of inputs)
• - CMOS stuck-open fault problem

3
Test generation

• Pseudo-exhaustive test sets
• Output function verification
• maximal parallel testability
• partial parallel testability
• Segment function verification

Output function verification
4
4
4
Segment function verification
4
216 65536 gtgt 4x16 64 gt 16
1111
0011

F
Pseudo- exhaustive sequential
Pseudo- exhaustive parallel
Exhaustive test
0101
4
Functional testing universal test sets
Output function verification (maximum parallelity)
Exhaustive test generation for n-bit adder Good
news Bit number n - arbitrary Test length -
always 8 (!)
Bad news The method is correct only for
ripple-carry adder
0-bit testing
1-bit testing
3-bit testing etc
2-bit testing
5
Testing carry-lookahead adder
General expressions
n-bit carry-lookahead adder
6
Testing carry-lookahead adder
R
1 1 0 0 1 1 0 0 1 1
0 0 1 1 0 0 1 1 0 0
1 1 0 0 1 1 1 1 1 0 0
1 1 1 1 1
1 0 0 1 1 0 1 1
1 1 1 0 1 1
0 1 1 0 1 1 1 0
1 1 1 0 0 1 1 1
1 0 1 1 1
1 0 1 1 0 1 0 1 1
1 1 1 0
0 1 1 0 1 1 1 1
1 1 0 0
Testing ? 0
Testing ? 1
For 3-bit carry lookahead adder for testing only
this part of the circuit at least 9 test patterns
are needed (i.e. pseudoexhaustive testing will
not work) Increase in the speed implies worse
testability
7
Test generation
Output function verification (partial parallelity)
F1
x1
0011- -
F1(x1, x2)
0011- 0
F2(x1, x3)
x2
010101
F3(x2, x3)
F3
F4(x2, x4)
x3
F2
010110
F5(x1, x4)
F4
x4
00-11-
F6(x3, x4)
F5
000111
Exhaustive testing - 16 Pseudo-exhaustive, full
parallel - 4 Pseudo-exhaustive, partially
parallel - 6
8
Structural Test Generation
Structural gate-level testing fault
sensitization

• A fault a/0 is sensitisized by the value 1 on a
  line a
• A test t 1101 is simulated, both without and
  with the fault a/0
• The fault is detected since the output values in
  the two cases are different
• A path from the faulty line a is sensitized (bold
  lines) to the primary output

9
Structural Test Generation

• Structural gate-level testing
• Path activation

Fault sensitisation x7,1 D Fault
propagation x2 1, x1 1, b 1, c 1 Line
justification x7 D 0 x3 1, x4 1 b 1
(already justified) c 1 (already justified)
1
1
Macro
1
d
1
1
a

2

71
D
D
D
1

e
3
7
72
b
1
1
4
y
D
D

5
73
c
1
6
Test pattern
Symbolic fault modeling
D 0 - if fault is missing D 1 - if fault
is present
10
Test generation

• Test generation for a bridging fault

Component F(x1,x2,,xn)
y
Activate a path
Bridge between leads 73 and 6
Wd
Defect
Macro
1
1
d
1

2
a

71
D
D
Fault manifestation Wd x6x7 1 x6 0, x7
1, x7,1 ?D Fault propagation x2
1, x1 1, b 1, c 1 Line justification b
1 x5 0
D

e
3
72
7
b
1
4
y
D
D

5
73
c
1
6
Wd
11
Test generation
Multiple path fault propagation
0
0
D
D
0
0
x1
x1
1
D
1
D
D
x2
y
x2
y
D
D
1
1
D
D
x3
x3
x4
x4
D
0
1
1
1
0
Single path activation is not possible
Three paths simultaneously activated
12
Test generation
D - algorithm (Roth, 1966)
Select a fault site, assign D Propagate D along
all available paths using D-cubes of
gates Backtracking, to find the inputs needed
Example
Fault site
D
1
D
1

2
4
1
Propagation D-cubes for AND-gate
3
1
1
D
D

4
2
1
3
13
Test generation
D - algorithm
Propagation of D-cubes in the circuit
Primitive D-cubes for NAND and c ? 0 a
b c 0 x D x 0 D
Singular cover for C NAND (A,B) a b c 1
1 0 x 0 1 0 x 1
4
1


2
6
5
Intersection of cubes Let have 2 D-cubes A
(a1, a2,... an) B (b1, b2,... bn) where ai, bj
? ?0,1,x,D,D) 1) x ? ai ai 2) If ai ? x and bi
? x then ai ? bi ai if bi ai or
ai ? bi ? otherwise 3) A ? B ? if for
any i ai ? bi ?

3
1 2
3 4 5 6 D-drive Primitive cube for x2 ? 1
D Propagate D through G4 1 D D Propagate
D through G6 1 D D 1 D Consistency
operation Intersect with G5 1 D 0
D 1 D
Propagation D-cubes for C NAND (A,B) a b
c 1 D D D 1 D D D D
14
Test generation
Multiple path fault propagation by DDs
Structural DD
x41
x3
x21
x12
y
x23
x33
x24
x42
x11
x31
x22
x32
0
Functional DD
D
0
x1
D
1
x3
x1
x4
x2
y
D
x2
y
1
D
x3
x3
x1
x4
x4
D
0
1
1
0
15
Test generation

• PODEM - algorithm (Goel, 1981)
• 1. Controllability measures are used during
  backtracking
• Decision gate
• The easiest input
• will be chosen at first
• Imply gate
• The most difficult input
• will be chosen at first
• 2. Backtracking ends always
• only at inputs
• 3. D-propagation on the basis of
• observability measures

0
0

1

1
16
Test generation

• FAN - algorithm (Fujiwara, 1983)
• 1. Special handling of fan-outs (by using
  counters)
• PODEM backtracking continues
• over fan-outs up to inputs
• FAN backtracking breaks off,
• the value is chosen
• on the basis of values in counters
• 2. Heuristics is introduced into D-propagation
• PODEM moves step by step (without predicting
  problems)
• FAN finds bottlenecks and makes appropriate
  decisions
• at the beginning, before starting D-propagation

1 (C 6)
Chosen value
0 (C 3)
1
0 (C 2)
17
Example Test Generation with SSBDDs
Testing Stuck-at-0 faults on paths
x11
x21
1
x11
y
x1
x21

x2
x12
x31
x4
x12
x31
x3
y

1
x4
Test pattern
x22
x32
x13

x13
x1 x2 x3 x4 y 1 1 0 - 1

x22
x32
0
Tested faults x12?0, x21?0
18
Example Test Generation with SSBDDs
Testing Stuck-at-0 faults on paths
x21
x11
y
x1

x2
x31
x4
x12
1
x3
y

1
x4
x22
x32
x13


Test pattern
0
x1 x2 x3 x4 y 1 0 1 1 1
Tested faults x12?0, x31?0, x4?0
19
Example Test Generation with SSBDDs
Testing Stuck-at-0 faults on paths
x21
x11
y
x1

x2
x31
x4
x12
x3
y

1
x4
x22
x32
x13
1


Test pattern
x1 x2 x3 x4 y 0 1 1 0 1
0
Tested faults x22?0, x32?0 Not tested
x13?1
20
Example Test Generation with SSBDDs
Testing Stuck-at-1 faults on paths
y
x21
x11
1
x11
x1
x21

x2
x31
x4
x12
x12
1
x31
x3
y

1
x4
x22
x32
x13
1
Test pattern

x13
x1 x2 x3 x4 y 0 0 1 1 0

x22
0
x32
Tested faults x12?1, x22?1 Not tested
x11?1
21
Example Test Generation with SSBDDs
Testing Stuck-at-1 faults on paths
y
x21
x11
1
x11
x1
x21

x2
x31
x4
x12
x12
1
x31
x3
y

1
x4
x22
x32
x13
1
Test pattern

x13
x1 x2 x3 x4 y 1 0 0 1 0

x22
0
x32
Tested faults x21?1, x31?1, x13?0
22
Example Test Generation with SSBDDs
Testing Stuck-at-1 faults on paths
y
x21
x11
1
x11
x1
x21

x2
x31
x4
x12
x12
1
x31
x3
y

1
x4
x22
x32
x13
1
Test pattern

x13
x1 x2 x3 x4 y 1 0 1 0 0

x22
0
x32
Not yet tested fault x32?1
Tested fault x4?1
23
Transformation of BDDs
y
y
x21
y
x2
x2
x11
x1
x1
SSBDD
x31
x4
x31
x4
x3
x4
x12
x12
x12
x22
x32
x22
x32
x22
x32
x13
x13
x13
y
y
x2
x2
x1
x1
Optimized BDD
x4
x3
x4
x3
BDD
x2
x3
x2
24
Example Test Generation with BDDs
Testing Stuck-at faults on inputs
y
x21
x11
x11
x1
x21
SSBDD

x2
x31
x4
x12
x12
x31
x3
y

1
x22
x32
x4
x13

x13
y
x2
x1
1

x22
BDD
x32
x1 x2 x3 x4 y D 1 0 - D
x4
x3
Test pair D0,1
x2
0
Tested faults x1?0, x1?1
25
Test generation
Test generation by using disjunctive normal forms
26
Multiple Fault Testing

• Multiple faults fenomena
• Multiple stuck-fault (MSF) model is a
  straightforward extension of the single
  stuck-fault (SSF) model where several lines can
  be simultaneously stuck
• If n - is the number of possible SSF sites,
  there are 2n possible SSFs, but there are 3n -1
  possible MSFs
• If we assume that the multiplicity of faults is
  no greater than k , then the number of possible
  MSFs is
• ?ki1 Cni2i
• The number of multiple faults is very big.
  However, their consideration is needed because of
  possible fault masking

27
Multiple Fault Testing

• Fault masking
• Let Tg be a test that detects a fault g
• A fault f functionally masks the fault g
  iff the multiple fault f, g is not
  detected by any pattern in Tg
• The test 011 is the only test
• that detects the fault c ? 0
• The same test does not detect
• the multiple fault c ? 0, a ? 1
• Thus a ? 1 masks c ? 0
• Let Tg ? T be the set of all tests in T
  that detect a fault g
• A fault f masks the fault g under a test T
  iff the multiple fault f , g is not detected
  by any test in Tg

Example
Fault a ? 1
Fault c ? 0
28
Multiple Fault Testing

• Circular fault masking
• Example
• The test T 1111, 0111, 1110, 1001, 1010,
  0101 detects every SSF
• The only test in T that detects the single
  faults b ? 1 and c ? 1 is 1001
• However, the multiple fault b?1, c?1 is not
  detected because under the test vector 1001, b ?
  1 masks c ? 1, and c ? 1 masks b ? 1

Multiple fault F may be not detected by a
complete test T for single faults because of
circular masking among the faults in F
1
a
1/0

0/1
b
1

0/1
0

0/1

0/1
1
c

1
d
1/0
29
Multiple Fault Testing
Testing multiple faults by pairs of patterns
11

• To test a path under condition of multiple
  faults, two pattern test is needed
• As the result, either the faults on the path
  under test are detected or the masking fault is
  detected
• Example
• The lower path from b to output is under test
• A pair of patterns is applied on b
• There is a masking fault c ? 1
• 1st pattern fault on b is masked
• 2nd pattern fault on c is detected

10
a

01
b
11

?1 faults
01 (00)

01

00
(11)
10 (11)
c

11
d
11(00)
The possible results 01 - No faults detected 00
- Either b ? 0 or c ? 1 detected 11 - The fault
b ? 1 is detected
30
Test generation
Testing multiple faults by groups of patterns
Multiple fault x1?1, x2?0, x3?1
An example where the method of test pairs
does not help
Fault masking
Fault detecting
T1
T2
T3
x3?1
x1?1
x2?0
31
Test generation
Method of pattern groups on DDs
x1
x2
x1
y

x2
Test group for testing a part of circuit
x3
x3
x4
x1
y

1
x4

x2
x3
x1 x2 x3 x4 y 1 1 0 - 1 0
1 0 - 0 1 0 0 - 0
-
x1

Disjunctive normal forms are trending to
explode DDs provide an alternative
32
Test generation
Test generation for sequential circuits
Fault sensitization Test pattern consists of an
input pattern and a state Fault propagation To
propagate a fault to the output, an input pattern
and a state is needed Line justification To
reach the needed state, an input sequence is
needed
y
x
CC
R
Time frame model
y
x
y
y
x
x
CC
CC
CC
R
R
R
33
Hierarchical Test Generation

• In high-level symbolic test generation the test
  properties of components are often described in
  form of fault-propagation modes
• These modes will usually contain
• a list of control signals such that the data on
  input lines is reproduced without logic
  transformation at the output lines - I-path, or
• a list of control signals that provide one-to-one
  mapping between data inputs and data outputs -
  F-path
• The I-paths and F-paths constitute connections
  for propagating test vectors from input ports (or
  any controllable points) to the inputs of the
  Module Under Test (MUT) and to propagate the test
  response to an output port (or any observable
  points)
• In the hierarchical approach, top-down and
  bottom-up strategies can be distinguished

34
Hierarchical Test Generation Approaches
Bottom-up approach
Top-down approach
A
A
System
System
a
a
D
D
B
B
c
C
C
c
a,c,D fixed x - free
a,c,D fixed x - free
a
ax
D
dx
A ax D dx C cx
A ax D B bx C cx
cx
c
Module
Module
35
Hierarchical Test Generation Approaches

• Bottom-up approach
• Pre-calculated tests for components generated on
  low-level will be assembled at a higher level
• It fits well to the uniform hierarchical approach
  to test, which covers both component testing and
  communication network testing
• However, the bottom-up algorithms ignore the
  incompleteness problem
• The constraints imposed by other modules and/or
  the network structure may prevent the local test
  solutions from being assembled into a global test
  
• The approach would work well only if the the
  corresponding testability demands were fulfilled

A
System
a
D
B
C
c
a,c,D fixed x - free
a
D
A ax D B bx C cx
c
Module
36
Hierarchical Test Generation Approaches
Top-down approach
A
System

• Top-down approach has been proposed to solve the
  test generation problem by deriving environmental
  constraints for low-level solutions.
• This method is more flexible since it does not
  narrow the search for the global test solution to
  pregenerated patterns for the system modules
• However the method is of little use when the
  system is still under development in a top-down
  fashion, or when canned local tests for modules
  or cores have to be applied

a
D
B
c
C
a,c,D fixed x - free
ax
dx
A ax D dx C cx
cx
Module
37
Test Generation
High-level test generation with DDs Conformity
test
Decision Diagram
Multiple paths activation in a single DD Control
function y3 is tested
R
2
0
y

0
4
Data path
1
R
2
0
0
2
y
y
R

R
3
1
1
2
1
IN

R
2
1
IN
2
R
1
0
3
y
R

R
2
1
2
1
IN
R
Control For D 0,1,2,3 y1 y2 y3 y4 00D2
Data Solution of R1 R1 ? IN ? R1 ? R1
R1
Test program
2
38
Test Generation
High-level test generation with DDs Scanning
test
Decision Diagram
Single path activation in a single DD Data
function R1 R2 is tested
R
2
0
y

0
4
Data path
1
R
2
0
0
2
y
y
R

R
3
1
1
2
1
IN

R
2
1
IN
2
R
1
0
3
y
R

R
2
1
2
1
IN
R
Control y1 y2 y3 y4 0032 Data For all
specified pairs of (R1, R2)
Test program
2
39
Test Generation
High-level path activation on DDs
Transparency functions on Decision Diagrams Y
C ? y3 2, R2 0 C - to be tested R1 B ?
y1 2, R3 0 R1 - to be justified
40
Test Generation
DD synthesis for control path
DD for the FSM
FSM state transitions and output functions
41
Test Generation
High-level DDs
Data path
Control path
42
Test Generation for Digital Systems
High-level test generation for data-path
(example)
t
t-1
t-2
t-3
Time
q4
q2
q1
q0
y
2
3
y
R
0
0
2
2


R
D
3


0

R
0
2
q2
Fault propagation
q1
y
2

• Test generation steps
• Fault manifestation
• Fault-effect propagation
• Constraints justification

1
y
0
3
C
D

R
0



A
D
3

1


0


A
R
R
D

B
D
1
1
2

2
Fault manifestation
Constraints justification
43
Test Generation for Digital Systems
Test generation step Fault-effect propagation
t
t-1
t-2
t-3
Time
q2
q1
q0
q4
0
y
y

Y,R
0
2
3
3
3
y
R
0
0
2
2
1


R
D
R
3
3


2
0
q y1 y2 y3
0

R
0
C
R
2
q2
0
2

1001
Fault propagation
q
q1
0
C
R
1
1
y
2
2
2120
1
R
0
2
y
0
3
0
C
D

R
0


3021


A
D
3
CR

1


2
0
2

4200


A
R
R
D

B
D
1
1
2

2
3
Fault manifestation
4

0112
Constraints justification
44
Test Generation for Digital Systems
Test generation step Line justification
Time t-1
Path activation procedures on DDs
0
0

y
Y,R
0
3
3
1
R
3
2
0
C
R
2
0
C
R
2
CR
2
0
q y1 y2 y3

1001
q
1
1
R
0
2120
2
0

3021
2

4200
3
4
45
Test Generation for Digital Systems
High-level test generation example
t
t-1
t-2
t-3
Time
Symbolic test sequence
q4
q2
q1
q0
y
2
3
y
R
0
0
2
2


R
D
3


0

R
0
2
q2
Fault propagation
q1
y
2
1
y
0
3
C
D

R
0



A
D
3

1


0


A
R
R
D

B
D
1
1
2

2
Fault manifestation
Constraints justification
46
Test Generation
Test program generation for a microprocessor
(example)
DD-model of the microprocessor
Instruction set
1,6
I
IN
A
3

• I1 MVI A,D A ? IN
• I2 MOV R,A R ? A
• I3 MOV M,R OUT ? R
• I4 MOV M,A OUT ? IA
• I5 MOV R,M R ? IN
• I6 MOV A,M A ? IN
• I7 ADD R A ? A R
• I8 ORA R A ? A ? R
• I9 ANA R A ? A ? R
• I10 CMA A,D A ? ? A

2,3,4,5
I
R
OUT
IN
4
7
A R
A
8
2
A ? R
I
A
R
9
A ? R
5
IN
10
? A
1,3,4,6-10
R
47
Test Generation
Test program generation for a microprocessor
(example)
DD-model of the microprocessor
Scanning test for adder Instruction sequence
I5 I1 I7 I4 for all needed pairs of (A,R)
1,6
I
IN
A
I4
3
OUT
2,3,4,5
I
R
OUT
IN
I7
4
A
7
A R
I1
A
A
8
R
IN(2)
2
A ? R
I
A
R
I5
R
9
A ? R
5
IN(1)
IN
Time
10
t
t - 1
t - 2
t - 3
? A
1,3,4,6-10
Observation Test Load
R
48
Test Generation
Test program generation for a microprocessor
(example)
Conformity test for decoder Instruction sequence
I5 I1 D I4 for all D??I1 - I10? at given A,R,IN
DD-model of the microprocessor
1,6
I
IN
A
Data generation
3
2,3,4,5
I
R
OUT
IN
4
7
A R
A
8
2
A ? R
I
A
R
9
A ? R
5
IN
10
? A
1,3,4,6-10
Data IN,A,R are generated so that the values of
all functions were different
R
49
Test Generation
Hierarchical approach with functional fault model