Transcript Test generation

2. Sissejuhatus teooriasse
1.
2.
Teooria: Boole’i differentsiaalvõrrand
Teooria: Binaarsed otsustusdiagrammid
4
Rike
2
Testi süntees
Test
1
Digitaalsüsteem
Diagnoos
Reaktsioon
Stiimul
Diagnostika
sõnastik
Testi analüüs – rikete simuleerimine
3
Technical University Tallinn,
ESTONIA
1
Fault Propagation Problem
Logic gate
Logic circuit
Path activation
Correct
signal
Fault
activation
1
0
&
1 0
Error
Fault
activation
x3 = 1
Fault
Stuck-at-0
Path
activation
Correct
signal
x1 0
x2 1
x4 0
x5
x6 0
x7
1 0
y
F (X)
Error
Fault“Stuck-at-1”
Technical University Tallinn,
ESTONIA
2
Boolean derivatives
Boolean function:
Y = F(x) = F(x1, x2, … , xn)
Boolean partial derivative:
F ( X )
 F ( x1 ,...xi ,...xn )  F ( x1 ,...x i ,...xn )
xi
F ( X )
 F ( x1 ,...xi  0,...xn )  F ( x1 ,...xi  1,...xn )
xi
Technical University Tallinn,
ESTONIA
3
Boolean Derivatives
Useful properties of Boolean derivatives:
If F(x) is independent of xi
F ( X )G( X )
G( X )
 F(X )
xi
xi
F ( X )  G( X )
G( X )
 F(X )
xi
xi
Näide:
x1x4 ( x2 x3  x2 x3 )
F ( X )  x1 x4 G( X )  x2 x3  x2 x3
F ( X )G( X )
G( X )
 x1 x4
x2
xi
These properties allow to simplify the Boolean differential equation
to be solved for generating test pattern for a fault at
xi
Technical University Tallinn,
ESTONIA
4
Boolean derivatives
F ( X )
0
xi
F ( X )
1
xi
-
if F(x) is independent of xi
-
if F(x) depends always on xi
F ( X )
 F ( x1 ,...xi  0,...xn )  F ( x1 ,...xi  1,...xn )
xi
Examples:
F ( X )  x1 ( x2 x3  x2 x3 ) :
F ( X )  x1  x2 :
F ( X )
 x1 x2  x1 x2  0
x3
F ( X )
 x2  x2  1
x1
Technical University Tallinn,
ESTONIA
5
Boolean vector derivatives
If multiple faults take place independent of xi
F ( X )
 F ( x1 ,...xi , x j ,...xn )  F ( x1 ,...xi , x j ,...xn )
( xi , x j )
Example:
x1
x2
y  F ( x1, x2 , x3 , x4 )  x1x2  x3 x4
&
1
x3
x4
&
F ( X )
 x1 x 4  x1 x4  x1 x4 ( x 2 x 3  x2 x3 )
( x2 , x3 )
Technical University Tallinn,
ESTONIA
6
Boolean vector derivatives
Calculation of the vector derivatives by Carnaugh maps:
y  F ( x1, x2 , x3 , x4 )  x1x2  x3 x4
x2
x2
x2
1 1
x1
1 1
1 1 1
1 1
x3
x
1
1 1
1 1 1
x4
x4
x1 x2  x3 x4
x1 x2  x3 x4
x3
=
x1
1 1
1 1
1
1 1
1
1 1
x3
x4
F ( X )
 x1 x 4  x1 x4  x1 x4 ( x 2 x3  x2 x3 )  1
( x2 , x3 )
Technical University Tallinn,
ESTONIA
7
Boolean vector derivatives
Interpretation of three solutions:
F ( X )
 x1 x 4  x1 x4  x1 x4 ( x 2 x3  x2 x3 )  1
( x2 , x3 )
x1 x4  1
x1
x2
Fault
1
&
1
Two paths activated
x3
x4
y
&
1
x1 x4  1
Single path activated
Fault
x1
x2
1
&
1
x3
x4
y
&
0
Technical University Tallinn,
ESTONIA
8
Derivatives for complex functions
Boolean derivative for a complex function:
Fk ( Fj ( X ), X )
xi
Fk Fj

Fj xi
y y x1 x3

x4 x1 x3 x4
Example:
x4
x3
x1
y
x2
Additional condition:
x2
0
x3
Technical University Tallinn,
ESTONIA
9
Research in ATI
© Raimund Ubar
Topological Idea of Test Generation
Fault
activation
Path
activation
x1
x2
x3 = 0
Fault
Stuck-at-1
Correct
signal
xx1
0
0
1
x4 0
x5
x6 0
x7
y
0 1
11
1
1
x2
xx3
0
y
F (X)
1
x4
Error
x5
0
x6
x7
0
y  x1  x2 ( x3  x4 x5 )  x6 x7
0
10
Research in ATI
© Raimund Ubar
Mapping Between Circuit and SSBDD
Each node in SSBDD represents a signal path:
x11
x6
x21 &
Signal path
x1
x2
x12
x31
x3
x4
x13
&
&
x7
x5
x22
x32
1
y
x11 1 x21
1
0
x12
x31
x4
x13
x22
x32
y
&
x8
0
Node x11 in SSBDD represents the path (x1, x11, x6, y ) in the circuit
The SAF-0(1) fault at the node x11 represents the SAF faults on the
lines x11, x6, y in the circuit  fault collapsing
32 faults (16 lines) in the circuit  16 faults (8 nodes) in SSBDD
11
Test Generation with BD and BDD
BDD:
BD:
y  x1x2  x3 ( x2 x4  x1 ( x4  ( x5  x2 x6 ))  x1 x3
y
x1
x2
x3
x2
x4
x1
x4
x
y
 ( x1 x2  x1 x3 ) x3 ( x2 x4 ) x1 x4 ( x2 x6 ) 5 
x5
x5
x5
 ( x1  x2 )(x1  x3 ) x3 ( x2  x4 ) x1 x4 ( x2  x6 ) 
 x1 x4 x3 x2  ...  1
x2
Test pattern:
x1
x2
x3
x4
x5
x6
y
0
1
-
0
D
-
D
x1
1
x6
x2
0
Technical University Tallinn,
ESTONIA
12
Research in ATI
© Raimund Ubar
Fault Analysis with SSBDDs
Algorithm:
1. Determine the activated path to find the fault candidates
2. Analyze the detectability of the each candidate fault
(each node represents a subset of real faults)
x1
x2
x11
x21
0
1
x12
x31
x3 0
x 1
y
&
&
4
x13
&
x11 1 x21
1
0
0
0
0
0
1
y
x12
x31
x4
x13
x22
x32
&
x22
x32
0
13
Research in ATI
© Raimund Ubar
Test Generation with SSBDDs
Test generation for: x110
x1
x2
x3
x4
10
1
x13
1
x11
x21 &
x12
x31
0
Structural BDD:
&
y
0
0
x12
10
1
x31
x4
y
1
0
x13
&
x22
x32
Test pattern:
x21
10
10
&
x11
1
0
x1 x2 x3 x4 y
1
1
0
-
1 0
x22
x32
Functional
BDD:
1
y
x1
1
x2
1
x4
x3
10
x2
1
0
14
Research in ATI
© Raimund Ubar
Functional Synthesis of BDDs
Shannon’s Expansion Theorem: y  F ( X )  xk F ( X ) xk 1  xk F ( X ) xk 0
y  x1 ( x2 ( x3  x4 )  x2 )  x1x3 x4
Using the Theorem
for BDD synthesis:
x1 F ( X ) x 0
1
y
y
x1
x2 ( x3  x4 )  x2
x3 x4
x3
x4
x3  x4
1
xk
F(X ) x
0
x2
x3
1
x4
0
1
F(X ) x
k 0
k 1
Research in ATI
© Raimund Ubar
Functional Synthesis of BDDs
Shannon’s Expansion Theorem: y  F ( X )  xk F ( X ) xk 1  xk F ( X ) xk 0
y  x1 ( x2 ( x3  x4 )  x2 )  x1x3 x4
y
x1
x3
x2
1
y
x1
x3
x4
x2
x3
1
x4
0
x3
x4
0
1
Research in ATI
© Raimund Ubar
BDDs for Logic Gates
Elementary BDDs:
x1
x2
x3
y
&
AND
x1 1 x2
x3
x1
x2
x3
1
0
x1
x2
x3
0
Given circuit:
x1
x2
x21
1
y
1
x2
NOR
1
x1
y
x1
x2
x3
x3
a
&
x22
x3
OR
y
SSBDD synthesis:
SSBDDs for a given circuit are built
by superposition of BDDs for gates
1
b
17
Research in ATI
© Raimund Ubar
Synthesis of SSBDD for a Circuit
Superposition of BDDs:
Given circuit:
x1
x2
x21
1
&
x22
x3
y
a
a
b
y
b
y
a
x22
1
b
x22
x3
x3
Structurally
Synthesized
BDD:
b
Compare to
Superposition of Boolean functions:
a
y
y  a & b  ( x1  x21 )(x22  x3 )
a
x1
x22
x21
x3
x1
x21
18
© Raimund Ubar
Research in ATI
Logic Operation with BDDs
Generation of BDDs
for
y  x1 x2  x3
19
Research in ATI
© Raimund Ubar
Boolean Operations with SSBDDs
x1
x2
x21 &
c
g
1
x52
x6
y
b
x51 &
x1
x22
x21
x3
x4
x52
x51
x6
e
&
x3
y
y=eg
1
x22
x4
x5
AND-operation:
a
&
d
OR-operation:
y=eg
y
x1
x22
x4
x52
x21
x3
x51
x6
20
Research in ATI
© Raimund Ubar
Properties of SSBDDs
Boolean function:
y
x1
x2
x3
x4
x5
y = x1x2  x3 (x4  x5x6)
#1
y
x1
x2
x3
x4
x6
#0
1-nodes of a 1-path represent
a term in the DNF: x3x5x6 = 1
x5
#1
x6
#0
0-nodes of a 0-path represent
a term in the CNF: x1x4x5 = 0
21
Research in ATI
© Raimund Ubar
Boolean Operations with SSBDDs
Boolean function:
y = x1x2  x3
y
x1
x2
x3
Inverted function (DeMorgan):
y = x1x2  x3 = ( x1  x2) x3
y
x3
x1
x2
Dual function:
Inverted dual function:
y*= (x1  x2) x3
x1
x3
y*
y * = x1x2  x3
x2
x1
y*
x2
x3
22
Research in ATI
© Raimund Ubar
Transformation Rules for SSBDDs
SSBDD
BOOLEAN ALGEBRA
Exchange of nodes:
y
y
x1
x2
Commutative law:
=
x2
x1
Node passing:
y
Idempotent law:
y
x1
x2
x2
y = x1  x2= x2  x1
=
x1
x1
y = x 1  x1  x2 =
= x 1  x2
x2
23
Research in ATI
© Raimund Ubar
Properties of SSBDDs
Exchange of nodes:
Boolean function:
y
y = x1x2  x3 (x4  x5x6)
x2
y
x1
x2
=
x2
x1
#1
y
x3
y
x1
x2
x1
x3
x4
#1
x4
x5
x6
x6
x5
#0
#0
24
Research in ATI
© Raimund Ubar
Properties of SSBDDs
Exchange of subgraphs:
Boolean function:
y
y = x1x2  x3 (x4  x5x6)
y
G1
=
G2
y
x1
x2
G1
#1
y
x3
G2
x3
x4
x1
x2
#1
x4
x5
x6
x5
x6
#0
#0
25
Research in ATI
© Raimund Ubar
Transformation Rules for SSBDDs
SSBDD
BOOLEAN ALGEBRA
Node passing:
y
Absorption law:
y
x1
x1
x2
y = x1  x1x2 = x1
x1
=
x1
x2
Distributive law:
y
x1
y
x2
x1
x2
x1
x3
=
x1
x3
y = x1x2  x1x3=
= x1(x2  x3)
26
Research in ATI
© Raimund Ubar
Transformation of SSBDDs to BDDs
y
SSBDD:
y
x11
x21
x12
x31
x13
x22
x1
x2
x4
BDD:
x2
x3
y
x1
x2
x4
x12
x31
x32
x13
x22
y
x1
x2
x4
x12
x3
x4
x32
x13
x22
x32
y
x3
x1
Optimized
BDD:
x2
x4
x3
x2
27
Research in ATI
© Raimund Ubar
Transformation Rules for SSBDDs
SSBDD
BOOLEAN ALGEBRA
Superposition:
y
x1
y
x2
x3
z
Assotiative law:
x3
x2
z
z
y
x1
=
=
x3
z
x1
y = x1  (x2  x3) =
= (x1  x2)  x3
x2
28
Research in ATI
© Raimund Ubar
Properties of SSBDDs
Graph related properties:

SSBDD is

planar

asyclic

traceable (Hamiltonian path)

for every internal node there
exists a 1-path and 0-path

homogenous
y
x11
x21 1
1
0
0
x12
x31
x4
x13
0
x22
1
1
x32
1
Research in ATI
© Raimund Ubar
BDDs for Flip-Flops
Elementary BDDs
S
J
C
D Flip-Flop
q
D
C
c
D
JK Flip-Flop
q
K
R
q’
S
R
RS Flip-Flop
S
C
R
q
c
S
q’
R
q  c( S  q' R)  cq'
SR  0
c
q’
q’
J
K
R
q’
U
U - unknown value
30
Research in ATI
© Raimund Ubar
Mapping Between Circuit and SSBDD
From circuit to set of SSBDDs
Fan-out stems
FFR
C
D
FFR
Y1
SSBDD2
SSBDD4
FFR
A
B
C
FFR
SSBDD3
FFR
SSBDD1
Y2
SSBDD5
E
Y
a
b
a
c
31
Research in ATI
© Raimund Ubar
Advantages of SSBDDs

Linear complexity: a circuit is represented as a system
of SSBDDs, where each fanout-free region (FFR) is
representred by a separate SSBDD

One-to-one correspondence
between the nodes in SSBDDs
and signal paths in the circuit

This allows easily to extend the
logic simulation with SSBDDs to
simulation of faults on signal
paths
32
Research in ATI
© Raimund Ubar
Shared SSBDDs - S3BDD

Extension of superposition
procedure beyond the fanout nodes of the circuit

Merging several functions
in the same graphs by
introducing multiple roots
Superpositioning of FFRs
Node of SSBDD  signal path up to fan-out stem
Input of SSBDD  circuit down to primary inputs
33
Research in ATI
© Raimund Ubar
S3BDDs and Fault Collapsing
19
7
3
4
&
11
1
2
1
&
&
14
12
10
6
17
15
13
8
1
&
1
&
11
20
17
7
18
15
12
10
16
1
1
5
14
1
&
1
16
9
19
17
14
20 11
15
12
3
1
Fault collapsing:
1
2
21
&
5
4
21
18
13
18
9
16
13
8
6
10
From 84 faults to 36 faults
For SSBDD: 50 faults
34
Research in ATI
© Raimund Ubar
Shared SSBDDs
7
Each node represents different paths
(path segments) in the circuit
3
4
&
1
11
&
14
1
SSSBDD for 19
Node
Path
14
140-19 (3)
11
110-19 (4)
7
7-19 (4)
15
150-170-19 (3)
12
120-141-170-19 (4)
3
3-111-141-170-19 (5)
4
4-111-141-170-19 (5)
5
1
2
1
&
10
14
11
15
&
1
16
9
19
1
20
1
21
17
15
13
8
7
17
14
11
&
1
&
6
19
12
1
18
&
12
3
4
35
Research in ATI
© Raimund Ubar
Synthesis of S3BDD for a Circuit
Superposition of BDDs:
Given circuit C17
Y1
Y1
Y1
G1
3
G1
X1
11
3
3
Each node in the SSMIBDD
represents a signal path in the
circuit
Testing a node in SSMIBDD
means testing a signal path in
the circuit
Y1
X1
X3
3
X2
21
Y1
X1
X3
3
X2
21
21
X2
2
3
11
X1
12
X4
11
X1
Y1
X1
X3
3
X2
X3
2
1
X4
Research in ATI
© Raimund Ubar
Synthesis of S3BDD for a Circuit
Given circuit C17
Superposition of BDDs:
Y2
3
Y2
G4
Two-output circuit
is represented
by a single SSBDD
with shared
subgraphs
Y1
X1
X3
3
X2
X3
G4
Y2
2
3
Y1
X1
X3
3
X2
X4
3
X4
Y2
X5
2
X5
2
X5
3
2
X3
X5
37
Research in ATI
© Raimund Ubar
3
Sequentional S3BDDs
T1
10
2
7
15
1
9
1
10 nodes,
20 stuck-at faults,
2,5 times less
12
&
1
8
13
17
&
18
1
&
16
&
T1
15
1
T1
9
8 13
25 nodes,
50 stuck-at faults
8
&
11
9
&
17 1
T3
T2
18
5
14
1
&
6
16
14
4
1
19
8
T3
T2
9
4
14
T3
26
1
9
T2
T1
T1
10
7
2
3
26
T3
19
2
1
23
24
3
T1
1
21
20
1
4
T2
5
14
6
1
22
11
25
20
12
22 1
21 23
24
&
T3
25
14
T2
26
T3
1
8
T2
GAIN:
2,5 times
in logic simulation,
2,5 2 = 6,25 times
in fault simulation
38
Research in ATI
© Raimund Ubar
Structured Interpretation of S3BDDs
3
10
2
T1
1
1
7
G
GT1
9
Nodes
3
2
T1
15
T1
T1
T1
3
9
2
T1
S3BDD represents two
subcircuits
Signal paths
3 – 15 – T1
2 – 9 – 10 – 15 – T1
T1 – 7 – 9 – 10 – 15 – T1
L
3
5
6
Each node in the S3BDD represents a
signal path in the circuit
39
Research in ATI
© Raimund Ubar
Structured Interpretation of S3BDDs
8
11
1
8 13
&
9
&
17 1
18
5
14
1
&
6
T3
T2
16
14
4
G
GT2
G26
1
20
12
22 1 25 T2
21 23
24
&
T3
T2
26
T3
19
Nodes
8
T2
9
14
4
T3
1
Signal paths
8 – 12 – 25 – T2
T2 – 5 – 21 – 23 – 26
9 – 11 – 18 – 20 – 21 – 23 – 26
14 – 17 – 18 – 20 – 21 – 23 – 26
4 – 19 – 20 – 21 – 23 – 26
T3 – 6 – 14 – 16 – 19 – 20 – 21 – 23 – 26
1 – 8 – 13 – 14 – 16 – 19 – 20 – 21 – 23 – 26
L
4
5
7
7
6
9
10
40
Research in ATI
© Raimund Ubar
BDDs and Complexity
Optimization (by ordering of nodes): BDDs for a 2-level AND-OR circuit
BDD optimization:
We start synthesis:
• from the most
repeated variable
2n nodes
22n - 2 nodes
Research in ATI
© Raimund Ubar
BDDs and Complexity
BDDs for an 8-bit data selector
Research in ATI
© Raimund Ubar
BDDs and Complexity
Elementary BDDs
S
J
C
D Flip-Flop
q
D
C
c
D
JK Flip-Flop
q
K
R
q’
R
RS Flip-Flop
S
C
R
q
c
S
q’
R
c
q’
q’
J
K
R
q’
U
q  c( S  q' R)  cq'
SR  0
S
U - unknown value
BDD optimization:
We start synthesis:
• from the most
important variable, or
• from the most
repeated variable
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