Hierarchical Fault Collapsing; Functional Equivalences and Dominances

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Hierarchical Fault Collapsing Functional
Equivalences and Dominances

• Vishwani D. Agrawal
• Rutgers University, Dept. of ECE
• vishwani02_at_yahoo.com
• http//cm.bell-labs.com/cm/cs/who/va
• va_at_agere.com
• mvatre_at_agere.com

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Test Vector Generation Flow

• DUT
• Generate fault list
• Collapse fault list
• Generate test vectors

Fault Model
Required fault coverage
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Background

• Single stuck-at fault model is the most popularly
  used model.
• Two faults f1 and f2 are equivalent if the same
  tests detect f1 and f2 (f1f2)
• If all tests of fault f1 also detect fault f2,
  then f2 is said to dominate f1 (f1?f2).

a0 a1
c0 c1
b0 b1
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Background

• Both equivalence and dominance relations are
  transitive in nature.
• (f1 ? f2) and (f2 ? f3) gt (f1 ? f3)
• If f1 dominates f2 and f2 dominates f1 then f1
  and f2 are equivalent.
• (f1 ? f2) and (f2 ? f1) gt (f1 f2)
• Number of faults in a 2-input AND gate reduces
  from 6 to 4 (by equivalence) and to 3 (by
  dominance) collapsing.
• Example c6288, faults 12576
  equ. 7744 (0.62), dom. 5824 (0.46)

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Problem Statement

• To devise a new method for fault collapsing with
  following attributes
• A single procedure for equivalence and dominance
• Global analysis (independence from direction, and
  other choices, in collapsing)
• Use functional equivalences and dominances
• Hierarchical fault collapsing (collapsing in
  large circuits using pre-collapsed sub networks)

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A New Dominance Graph Model

• A fault in the circuit is represented by a node
  in the graph.
• A directed edge from f2 to f1 indicates that f1
  dominates f2 (f2 ?f1).
• Edges can represent either structural or
  functional relations.

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Computational Model

• Graph is represented as a connectivity matrix
• Each fault is assumed to be equivalent to itself
• Treats functional and structural relations
  identically

• (f1 ? f2) and (f2 ?f1) gt f2 f1. Appear as
  symmetrical components in the matrix (e.g.,
  a0,b0,c0)
• faults 6 (dimension of dominance matrix)

2-input AND gate
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Transitive Closure

• Transitive closure (TC) of the dominance matrix
  gives all dominance relations between faults.
• TC is computed by the O(n3) Floyd-Warshall
  algorithm, where n is the dimension of the
  dominance matrix.

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Transitive Closure

• (F1 ? F2) and (F2 ? F3) gt (F1 ? F3)

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Example
A
D
E
B
C
Dominance Graph
A0
A1
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Finding Functional Equivalences
f1
f0
Always 0
f2
f1
Always 0
f2
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XOR Circuit
c1
h1
g1
m0
g0
i1
f1
Functional Equivalences (c1,f1), (g1,h1,i1),
(g0,m0)
Also (d1,f0) and (e1,c0) not used here
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Dominance matrix (XOR)
(24x24)
Functional equivalences shown as boxed entries
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Transitive Closure (XOR)
j0 k0 m1 f1 f0c1 a0
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Results for XOR Circuit
faults Eq. Faults Dom. faults
24 16 13
 
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Design Hierarchy

• Large designs are modular and hierarchical.
• Advantageous to store the fault information of
  repeated blocks in a library.
• When configured as a library cell the fault list
  includes cell PI PO faults for transitivity.

Top module
B1
B1
B0
C0
C0
C0
C0
C1
C1
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XOR Library Cell

• Useful for hierarchical fault collapsing
• Dimension of the matrix 14

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8-bit Ripple Carry Adder (RCA)
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Fault Collapsing in 8-bit RCAUsing Functional
Equivalences
Number of collapsed faults Number of collapsed faults Number of collapsed faults Number of collapsed faults
Flat structural only Flat structural only Hierarchical with functional Hierarchical with functional
Equ. Dom. Equ. Dom.
xor cell 24 16(0.63) 13(0.54) 12(0.50) 10(0.42)
Full-adder 60 38(0.63) 30(0.50) 30(0.50) 24(0.40)
8-bit adder 466 290(0.62) 226(0.49) 226(0.49) 178(0.38)
Circuit name
All faults
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ISCAS85 Circuits
Circuit name Total faults Equivalence fault set size Equivalence fault set size Dominance fault set size Dominance fault set size
Circuit name Total faults Graph method Other programs Graph method Fastest
C17 34 22 22 16 16
C432 864 524 524 449 449
C432exp 1044 560 632 449 503
C499 998 758 758 706 706
C499exp 2710 1158 1574 898 1210
C1355 2710 1574 1574 1210 1210
C1908 3816 1879 1879 1566 1566
C2670 5276 2747 2747 2317 2318
C3540 7080 3428 3428 2786 2794
C5315 10630 5350 5350 4492 4500
C6288 12576 7744 7744 5824 5824
C7552 15012 7550 7550 6132 6134
Fastest, Gentest, Hitec, TetraMax
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Finding Dominances
f1
Always 0
f0
f2
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Fault Collapsing in 8-bit RCAUsing Functional
Dominances
Number of collapsed faults Number of collapsed faults Number of collapsed faults Number of collapsed faults
Flat structural only Flat structural only Hierarchical with functional Hierarchical with functional
Equ. Dom. Equ. Dom.
xor cell 24 16(0.63) 13(0.54) 10(0.41) 4(0.17)
Full-adder 60 38(0.63) 30(0.50) 26(0.43) 14(0.23)
8-bit adder 466 290(0.62) 226(0.49) 194(0.42) 112(0.24)
Circuit name
All faults
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Conclusion

• A new algorithm for global fault collapsing
• With functional equivalence number of faults for
  ATPG reduces considerably
• Further reduction with functional dominances
  (Caution fault coverage not correct when
  redundant faults are present)
• Library based hierarchical fault collapsing is a
  new concept
• Further studies are being carried out on
  independent fault sets
• Reference Prasad et al., ITC-02, pp. 391-397