1. RALF: Reliability Analysis for Logic Faults – An Exact Algorithm and its Applications
Samuel Luckenbill1, Ju-Yueh Lee2, Yu Hu3,
Rupak Majumdar1, and Lei He2
1Computer Science Dept., UCLA
2Electrical Engineering Dept., UCLA
3Electrical Engineering Dept., University of Alberta, Edmonton Canada

2. Outline RALF Overview Circuit Representation Algorithms
Experimental results

3. RALF Features
Single-gate criticality: The probability that a flipped bit at one gate will affect the output
Full-chip fault rate: The average criticality over all gates in a circuit

4. Applications for RALF Circuit optimization for reliability
Random pattern-resistant fault identification to enhance testability
Optimality studies of approximate algorithms

5. RALF System Exact symbolic algorithm
Compiles miter to d-DNNF (similar to BDD)
Computes criticality in one pass over d-DNNF
Circuit
Miter
CNF
d-DNNF
Fault Rate

6. Miter-Based Calculation
Criticality of G: Fraction of assignments to primary inputs Xi for which
O = 1

7. Compiled Circuit Representation
Deterministic Decomposable Negation Normal Form (d-DNNF)
A subset of NNF which satisfies
Decomposability
Determinism

8. Why d-DNNF? Almost as powerful as BDD
Polytime operations include SAT, model counting, and model enumeration
Usually more concise than BDD and faster to compile
Determinism and decomposability make the criticality computation efficient
Compiler:

9. Negation Normal Form A B  B A C  D D  C and or
A circuit representation where all of the inverters have been pushed to the inputs and all internal nodes are either AND or OR.

10. Decomposability or and and or or or or and and and and and and and and
No two children of AND share a variable or
and
and
A,B
C,D or or or or
Decomposability only applies to AND nodes. An AND node is decomp. If its children do not mention the same variable. An NNF is decomposable if every AND node in the graph is decomposable.
and
and
and
and
and
and
and
and A B
 B A C
 D D
 C

11. Determinism or and and or or or or and and and and and and and and A
No two children of OR share a satisfying assignment or
and
and or or or or
Determinism on the other hand is a property of OR nodes. An OR node is deterministic if its children do not share a model and we say that an NNF is deterministic if every OR node in the graph is deterministic.
and
and
and
and
and
and
and
and A B
 B A C
 D D
 C

12. Criticality Algorithm
d-DNNF is a representation of the miter
Invariant: at each node, we compute the probability that the circuit below it evaluates to 1
Value computed at root is the criticality of the faulty node in the miter
Computation is linear in d-DNNF size

13. Evaluating d-DNNF α α β β Pr(L) 1 - Pr(L)
(e.g. 0.5 for a uniform distribution over the inputs) L
1 - Pr(L) L
The value of a leaf node is simply the probability that the primary input it represents is set to true
The value of an AND node is the product of its children’s values
The value of an OR node is the sum of its children’s values
AND
Pr(AND) = Pr(α) * Pr(β)(Requires Decomposability) α β OR
Pr(OR) = Pr(α) + Pr(β)
(Requires Determinism) α β

14. Circuit Characteristics
Tractability
Circuit Characteristics
Performance
Name
Gates
Inputs
d-DNNF
Nodes
BDD Nodes
Compile Time
Total Run Time i7
581
199
79637 -
138.24
mult32a
535 34
166976
118.85 i6
455
138
67295
895599
58.31 i5
402
133
507965
59.99 b9
296 41
725729
67.97 i4
292
192
20231
3.78
my_adder
259 33
29865
116621
10.03
cht
244 47
175055
8.55 i2
238
201
103434
398017
4.84
lal
234 26
523283
67.77

15. Random-Pattern Resistant Faults
Logic Masking

16. Detection of Random-Pattern Resistant Faults

17. Fidelity of Monte Carlo Simulation

18. Conclusion
RALF performs surprisingly well on MCNC circuits, despite being an exact algorithm
RALF uses d-DNNF, a less powerful but usually more succinct circuit representation than BDD.
For criticality and fault-rate computation, Monte Carlo simulation is good enough for most circuits

19. Thank you