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., UCLA2Electrical Engineering Dept., UCLA

3Electrical Engineering Dept., University of Alberta, Edmonton Canada

Outline

RALF Overview Circuit Representation Algorithms Experimental results

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

Applications for RALF

Circuit optimization for reliability

Random pattern-resistant fault identification to enhance testability

Optimality studies of approximate algorithms

RALF System

CNF

Miter

Circuit

d-DNNF

Fault Rate

Exact symbolic algorithm Compiles miter to d-DNNF (similar to BDD) Computes criticality in one pass over d-DNNF

Miter-Based Calculation

Criticality of G: Fraction of assignments to primary inputs Xi for which O = 1

Compiled Circuit Representation

Deterministic Decomposable Negation Normal Form (d-DNNF)

A subset of NNF which satisfies Decomposability Determinism

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: http://reasoning.cs.ucla.edu/c2d

A B B A C D D C

and and and and and and and and

or or or or

and and

or

Negation Normal Form

Decomposability

A B B A C D D C

and and and and and and and and

or or or or

and andor

A,B C,D

No two children of AND share a variable

Determinism

A B B A C D D C

and and and and and and and and

or or or or

and andor

No two children of OR share a satisfying assignment

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

Evaluating d-DNNF

L

AND

OR

Pr(L) (e.g. 0.5 for a uniform distribution over the inputs)

Pr(AND) = Pr(α) * Pr(β)(Requires Decomposability)α β

Pr(OR) = Pr(α) + Pr(β)(Requires Determinism)α β

L 1 - Pr(L)

Tractability

Circuit Characteristics Performance

Name Gates Inputs d-DNNFNodes

BDD Nodes

Compile Time

Total Run Time

i7 581 199 79637 - 138.24mult32a 535 34 166976 - 118.85

i6 455 138 67295 895599 58.31i5 402 133 507965 - 59.99b9 296 41 725729 - 67.97i4 292 192 20231 - 3.78

my_adder

259 33 29865 116621 10.03

cht 244 47 175055208494

88.55

i2 238 201 103434 398017 4.84lal 234 26 523283 - 67.77

Random-Pattern Resistant Faults

Logic Masking

Fault

Detection of Random-Pattern Resistant Faults

Fidelity of Monte Carlo Simulation

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

Thank you