Static and Dynamic Fault Diagnosis

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Static and Dynamic Fault Diagnosis

• Richard Beigel
• Univ. Illinois at Chicago
• and DIMACS

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Nonstandard computing architectures

• Perceptrons and small-depth circuits
• Optically interconnected multiprocessors
• DNA computing

Self-diagnosing Systems
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Brief history of system-level fault diagnosis

• Preparata et al 67
• static, nonadaptive
• Nakajima 81
• static, adaptive, serial
• Hakimi Nakajima 84
• static, adaptive, parallel

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Recent advances in system-level diagnosis

• Distributed diagnosis
• Diagnosing intermittent faults
• Diagnosis with errors
• Fast parallel diagnosis of static faults
• Ongoing diagnosis and repair of dynamic faults

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Fault diagnosis problem

• Given n processors
• a primitive by which each processor can test any
  other
• a reliable external controller that observes test
  results
• Determine which are good and which are faulty
• Assume perfect communication in a complete network

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Whats so hard about that?
Say Ah
Ha Ha!
OK, you pass
Faulty processors may give incorrect test results
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Possible test results
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A majority of processors must be goodfor
diagnosis to be possible
Were all good Theyre all faulty
Were all good Theyre all faulty
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Serial diagnosis of static faults

• n processors, at most t faults, t lt n/2
• Nonadaptive diagnosis
• n(t1) tests are necessary and sufficient
• Preparata et al 67
• Adaptive diagnosis
• nt-1 tests are necessary and sufficient
• Nakajima 81

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Distributed diagnosis of static faults

• In the distributed diagnosis model there is no
  central controller, and all good processors must
  learn the status of the other processors.
• Distributed diagnosis is reducible to the
  cooperative collect problem, and can be solved
  with tests Aspnes-Hurwood 96

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INTERMITTENT FAULTS AND ERRORS

• Work in progress by Beigel and Fu

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Intermittent faults

• An intermittent fault may appear faulty in some
  tests and good in others
• We cannot hope to diagnose intermittent faults as
  such because they might exhibit consistent
  behavior in all tests
• Goal correctly diagnose all other processors

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Errors

• An error is a misdiagnosis by a good processor.
• Note the similarity to an intermittent fault

faulty
good
good
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Results

• In rounds, we can perform static diagnosis
  assuming that a majority of the processors are
  good and at most t of them are intermittently
  faulty.
• In rounds, we can perform static diagnosis in
  the presence of errors. Assuming at most t
  errors per round, the results will be within
  of a correct diagnosis.

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PARALLEL DIAGNOSIS OF STATIC FAULTS

• Perform many tests simultaneously

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Parallel diagnosis of static faults

• 84 Hakimi Schmeichel O(n/logn)
• 90 S H Otsuka Sullivan O(logn)
• 89 Beigel Kosaraju Sullivan O(1)
• 93 Beigel Margulis Spielman 32
• 94 Beigel Hurwood Kahale 10
• best lower bound 5

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Digraphs

• tester testee
• testing round directed matching

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SHOS 90 generates a large mutual admiration
society

• MAS strongly connected component with all good
  edges
• Either
• all nodes good, or
• all nodes faulty

g
g
g
g
g
g
g
g
g
g
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SHOS 90O(logn) pairing algorithm

• Pair up processors

• Pair up pairs

• Pair up fours

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What about processors that dont like each other?

• Build one chain for each good processor we found
  (4 rounds)
• Most chains must have a good processor in each
  level (count!)
• Total 4 1 rounds

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Beigel-Margulis-Spielman 94

• non (32 rounds)
• Find several MASs of size including
  at least one good MAS
• Large MASs test each other and all remaining
  processors in 4 rounds

• constructive (84 rounds)
• Find several MASs of size including
  at least one good MAS
• Large MASs test each other and all remaining
  processors in 6 rounds

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Expander graphs guarantee a good big MAS

• In the Cayley graphs of Margulis and LPS with
  p37, every n/2-node induced subgraph contains a
  strong component of size
• (cf Alon Chung 88, who find long paths)
• degree of undirected graph 38
• 78 directed matchings cover graph
• 78 6 84 rounds

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Random graphs guarantee a good big MAS

• If G consists of 14 directed Hamiltonian paths on
  n vertices then, whp, every n/2-node induced
  subgraph contains a strong component of size
• 28 directed matchings cover graph
• 28 4 32 rounds

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Beigel-Hurwood-Kahale 95 speeds up BMS 94

• In k1 rounds build MASs of size
• also build one chain of dont-likes
• each MAS can be in simultaneous tests
• Perform Gs directed matchings in 1 round
• Process chain in 2 or 3 more rounds
• Constructive 13 rounds. Non 10 rounds.

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Lower boundUpper bound for smaller t

• n processors, at most t faults
• If 5 rounds are necessary
• If 4 rounds suffice
• algorithm uses lower-degree expanders

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DIAGNOSIS AND REPAIR OF DYNAMIC FAULTS

• Processors fail each round,
• but algorithm may order repairs

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Ongoing diagnosis and repair of dynamic faults

• Processors may fail each round, but algorithm may
  order repairs
• In each round
• 1. perform tests
• 2. direct that up to t processors are repaired
• 3. at most t processors fail
• Goal bound number of faults at all times

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Results for n processorsat most t failures per
round

• When t gt 70 and n gt 376tlogt 50t, we can
  maintain n - 64tlogt - 10t good processors at all
  times
• This works even if the number of faults exceeds
  n/2
• When n 640 and t 1, we can maintain 520 good
  processors at all times.

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Whys this hard?

• We cant determine the status of a chosen
  processor because its testers might fail right
  before we choose them

• Mutual admiration societies dont work either

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SIFT and WINNOW

• SIFT finds a large set G consisting of processors
  that were good when SIFT started running, and a
  small set F containing some faulty processors
• WINNOW uses G to diagnose most of the faulty
  processors in F
• Algorithm SIFT, WINNOW, repair, repeat

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SIFT algorithm

• Let r 2logt
• In 2r rounds form undirected hypercubes of size
• Put MASs into G, others into F
• MASs must have been entirely good at start of
  SIFT, and are still mostly good

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WINNOW algorithm

• Choose a processor P in F
• For 2logt rounds,
• test P and every processor that has tested P so
  far, using testers in G
• If the tests always call P faulty but dont call
  any of the others faulty then we can be sure that
  P really is faulty
• Most old faults are diagnosed, but 4tlogt new
  ones could accumulate.

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Summary

• We have efficient algorithms for
• diagnosis in the presence of a small number of
  intermittent faults
• diagnosis with a small number of diagnosis errors
• parallel fault diagnosis
• ongoing diagnosis of dynamic faults