Local Tolerance to Unbounded Byzantine Faults

1. Mikhail Nesterenko Kent State University Anish Arora Ohio State University Local Tolerance to Unbounded Byzantine Faults

2. faulty affected • localize tolerance to unbounded complex faults Tolerating Faults in System of Large Scale • large system size presents uniquechallenges to ensuring dependability: • faults occur often • multiple regions can be affected by faults • faults may interact unpredictably • faults can be spatially/temporally unbounded & complex • how to tolerate such faults?

3. 2 3 4 5 1 sink Execution model & Example problems • execution model • asynchronous • interleaving • communication via shared registers • examples • graph coloring – color (assign numbers) vertices of a graph so that colors of adjacent onse do not match • if graph has degree d, can always color in d+1 colors • routing – assign parent to each process such that there is a path from each process to the sink (destination)

4. Outline • fault containment & tolerance • strict fault containment • strict fault tolerance • strict stabilization • examples of strictly fault tolerant programs • graph coloring • dining philosophers • routing • limits of strict fault containment • critique and further directions

5. Spatial Fault Hierarchy • bounded faults – processes outside certain locality of a fault perform correctly (according to specification) • unbounded faults – process performs correctly in spite of faults outside its locality • unbounded Byzantine faults - each process behaves correctly regardless of actions outside its locality if a program is tolerant to unbounded Byzantine faults, it is also tolerant to bounded and unbounded faults of any fault class

6. Containment of Unbounded Faults • What does it mean for an individual process to perform correctly? • Proposition 4. P is strictly fault containing if there exists a constant l such that for each process p there exists and invariant I.p which is closed with respect to Byzantine actions of processes whose distance to p is greater than l • what is the form of this invariant? • can it include variables outside locality? • can you always come up with an invariant of this form?

7. Tolerance Inside Locality • What if faults occur inside the containment locality? • can achieve additional tolerance • two process specifications • ideal (no faults) • tolerant (faults of some class present) • example – safety is never violated • which spec do processes outside fault locality satisfy?

8. Strict Stabilization • stabilization – special case of tolerantspec – eventual satisfaction of ideal spec when (transient) faults stop occurring • strict stabilization – process p eventuallysatisfies ideal spec regardless of behaviorof processes outside its locality • what is the difference between traditional stabilization and strict stabilization? • is strict containment required for strict stabilization? • more formally:

9. Vertex Coloring Program (PVC) • Lemma 2. when node has a neighbor with matchingcolor it can select a new color without affecting any of its neighbors • Invariant: • Theorem 1.PVC is strictly fault-containing and strictly stabilizing(with locality of 1) Byzantine node nodes that may recolor following Byzantine

10. cycle of requesting process thinking(T) hungry (H) eating (E) Dining Philosophers Problem (DP) [D72] • graph of processes, each may request to eat • properties • no two neighbors eat together • each requesting process eats eventually

11. T T T a b c executes T T H a b c T E T a b c T T T a b c E E T a b c DP: Fault-Free Operation [CM84] actions: • if thinking, needs to eat & all parents thinking become hungry • if hungry & no neighbors eating  eat • when finished  think & become child of each neighbor b eats &gives upprivilege a & c eat

12. T H E E T H H H T T T E Dining Philosophers Program (PDP) • a hungry faulty process may block immediate thinking neighbors • an eating faulty process may block hungry neighbors and their thinking neighbors

13. Dining Philosophers Program (PDP) Lemma 4. non-Byzantine eating process eventually thinks Lemma 5. a hungry process whose immediate neighborhood is not Byzantine eventually eats Lemma 6. If a Byzantine process is at least 2 hops away a thinking process eventually becomes hungry Invariant Theorem 2.PDP is strictly fault-containing and strictly stabilizing(with locality of 2)

14. Limits of Containment s1ands2differ in values of a process at least r away from p s1 σ is in p’s spec s2 • Theorem 3. the containment radius of a solution to an r-restrictive problem is at least r • graph coloring and dining-philosophers are 1-restrictive • routing is restrictive for arbitrary r

15. Critique and Further Research • interesting and useful examples of strict containment • geometric spanners, spanners of fixed degree • low-atomicity dining-philosophers • ?? • better bounds on containment • r-restriction is obvious but too crude a bound for containment • some non-containing problems appear “almost” the same as containing • example: • maximal independent set – 1-containing • maximal independent set with distance of at most 2 – not containing for any l