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Parametrized Model Checking of Fault Tolerant Distributed Algorithms by Abstraction (part 1)

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Igor Konnov, Vienna University of Technology

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Parametrized Model Checking of Fault Tolerant Distributed Algorithms by Abstraction (part 1)

  1. 1. Model Checking of Fault-Tolerant Distributed Algorithms Part III: Parameterized Model Checking of Fault-tolerant Distributed Algorithms by Abstraction Annu Gmeiner Igor Konnov Ulrich Schmid Helmut Veith Josef Widder TMPA 2014, Kostroma, Russia Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 1 / 1
  2. 2. Fault-tolerant DAs: Model Checking Challenges unbounded data types counting how many messages have been received parameterization in multiple parameters among n processes f ≤ t are faulty with n > 3t contrast to concurrent programs fault tolerance against adverse environments degrees of concurrency many degrees of partial synchrony continuous time fault-tolerant clock synchronization Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 2 / 1
  3. 3. Distributed algorithms: computational model and faults In previous parts, we considered algorithms operating in the classic model by [Fischer, Lynch, Paterson’85] Environment: Asynchronous processes (interleaving semantics) Reliable asynchronous message passing (non-blocking send and receive) Faults: crashes and clean crashes, omission faults, symmetric faults, Byzantine faults Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 3 / 1
  4. 4. Model checking problem for fault-tolerant DA algorithms Parameterized model checking problem: given a distributed algorithm and spec. ϕ show for all n, t, and f satisfying n > 3t ∧ t ≥ f ≥ 0 M(n, t, f ) |= ϕ every M(n, t, f ) is a system of n − f correct processes n ? ? ? t n ? ? ? t f Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 4 / 1
  5. 5. Model checking problem for fault-tolerant DA algorithms Parameterized model checking problem: given a distributed algorithm and spec. ϕ show for all n, t, and f satisfying resilience condition M(n, t, f ) |= ϕ every M(n, t, f ) is a system of N(n, f ) correct processes n ? ? ? t n ? ? ? t f Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 4 / 1
  6. 6. Properties in Linear Temporal Logic Unforgeability (U). If vi = 0 for all correct processes i, then for all correct processes j, acceptj remains 0 forever. G n−f i=1 vi = 0 → G n−f j=1 acceptj = 0 Completeness (C). If vi = 1 for all correct processes i, then there is a correct process j that eventually sets acceptj to 1. G n−f i=1 vi = 1 → F n−f j=1 acceptj = 1 Relay (R). If a correct process i sets accepti to 1, then eventually all correct processes j set acceptj to 1. G n−f i=1 accepti = 1 → F n−f j=1 acceptj = 1 Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 5 / 1
  7. 7. Properties in Linear Temporal Logic Unforgeability (U). If vi = 0 for all correct processes i, then for all correct processes j, acceptj remains 0 forever. G n−f i=1 vi = 0 → G n−f j=1 acceptj = 0 Safety Completeness (C). If vi = 1 for all correct processes i, then there is a correct process j that eventually sets acceptj to 1. G n−f i=1 vi = 1 → F n−f j=1 acceptj = 1 Liveness Relay (R). If a correct process i sets accepti to 1, then eventually all correct processes j set acceptj to 1. G n−f i=1 accepti = 1 → F n−f j=1 acceptj = 1 Liveness Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 5 / 1
  8. 8. Threshold-guarded fault-tolerant distributed algorithms Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 6 / 1
  9. 9. Threshold-guarded FTDAs Fault-free construct: quantified guards (t=f=0) Existential Guard if received m from some process then ... Universal Guard if received m from all processes then ... These guards allow one to treat the processes in a parameterized way Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 7 / 1
  10. 10. Threshold-guarded FTDAs Fault-free construct: quantified guards (t=f=0) Existential Guard if received m from some process then ... Universal Guard if received m from all processes then ... These guards allow one to treat the processes in a parameterized way what if faults might occur? Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 7 / 1
  11. 11. Threshold-guarded FTDAs Fault-free construct: quantified guards (t=f=0) Existential Guard if received m from some process then ... Universal Guard if received m from all processes then ... These guards allow one to treat the processes in a parameterized way what if faults might occur? Fault-Tolerant Algorithms: n processes, at most t are Byzantine Threshold Guard if received m from n − t processes then ... (the processes cannot refer to f!) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 7 / 1
  12. 12. Threshold-based fault-tolerant distributed algorithms The parameters (n, t, f ) are fixed in each run Main loop with the body executed atomically Processes are anonymous (no identifiers) Receiving messages, counting them and comparing to thresholds, e.g., if received <ECHO> from t + 1 distinct processes then ... Sending messages to all processes, e.g., send <ECHO> to all Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 8 / 1
  13. 13. Control Flow Automata Variables of process i vi : {0 , 1} init with 0 or 1 accepti : {0 , 1} init with 0 An indivisible step: i f vi = 1 then send ( echo ) to all ; i f received (echo) from at l e a s t t + 1 distinct processes and not sent ( echo ) before then send ( echo ) to all ; i f received ( echo ) from at l e a s t n - t distinct processes then accepti := 1; n − f copies of the process qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 q8 qF nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := SE sv := AC Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 9 / 1
  14. 14. Counting argument in threshold-guarded algorithms n t f if received m from t + 1 processes then ... t + 1 Correct processes count distinct incoming messages Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 10 / 1
  15. 15. Counting argument in threshold-guarded algorithms n t f if received m from t + 1 processes then ... t + 1 Correct processes count distinct incoming messages Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 10 / 1
  16. 16. Counting argument in threshold-guarded algorithms n t f if received m from t + 1 processes then ... t + 1 at least one non-faulty sent the message Correct processes count distinct incoming messages Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 10 / 1
  17. 17. qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 q8 qF nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := SE sv := AC concrete values are not important thresholds are essential: 0, 1, t + 1, n − t Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 11 / 1
  18. 18. qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 q8 qF nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := SE sv := AC concrete values are not important thresholds are essential: 0, 1, t + 1, n − t intervals with symbolic boundaries: I0 = [0, 1) I1 = [1, t + 1) It+1 = [t + 1, n − t) In−t = [n − t, ∞) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 11 / 1
  19. 19. qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 q8 qF nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := SE sv := AC concrete values are not important thresholds are essential: 0, 1, t + 1, n − t intervals with symbolic boundaries: I0 = [0, 1) I1 = [1, t + 1) It+1 = [t + 1, n − t) In−t = [n − t, ∞) Parameteric Interval Abstraction (PIA) Similar to interval abstraction: [t + 1, n − t) rather than [4, 10). Total order: 0 < 1 < t + 1 < n − t for all parameters satisfying RC: n > 3t, t ≥ f ≥ 0. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 11 / 1
  20. 20. Technical challenges We have to reduce the verification of an infinite number of instances where 1 the process code is parameterized 2 the number of processes is parameterized to one finite state model checking instance Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 12 / 1
  21. 21. Technical challenges We have to reduce the verification of an infinite number of instances where 1 the process code is parameterized 2 the number of processes is parameterized to one finite state model checking instance We do that by: 1 PIA data abstraction 2 PIA counter abstraction Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 12 / 1
  22. 22. Technical challenges We have to reduce the verification of an infinite number of instances where 1 the process code is parameterized 2 the number of processes is parameterized to one finite state model checking instance We do that by: 1 PIA data abstraction 2 PIA counter abstraction abstraction is an over approximation ⇒ possible abstract behavior that does not correspond to a concrete behavior. 3 Refining spurious counter-examples Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 12 / 1
  23. 23. Abstraction overview Parameterized family M(n, t, f ) = P(n, t, f ) · · · P(n, t, f ) N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} extract Parametric Interval Domain D parametric interval data abstraction Uniform parameterized family ˆM(n, t, f ) = ˆP · · · ˆP N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} P does not depend on n, t, f P simulates P(n, t, f ) change representation Counter representation parametric interval counter abstraction one abstract system A that simulates for every n, t, f the behavior of M(n, t, f ) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 13 / 1
  24. 24. Abstraction overview Parameterized family M(n, t, f ) = P(n, t, f ) · · · P(n, t, f ) N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} extract Parametric Interval Domain D parametric interval data abstraction Uniform parameterized family ˆM(n, t, f ) = ˆP · · · ˆP N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} P does not depend on n, t, f P simulates P(n, t, f ) change representation Counter representation parametric interval counter abstraction one abstract system A that simulates for every n, t, f the behavior of M(n, t, f ) finite-state model checkin replay the counter-example refine the system Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 13 / 1
  25. 25. Data abstraction Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 14 / 1
  26. 26. qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 q8 qF nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := SE sv := AC concrete values are not important thresholds are essential: 0, 1, t + 1, n − t intervals with symbolic boundaries: I0 = [0, 1) I1 = [1, t + 1) It+1 = [t + 1, n − t) In−t = [n − t, ∞) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 15 / 1
  27. 27. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1 It+1 In−t Concrete t + 1 ≤ x Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  28. 28. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  29. 29. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  30. 30. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, is abstracted as: x = I0 ∧ x = I1 . . . Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  31. 31. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, is abstracted as: x = I0 ∧ x = I1 ∨x = I1 ∧ (x = I1 ∨ x = It+1) . . . Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  32. 32. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, is abstracted as: x = I0 ∧ x = I1 ∨x = I1 ∧ (x = I1 ∨ x = It+1) ∨x = It+1 ∧ (x = It+1 ∨ x = In−t) . . . Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  33. 33. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1I0 I1 It+1 In−t Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, is abstracted as: x = I0 ∧ x = I1 ∨x = I1 ∧ (x = I1 ∨ x = It+1) ∨x = It+1 ∧ (x = It+1 ∨ x = In−t) ∨x = In−t ∧ x = In−t Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  34. 34. Abstract operations Concrete: Abstract: 0 1 t + 1 n − t above · · · I0 I1 Concrete t + 1 ≤ x is abstracted as x = It+1 ∨ x = In−t. Concrete x = x + 1, is abstracted as: x = I0 ∧ x = I1 ∨x = I1 ∧ (x = I1 ∨ x = It+1) ∨x = It+1 ∧ (x = It+1 ∨ x = In−t) ∨x = In−t ∧ x = In−t abstract increase may keep the same value! Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 16 / 1
  35. 35. Abstract CFA qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 17 / 1
  36. 36. Abstract CFA qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd = I0 ∧ nsnt = I0 ∧ (nrcvd = I0 ∨ nrcvd = I1) ∨ . . . ¬(t + 1 ≤ nrcvd) nrcvd = It+1 ∨ nrcvd = In−t sv = V0 ¬(sv = V0) nsnt = I1 ∧ (nsnt = I1 ∨ nsnt = It+1) ∨ . . . n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 17 / 1
  37. 37. Abstraction overview Parameterized family M(n, t, f ) = P(n, t, f ) · · · P(n, t, f ) N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} extract Parametric Interval Domain D parametric interval data abstraction Uniform parameterized family ˆM(n, t, f ) = ˆP · · · ˆP N(n,t,f ) processes : n > 3t, t ≥ f , f ≥ 0} P does not depend on n, t, f P simulates P(n, t, f ) change representation Counter representation parametric interval counter abstraction one abstract system A that simulates for every n, t, f the behavior of M(n, t, f ) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 18 / 1
  38. 38. Counter abstraction Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 19 / 1
  39. 39. Classic (0, 1, ∞)-counter abstraction Pnueli, Xu, and Zuck (2001) introduced (0, 1, ∞)-counter abstraction: finitely many local states, e.g., {N, T, C}. based on counter representation: for each local states count how many processes are in it Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 20 / 1
  40. 40. Classic (0, 1, ∞)-counter abstraction Pnueli, Xu, and Zuck (2001) introduced (0, 1, ∞)-counter abstraction: finitely many local states, e.g., {N, T, C}. based on counter representation: for each local states count how many processes are in it abstract the number of processes in every state, e.g., K : C → 0, T → 1, N → “many”. perfectly reflects mutual exclusion properties e.g., G (K(C) = 0 ∨ K(C) = 1). Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 20 / 1
  41. 41. Limits of (0, 1, ∞)-counter abstraction Our parametric data + counter abstraction: we require finer counting of processes: t + 1 processes in a specific state can force global progress, t processes cannot mapping t, t + 1, and n − t to “many” is too coarse. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 21 / 1
  42. 42. Limits of (0, 1, ∞)-counter abstraction Our parametric data + counter abstraction: we require finer counting of processes: t + 1 processes in a specific state can force global progress, t processes cannot mapping t, t + 1, and n − t to “many” is too coarse. starting point of our approach... Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 21 / 1
  43. 43. Data + counter abstraction over parametric intervals n = 6, t = 1, f = 1 t + 1 = 2, n − t = 5 nr. processes (counters) received received sent accepted • 0 • 0 • 1 • 1 • 2 • 2 • 3 • 3 • 4 • 4 • 5 • 5 • 6 • 6 •0 •1 •2 •3 •4 •5 •6 • Local state is (sv, nrcvd), where sv ∈ {sent, accepted} and 0 ≤ rcvd ≤ n 3 processes at (sent, received=3) 1 process at (accepted, received=5) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 22 / 1
  44. 44. Data + counter abstraction over parametric intervals n = 6, t = 1, f = 1 t + 1 = 2, n − t = 5 nr. processes (counters) received received sent accepted • 0 • 0 • 1 • 1 • 2 • 2 • 3 • 3 • 4 • 4 • 5 • 5 • 6 • 6 •0 •1 •2 •3 •4 •5 •6 • Local state is (sv, nrcvd), where sv ∈ {sent, accepted} and 0 ≤ rcvd ≤ n Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 22 / 1
  45. 45. Data + counter abstraction over parametric intervals n = 6, t = 1, f = 1 t + 1 = 2, n − t = 5 nr. processes (counters) received received sent accepted • 0 • 0 • 1 • 1 • 2 • 2 • 3 • 3 • 4 • 4 • 5 • 5 • 6 • 6 •0 •1 •2 •3 •4 •5 •6 • Local state is (sv, nrcvd), where sv ∈ {sent, accepted} and 0 ≤ rcvd ≤ n Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 22 / 1
  46. 46. Data + counter abstraction over parametric intervals XXXXXXn = 6, XXXXXXt = 1, XXXXXXf = 1 n 3 · t ∧ t ≥ f Parametricintervals: I0 = [0, 1) I1 = [1, t + 1) It+1 = [t + 1, n − t) In−t = [n − t, ∞) nr. processes (counters) received received sent accepted • • • • I0 I1 It+1 In−t • • • • I0 I1 It+1 In−t • • • • I0 I1 It+1 In−t A local state is (sv, nrcvd), where sv ∈ {sent, accepted} and nrcvd ∈ {I0, I1, It+1, In−t} Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 22 / 1
  47. 47. Data + counter abstraction over parametric intervals n 3 · t ∧ t ≥ f Parametricintervals: I0 = [0, 1) I1 = [1, t + 1) It+1 = [t + 1, n − t) In−t = [n − t, ∞) nr. processes (counters) received received sent accepted • • • • I0 I1 It+1 In−t • • • • I0 I1 It+1 In−t • • • • I0 I1 It+1 In−t when all correct processes accepted, all non-zero counters are in this area A local state is (sv, nrcvd), where sv ∈ {sent, accepted} and nrcvd ∈ {I0, I1, It+1, In−t} Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 22 / 1
  48. 48. Abstraction refinement Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 23 / 1
  49. 49. Spurious behavior abstraction adds behaviors (e.g., x’=x+1 may lead to x’ being equal to x) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 24 / 1
  50. 50. Spurious behavior abstraction adds behaviors (e.g., x’=x+1 may lead to x’ being equal to x) ⇒ specs that hold in concrete system may be violated in abstract system spurious counterexamples we have to reduce the behaviors of the abstract system make it more concrete . . . based on the counterexamples = CEGAR Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 24 / 1
  51. 51. Spurious behavior abstraction adds behaviors (e.g., x’=x+1 may lead to x’ being equal to x) ⇒ specs that hold in concrete system may be violated in abstract system spurious counterexamples we have to reduce the behaviors of the abstract system make it more concrete . . . based on the counterexamples = CEGAR Three sources of spurious behavior # processes decreasing or increasing # messages sent = # processes which have sent a message unfair loops Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 24 / 1
  52. 52. Spurious behavior abstraction adds behaviors (e.g., x’=x+1 may lead to x’ being equal to x) ⇒ specs that hold in concrete system may be violated in abstract system spurious counterexamples we have to reduce the behaviors of the abstract system make it more concrete . . . based on the counterexamples = CEGAR Three sources of spurious behavior # processes decreasing or increasing # messages sent = # processes which have sent a message unfair loops . . . and a new abstraction phenomenon Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 24 / 1
  53. 53. Parametric abst. refinement — uniformly spurious paths Classic case: Concrete Abstract Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 25 / 1
  54. 54. Parametric abst. refinement — uniformly spurious paths Classic case: Concrete Abstract Our case: Concrete n2 , t2 , f2 Concrete n1 , t1 , f1 Abstract ··· ··· Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 25 / 1
  55. 55. CEGAR — automated workflow Model Checking Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 26 / 1
  56. 56. CEGAR — automated workflow Model Checking correct Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 26 / 1
  57. 57. CEGAR — automated workflow Model Checking correct Abstraction refinement using SMT counterexample Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 26 / 1
  58. 58. CEGAR — automated workflow Model Checking correct Abstraction refinement using SMT counterexample CE feasible: bug Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 26 / 1
  59. 59. CEGAR — automated workflow Model Checking correct Abstraction refinement using SMT counterexample CE feasible: bug CE spurious: refined abstraction Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 26 / 1
  60. 60. What is SMT? recall SAT: given a Boolean formula, e.g., (¬a ∨ ¬b ∨ c) ∧ (¬a ∨ b ∨ d ∨ e) is there an assignment of true and false to variables a, b, c, d, e such that the formula evaluates to true? Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 27 / 1
  61. 61. What is SMT? recall SAT: given a Boolean formula, e.g., (¬a ∨ ¬b ∨ c) ∧ (¬a ∨ b ∨ d ∨ e) is there an assignment of true and false to variables a, b, c, d, e such that the formula evaluates to true? Satisfiability Modulo Theories (SMT) : here just linear arithmetics given a formula, e.g., x = y ∧ y = z ∧ u = x ∧ (x + y ≤ 1 ∧ 2x + y = 1) ∨ 3x + 2y ≥ 3 is there an assignment of values to u, x, y, z such that formula evaluates to true? practically efficient tools: Yices, Z3 Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 27 / 1
  62. 62. Counter example: losing processes Output of data abstraction: 16 local states: L = {(sv, ˆnrcvd) with sv ∈ {v0, v1, sent, accepted} and ˆrcvd ∈ {I0, I1, It+1, In−t}} An abstract global state is (ˆk, ˆnsnt), where ˆnsnt ∈ {I0, I1, It+1, In−t} and ˆk : L → {I0, I1, It+1, In−t} Consider an abstract trace: ˆnsnt1 = I0 ˆk1( ) =   In−t , if = (v1, I0) I0, otherwise ˆnsnt2 = I1 ˆk2( ) =   In−t , if = (v1, I0) I1, if = (sent, I0) I0, otherwise ˆnsnt3 = It+1 ˆk3( ) =   In−t , if = (v1, I0) It+1, if = (sent, I0) I0, otherwise Encode the last state in SMT as a conjunction T of the constraints: resilience condition n 3t ∧ t ≥ f ∧ f ≥ 0 zero counters (i = 4 ∧ i = 8) → 0 ≤ k3[i] 1 non-zero counters n − t ≤ k3[4] ∧ t + 1 ≤ k3[8] n − t system size n − f = k3[0] + k3[1] + · · · + k3[15] UNSAT Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 28 / 1
  63. 63. Remove transitions We ask the SMT solver: is there a satisfiable assignment for T? if yes, then the state is OK, may be part of a real counterexample if not, then the state is spurious remove transitions to that state in the abstract system Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 29 / 1
  64. 64. Liveness distributed algorithm requires reliable communication every message sent is eventually received ¬in transit ≡ [∀i. nrcvdi ≥ nsnt] fairness F G ¬in transit necessary to verify liveness, e.g., F G ¬in transit → G ([∀i. svi = v1] → F [∀i. svi = accept]) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 30 / 1
  65. 65. Liveness distributed algorithm requires reliable communication every message sent is eventually received ¬in transit ≡ [∀i. nrcvdi ≥ nsnt] fairness F G ¬in transit necessary to verify liveness, e.g., F G ¬in transit → G ([∀i. svi = v1] → F [∀i. svi = accept]) counter example (lasso): ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit s1 ¬in transit s2 sk s3 · · · · · · · · · Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 30 / 1
  66. 66. Liveness — fairness suppression ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit s1 ¬in transit s2 sk s3 · · · · · · · · · if there is a spurious sj (all its concretizations violate ¬in transit), then the loop is spurious. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 31 / 1
  67. 67. Liveness — fairness suppression ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit s1 ¬in transit s2 sk s3 · · · · · · · · · if there is a spurious sj (all its concretizations violate ¬in transit), then the loop is spurious. refine fairness to F G ¬in transit ∧ G F 1≤j≤k “out of sj Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 31 / 1
  68. 68. Liveness — fairness suppression ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit ¬in transit s1 ¬in transit s2 sk s3 · · · · · · · · · if there is a spurious sj (all its concretizations violate ¬in transit), then the loop is spurious. refine fairness to F G ¬in transit ∧ G F 1≤j≤k “out of sj Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 31 / 1
  69. 69. experimental evaluation Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 32 / 1
  70. 70. Concrete vs. parameterized (Byzantine case) Time to check relay (sec, logscale) Memory to check relay (MB, logscale) Parameterized model checking performs well (the red line). Experiments for fixed parameters quickly degrade (n = 9 runs out of memory). We found counter-examples for the cases n = 3t and f t, where the resilience condition is violated. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 33 / 1
  71. 71. Experimental results at a glance Algorithm Fault Resilience Property Valid? #Refinements Time ST87 Byz n 3t U 0 4 sec. ST87 Byz n 3t C 10 32 sec. ST87 Byz n 3t R 10 24 sec. ST87 Symm n 2t U 0 1 sec. ST87 Symm n 2t C 2 3 sec. ST87 Symm n 2t R 12 16 sec. ST87 Omit n 2t U 0 1 sec. ST87 Omit n 2t C 5 6 sec. ST87 Omit n 2t R 5 10 sec. ST87 Clean n t U 0 2 sec. ST87 Clean n t C 4 8 sec. ST87 Clean n t R 13 31 sec. CT96 Clean n t U 0 1 sec. CT96 Clean n t A 0 1 sec. CT96 Clean n t R 0 1 sec. CT96 Clean n t C 0 1 sec. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 34 / 1
  72. 72. When resilience condition is wrong... Algorithm Fault Resilience Property Valid? #Refinements Time ST87 Byz n 3t ∧ f ≤ t+1 U 9 56 sec. ST87 Byz n 3t ∧ f ≤ t+1 C 11 52 sec. ST87 Byz n 3t ∧ f ≤ t+1 R 10 17 sec. ST87 Byz n ≥ 3t ∧ f ≤ t U 0 5 sec. ST87 Byz n ≥ 3t ∧ f ≤ t C 9 32 sec. ST87 Byz n ≥ 3t ∧ f ≤ t R 30 78 sec. ST87 Symm n 2t ∧ f ≤ t+1 U 0 2 sec. ST87 Symm n 2t ∧ f ≤ t+1 C 2 4 sec. ST87 Symm n 2t ∧ f ≤ t+1 R 8 12 sec. ST87 Omit n ≥ 2t ∧ f ≤ t U 0 1 sec. ST87 Omit n ≥ 2t ∧ f ≤ t C 0 2 sec. ST87 Omit n ≥ 2t ∧ f ≤ t R 0 2 sec. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 35 / 1
  73. 73. Summary of results Abstraction tailored for distributed algorithms threshold-based fault-tolerant allows to express different fault assumptions Verification of threshold-based fault-tolerant algorithms with threshold guards that are widely used Byzantine faults (and other) for all system sizes Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 36 / 1
  74. 74. Related work: non-parameterized Model checking of the small size instances: clock synchronization [Steiner, Rushby, Sorea, Pfeifer 2004] consensus [Tsuchiya, Schiper 2011] asynchronous agreement, folklore broadcast, condition-based consensus [John, Konnov, Schmid, Veith, Widder 2013] and more... Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 37 / 1
  75. 75. Related work: parameterized case Regular model checking of fault-tolerant distributed protocols: [Fisman, Kupferman, Lustig 2008] “First-shot” theoretical framework. No guards like x ≥ t + 1, only x ≥ 1. No implementation. Manual analysis applied to folklore broadcast (crash faults). Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 38 / 1
  76. 76. Related work: parameterized case Regular model checking of fault-tolerant distributed protocols: [Fisman, Kupferman, Lustig 2008] “First-shot” theoretical framework. No guards like x ≥ t + 1, only x ≥ 1. No implementation. Manual analysis applied to folklore broadcast (crash faults). Backward reachability using SMT with arrays: [Alberti, Ghilardi, Pagani, Ranise, Rossi 2010-2012] Implementation. Experiments on Chandra-Toueg 1990. No resilience conditions like n 3t. Safety only. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 38 / 1
  77. 77. Our current work Discrete synchronous Discrete partially synchronous Discrete asynchronous Continuous synchronous Continuous partially synchronous One instance/ finite payload Many inst./ finite payload Many inst./ unbounded payload Messages with reals core of {ST87, BT87, CT96}, MA06 (common), MR04 (binary) one-shot broadcast, c.b.consensus Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 39 / 1
  78. 78. Future work: threshold guards + orthogonal features Discrete synchronous Discrete partially synchronous Discrete asynchronous Continuous synchronous Continuous partially synchronous One instance/ finite payload Many inst./ finite payload Many inst./ unbounded payload Messages with reals core of {ST87, BT87, CT96}, MA06 (common), MR04 (binary) one-shot broadcast, c.b.consensus DHM12 ST87 AK00 CT96 (failure detector) DLS86, MA06, L98 (Paxos) ST87, BT87, CT96, DAs with failure-detectors DLPSW86 DFLPS13 WS07 ST87 (JACM) FSFK06 WS09 clock sync broadcast approx. agreement Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 39 / 1
  79. 79. Thank you! http://forsyte.at/software/bymc Doctoral College: Vienna, Graz, Linz http://logic-cs.at Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 40 / 1
  80. 80. the implementation Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 41 / 1
  81. 81. Tool Chain: ByMC Parametric Promela code static analysis + Yices Parametric Interval Domain D Parametric data abstraction with Yices Parametric Promela code Parametric counter ab- straction with Yices normal Promela code Spin property holds counterexample Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 42 / 1
  82. 82. Tool Chain: ByMC Parametric Promela code static analysis + Yices Parametric Interval Domain D Parametric data abstraction with Yices Parametric Promela code Parametric counter ab- straction with Yices normal Promela code Spin property holds counterexample Refine Concrete counter representation (VASS) SMT formula Yices counterexample feasible unsat sat Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 42 / 1
  83. 83. Tool Chain: ByMC Parametric Promela code static analysis + Yices Parametric Interval Domain D Parametric data abstraction with Yices Parametric Promela code Parametric counter ab- straction with Yices normal Promela code Spin property holds counterexample Refine Concrete counter representation (VASS) SMT formula Yices counterexample feasible invariant candidates (by the user) unsat sat Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 42 / 1
  84. 84. Experimental setup The tool (source code in OCaml), the code of the distributed algorithms in Parametric Promela, and a virtual machine with full setup are available at: http://forsyte.at/software/bymc Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 43 / 1
  85. 85. Running the tool — concrete case user specifies parameter value useful to check whether the code behaves as expected $bymc/verifyco-spin N=4,T=1,F=1 bcast-byz.pml relay model checking problem in directory “./x/spin-bcast-byz-relay-N=4,T=1,F=1” in concrete.prm parameters are replaced by numbers process prototype is replaced with N − F = 3 active processes Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 44 / 1
  86. 86. Running the tool — parameterized model checking PIA data and counter abstraction finite-state model checking on abstract model $bymc/verifypa-spin bcast-omit.pml relay model checking problem in directory “./x/bcast-byz-relay-yymmdd-HHMM.*” directory contains abs-interval.prm: result of the data abstraction; abs-counter.prm: result of the counter abstraction; abs-vass.prm: auxiliary abstraction for abstraction refinement; mc.out: the last output by Spin; cex.trace: the counterexample (if there is one); yices.log: communication log with Yices. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 45 / 1
  87. 87. Fairness, Refinement, and Invariants In the Byzantine case we have in transit : ∀i. (nrcvdi ≥ nsnt) and G F ¬in transit. In this case communication fairness implies computation fairness. But in the abstract version nsnt can deviate from the number of processes who sent the echo message. In this case the user formulates a simple state invariant candidate, e.g., nsnt = K([sv = SE ∨ sv = AC]) (on the level of the original concrete system). The tool checks automatically, whether the candidate is actually a state invariant. After the abstraction the abstract version of the invariant restricts the behavior of the abstract transition system. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 46 / 1
  88. 88. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 47 / 1
  89. 89. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 47 / 1
  90. 90. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. refine justice to G F ¬in transit ∧ G F 1≤j≤k ¬at(sj ) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 47 / 1
  91. 91. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. refine justice to G F ¬in transit ∧ G F 1≤j≤k ¬at(sj ) . . . we use unsat cores to refine several loops at once Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 47 / 1
  92. 92. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 48 / 1
  93. 93. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 48 / 1
  94. 94. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. refine justice to G F ¬in transit ∧ G F 1≤j≤k ¬at(sj ) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 48 / 1
  95. 95. Parametric abstraction refinement — justice suppression justice G F ¬in transit necessary to verify liveness counter example: in transit in transit in transit in transit in transit in transit in transit s1 in transit s2 sk s3 · · · · · · · · · if ∀j all concretizations of sj violate ¬in transit, then CE is spurious. refine justice to G F ¬in transit ∧ G F 1≤j≤k ¬at(sj ) . . . we use unsat cores to refine several loops at once Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 48 / 1
  96. 96. asynchronous reliable broadcast (srikanth toueg 1987) the core of the classic broadcast algorithm from the da literature. it solves an agreement problem depending on the inputs vi . Variables of process i vi : {0 , 1} init with 0 or 1 accepti : {0 , 1} init with 0 An indivisible step: i f vi = 1 then send ( echo ) to all ; i f received (echo) from at l e a s t t + 1 distinct processes and not sent ( echo ) before then send ( echo ) to all ; i f received ( echo ) from at l e a s t n - t distinct processes then accepti := 1; Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 49 / 1
  97. 97. asynchronous reliable broadcast (srikanth toueg 1987) the core of the classic broadcast algorithm from the da literature. it solves an agreement problem depending on the inputs vi . Variables of process i vi : {0 , 1} init with 0 or 1 accepti : {0 , 1} init with 0 An indivisible step: i f vi = 1 then send ( echo ) to all ; i f received (echo) from at l e a s t t + 1 distinct processes and not sent ( echo ) before then send ( echo ) to all ; i f received ( echo ) from at l e a s t n - t distinct processes then accepti := 1; asynchronous t byzantine faults correct if n 3t resilience condition rc parameterized process skeleton p(n, t) Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 49 / 1
  98. 98. Abstract CFA qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 50 / 1
  99. 99. Abstract CFA qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd := z where (nrcvd ≤ z ∧ z ≤ nsnt + f ) ¬(t + 1 ≤ nrcvd) t + 1 ≤ nrcvd sv = V0 ¬(sv = V0) inc nsnt n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC qI q0 q1 q2 q3 sv = V1 ¬(sv = V1) inc nsnt sv := SE q4 q5 q6 q7 nrcvd = I0 ∧ nsnt = I0 ∧ (nrcvd = I0 ∨ nrcvd = I1) ∨ . . . ¬(t + 1 ≤ nrcvd) nrcvd = It+1 ∨ nrcvd = In−t sv = V0 ¬(sv = V0) nsnt = I1 ∧ (nsnt = I1 ∨ nsnt = It+1) ∨ . . . n − t ≤ nrcvd ¬(n − t ≤ nrcvd) sv := AC Igor Konnov (www.forsyte.at) Checking Fault-Tolerant Distributed Algos TMPA’14, Nov. 2014 50 / 1

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