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Verification and Synthesis of Symmetric Uni-Rings for Leads-To Properties Ali Ebnenasir aebnenas@mtu.edu Department of Computer Science College of Computing Michigan Technological University Houghton MI 49931 http://asd.cs.mtu.edu/

  1. Verification and Synthesis of Symmetric Uni-Rings for Leads-To Properties Ali Ebnenasir aebnenas@mtu.edu Department of Computer Science College of Computing Michigan Technological University Houghton MI 49931 http://asd.cs.mtu.edu/

  2. Parameterized Distributed Systems (PDS) Family 2: just one process Dijkstra’s Token Ring for mutual exclusion: 𝛒 2 : Template process 2 P N-1 P 0 Action 0 : x 0 = x N-1 Γ  x 0 := x N-1 + 1 x N-1 x 0 Read/Write - Process P i has x i ∈ β„€ N = {0, 1, …, N-1} P 1 - N denotes the total number of processes - Addition and subtraction are done in modulo N x 1 𝛒 1 : Template process 1 Q = βˆ€ i ∈ β„€ N : ( (x i -1 = x i ) ∨ (x i -1 = x i +1)) Action i : x i β‰  x i-1 Γ  x i := x i-1 Legend : . . . Process/Node Family 1: N-1 symmetric processes Read from

  3. Significance From System on Chip, to multithreaded programs and large scale network protocols.

  4. Example: Agreement on Parity ● Parity on a fully symmetric unidirectional ring (uni-ring); i.e., k =1 ● Ring size: N>2, arbitrary but finite; i.e., 𝛒 1 = {P 0 , …, P N-1 } ● Process P i : has a variable x i capturing set of writeable variables x i ∈ β„€ 4 = {0, 1, 2, 3} ● ● Topology: uni-ring (read/write restrictions) read x i -1 and x i ; write x i ● ● Conjunctive state predicate Q = βˆ€ i ∈ β„€ N : ((|x i -1 - x i | mod 2) = 0) ● Template Process: P 0 x 0 P 1 A i : (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 x 1 x N-1 P 2 x 2

  5. Proposed Method Most existing methods for verification and synthesis: Correctness of a finite abstract model β‡’ Correctness of PDS We propose a method based on characterization of global failures in local state spaces of template processes in a topology-specific fashion. Absence of local characterizations β‡’ Correctness of PDS

  6. Topology and Property-Specific Synthesis of R ↝ Q Variable x i and its domain Parameterized Actions Synthesize Algorithm for Uni-Ring R = βˆ€ i ∈ β„€ N : r(x i -1 - x i ) Satisfies R ↝ Q for an arbitrary (but finite) Q = βˆ€ i ∈ β„€ N : q(x i -1 - x i ) # of processes Specifications: Linear Temporal Logic ● Leads-To: ☐ ( R β‡’ β—‡ Q ) ≣ ( R ↝ Q ) ● E.g., ( true ↝ Q ) ● Note : From any global state in R, the entire ring eventually reaches a global state in Q ; i.e., global liveness.

  7. Failures of Leads-To in PDS β€’ Reaching global deadlocks and/or global livelocksfrom R β€’ Verification of deadlock-freedom is decidable. [ICDCS 2012] [ICDCS 2012] A. Farahat and A. Ebnenasir, β€œ Local reasoning for global convergence of parameterized rings ,” in IEEE International Conference on Distributed Computing Systems (ICDCS), 2012, pp. 496–505.

  8. Livelock-Freedom in Uni-Rings ● Theorem 1: Verifying livelock-freedom in symmetric uni-rings is undecidable, even for deterministic , constant-space and self-disabling processes. [SSS’13, ACM TOCL’19] ● Self-disabling process: once it executes an action, it disables itself until enabled again by its predecessor. β€’ Corollary: Verifying R ↝ Q in symmetric uni-rings is undecidable. (Proof in the paper) [SSS’13] Alex Klinkhamer and Ali Ebnenasir, Verfiying Livelock Freedom of Parameterized Rings and Chains , 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2013). [ACM TOCL’19] A. Klinkhamer and A. Ebnenasir, β€œ On the verification of livelock-freedom and self-stabilization on parameterized rings ,” ACM Transactions on Computational Logic, vol. 20, no. 3, pp. 1–36, 2019.

  9. Local Characterization of Global Livelocks in Uni-Rings: Can we detect and construct global livelocks just by analyzing the local state transition system of the template process?

  10. Action Graphs ● Protocols can be represented as labeled directed multi-graphs in the local state space of the template process Vertices : values in the domain of x i ∈ {0, 1, 2, 3} ● Arcs : each arc (a, b, c) represents a local update of x i to c if x i-1 =a and x i = b ● E.g., ( 0 , 1 , 2 ) means if x i-1 =0 and x i = 1 then update x i to 2 ● (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 1 |3 0|2 0 2 1 3 1|3 0|2

  11. Action Graphs ● Protocols can be represented as labeled directed multi-graphs in the local state space of the template process Vertices : values in the domain of x i ∈ {0, 1, 2, 3} ● Arcs : each arc (a, b, c) represents a local update of x i to c if x i-1 =a and x i = b ● E.g., ( 0 , 1 , 2 ) means if x i-1 =0 and x i = 1 then update x i to 2 ● (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 ( 0 , 1 , 2 ) 1 |3 0|2 0 2 1 3 1|3 0|2

  12. Action Graphs ● Protocols can be represented as labeled directed multi-graphs in the local state space of the template process Vertices : values in the domain of x i ∈ {0, 1, 2, 3} ● Arcs : each arc (a, b, c) represents a local update of x i to c if x i-1 =a and x i = b ● E.g., ( 0 , 1 , 2 ) means if x i-1 =0 and x i = 1 then update x i to 2 ● (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 ( 0 , 1 , 2 ) x i-1 1 |3 0|2 0 2 1 3 1|3 0|2

  13. Action Graphs ● Protocols can be represented as labeled directed multi-graphs in the local state space of the template process Vertices : values in the domain of x i ∈ {0, 1, 2, 3} ● Arcs : each arc (a, b, c) represents a local update of x i to c if x i-1 =a and x i = b ● E.g., ( 0 , 1 , 2 ) means if x i-1 =0 and x i = 1 then update x i to 2 ● (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 ( 0 , 1 , 2 ) x i x i-1 1 |3 0|2 0 2 1 3 1|3 0|2

  14. Action Graphs ● Protocols can be represented as labeled directed multi-graphs in the local state space of the template process Vertices : values in the domain of x i ∈ {0, 1, 2, 3} ● Arcs : each arc (a, b, c) represents a local update of x i to c if x i-1 =a and x i = b ● E.g., ( 0 , 1 , 2 ) means if x i-1 =0 and x i = 1 then update x i to 2 ● (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 ( 0 , 1 , 2 ) x i x i-1 set x i to 1 |3 0|2 0 2 1 3 1|3 0|2

  15. Enabling Actions ● An action of a process P i may potentially enable another action of P i (0, 3 , 2 ), (1, 2 , 3 ), (2, 1 , 0 ) (3, 0 , 1 ) 1| 3 0| 2 0 2 1 3 1 |3 0 |2

  16. Closed Walks in Action Graphs ● Propagation of enablement as closed walk: sequence of consecutive actions A 0 : (|x i -1 - x i | mod 2) β‰  0 Γ  x i := x i-1 βŠ• 4 2 (0, 3 , 2 ), (1, 2 , 3 ), (2, 1 , 0 ) (3, 0 , 1 ) 1| 3 0| 2 0 2 1 3 1 |3 0 |2 Closed Walks 2 Closed Walks 1

  17. Enabling Closed Walks ● A closed walk enabling another closed walk. 0| 2 0 2 1 3 Closed walk 1: (1, 2 , 3 ), Closed walk 2:

  18. Enabling Closed Walks ● A closed walk enabling another closed walk. 1| 3 0| 2 0 2 1 3 Closed walk 1: (1, 2 , 3 ), Closed walk 2: (0, 3 , 2 ),

  19. Enabling Closed Walks ● A closed walk enabling another closed walk. 1| 3 0| 2 0 2 1 3 0 |2 Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ),

  20. Enabling Closed Walks ● A closed walk enabling another closed walk. 1| 3 0| 2 0 2 1 3 1 |3 0 |2 Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  21. Enabling Closed Walks ● A closed walk enabling another closed walk. 1| 3 0| 2 0 2 1 3 1 |3 0 |2 Closed walk 1 enables closed walk 2. Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  22. Enabling Closed Walks ● A closed walk enabling another closed walk. 1| 3 0| 2 0 2 1 3 1 |3 0 |2 A closed walk of length n enables another closed walk of length n iff j -th action of the first walk enables the j -th action of the second walk, for 1≀ j ≀ n Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  23. Circularly Enabling Closed Walks β€’ Closed walk 2 also enables closed walk 1. 1| 3 0 2 1 3 Closed walk 1: Closed walk 2: (0, 3 , 2 ),

  24. Circularly Enabling Closed Walks β€’ Closed walk 2 also enables closed walk 1. 1| 3 0| 2 0 2 1 3 Closed walk 1: (1, 2 , 3 ), Closed walk 2: (0, 3 , 2 ),

  25. Circularly Enabling Closed Walks β€’ Closed walk 2 also enables closed walk 1. 1| 3 0| 2 0 2 1 3 1 |3 Closed walk 1: (1, 2 , 3 ), Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  26. Circularly Enabling Closed Walks β€’ Closed walk 2 also enables closed walk 1. 1| 3 0| 2 0 2 1 3 1 |3 0 |2 Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  27. Circularly Enabling Closed Walks 1| 3 0| 2 0 2 1 3 1 |3 0 |2 2 circularly enabling closed walks, each of length 2. Closed walk 1: (1, 2 , 3 ), (3, 0 , 1 ) Closed walk 2: (0, 3 , 2 ), (2, 1 , 0 )

  28. Local Characterization of Global Livelocks Theorem 2 : There are m closed walks, each of length n , in the action graph that enable each other circularly (m > 1 and n β‰₯1) if and only if A uni-ring of symmetric, constant-space, deterministic and self-disabling processes has a livelock for a ring size ( m Γ— n ) [SSS’13] Alex Klinkhamer and Ali Ebnenasir, Verfiying Livelock Freedom of Parameterized Rings and Chains , 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2013). [ACM TOCL’19] A. Klinkhamer and A. Ebnenasir, β€œ On the verification of livelock-freedom and self-stabilization on parameterized rings ,” ACM Transactions on Computational Logic, vol. 20, no. 3, pp. 1–36, 2019.

  29. Synthesize a PDS using just locality and action graphs?

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