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Modeling and Analyzing Periodic Distributed Computations Anurag Agarwal Vijay Garg (garg@ece.utexas.edu)garg@ece.utexas.edu Vinit Ogale The University of Texas at Austin SSS2010

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Motivation Many distributed computations are infinite o E.g., various reactive systems, servers Correctness Specifications o Safety: Nothing "bad" will ever happen o Liveness: Something "good" will happen eventually Specified using Temporal Logic [Pnueli 77] Runtime verification of properties How can one detect violation of liveness properties?

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Difficulties Can observe only a finite execution How to exhibit the offending execution? How to detect if a global state is reachable in an infinite computation?

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Recurrent Global State: Liveness violation Two identical states with P1 being hungry P1 does not eat in the intermediate computation

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Overview Predicate Detection in Partial Order Recurrent global states Modeling infinite computations as d-diagrams Vector clocks in d-diagrams Predicate detection in d-diagrams Related work Conclusions and Future Work

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Predicate Detection Predicate: A property expressed using variables on processes. e.g., more than one process is in critical section Predicate detection: Determining if an execution trace satisfies the predicate 6 Predicate detection trace predicate Yes No Program

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Trace Model: Total Order Total order: interleaving of events in a trace –Temporal Rover [Drusinsky 03], –Java-MaC [Kim, Kannan, Lee, Sokolsky, and Viswanathan 04], –JPaX [Havelund and Rosu 04] –PET [Gunter, Kurshan, Peled 00] Low computational complexity 7

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Partial Order Traces Predicate Detection –Exponential time algorithm for general predicate [Cooper and Marzullo 91] –NP-complete for simple boolean expressions (2-CNF) [Mittal and Garg 01] –Efficient algorithms for linear predicates [Chase, Garg 95], relational predicates [Tomlinson, Garg 93], Temporal Logic [Ogale Garg 07] 8 {} {e 1 } {f 1 } {e 1, f 1 } {e 2, e 1, f 1 } {e 2, e 1, f 2, f 1 } {e 1, f 2, f 1 } e1e1 e2e2 f1f1 f2f2 P1P1 P2P2 {e2,e1}{e2,e1} Computation Corresponding computation lattice

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Recurrent global states (Consistent) Global states which occur more than once

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D-diagram A d-diagram is a finite representation of an infinite periodic distributed computation ; (V,R,F,B) V: set of vertices R: recurrent vertices (infinite instances) F: forward edges for all i:e i -> f i B: shift edges for all i: e i -> f i + 1

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Examples Set of natural numbers under natural order Set of natural numbers with no order

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Unrolling the d-diagram A directed graph can be generated by "unrolling" a d-diagram ie. creating infinite instances of the recurrent vertices and generating the appropriate edges between them.

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Finite Width Posets Lemma: A directed graph G defined by a d- diagram has finite width iff for every recurrent vertex there exists a cycle that includes a shift- edge.

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Shift-of-a-cut A consistent cut in the graph can be "shifted" forwards or backwards with respect to some recurrent vertices to generate another consistent cut.

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Overview Predicate Detection in Partial Order Recurrent global states Modeling infinite computations as d-diagrams Vector clocks in d-diagrams Predicate detection in d-diagrams Related work Conclusions and Future Work

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Vector Clocks in a Distributed System e happened before f iff V(e) < V(f) [Fidge 89, Mattern 89] P1P1 (1,0,0)(2,1,0)(3,1,0) P2P2 (0,1,0)(0,2,0) P3P3 (0,0,1)(0,0,2)(2,1,3) How do we timestamp infinite sets of events?

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Vector Clocks for d-diagrams J(e) = least consistent cut that includes e Theorem: For a recurrent vertex e, J(e) is guaranteed to stabilize after shift-diameter of the d-diagram. J(e i+1 ) can be derived from J(e i ) by shifting the cut. Shift-diameter of a d-diagram: Maximum number of shift-edges in the shortest path between any two vertices Lemma: On a d-diagram of a computation, shift-diameter is at most 2N.

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Vector Clock for d-diagrams PV(e) = (V(e 1 ), V(e 2 ),..., V(e n ); I(e)) where V(e i ) = vector timestamp of e i I(e) = V(e i+1 ) - V(e i ) n = shift diameter Given p-timestamp, V(e n+j ) = V(e n ) + j * I(e) I(a) = [2,2] PV(a) = ([1,0], [3,0], [5,2]; [2,2]

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Detecting global predicates Theorem: Sufficient to detect predicates on a finite part of the computation obtained by unrolling the d-diagram some number of times. Hence o If a predicate is never true in finite part, then it'll never be true in the infinite computation o If a predicate becomes true on recurrent events, then it'll be true infinitely often during the computation. Stopping Rule: the number of unrollings required is less than N (number of processes)

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Recurrent Global State Detection Step 1: Ensure that the computation can be replayed (Deterministic Replay) [LM 87] Step 2: Compute a global state G (Global Snapshot Algorithm) [CL85]. Let the vector clock be Y. Step 3: Replay the computation detecting the first global state H that matches G (Conjunctive Predicate Detection [GW92] with vector clock Z) Step 4: Return (G,H) if Y != Z

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Related Work Global Predicate Detection on happened-before model (e.g. conjunctive, linear, temporal logic predicates) – interpretation over finite traces Petri Nets – modeling and analyzing concurrent systems (versus a single computation) Message Sequence Charts (MSC) – incomparable to d-diagrams (e.g. require a message sent in a MSC to be received in the same MSC node)

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Conclusions Recurrent global states D-diagram as model of an infinite periodic poset suitable for distributed computation Algorithm to timestamp events in a d-diagram Algorithm to detect global predicates in a d-diagram

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Future Work Minimum unrolling Detecting general temporal logic formulas on d-diagrams

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Questions? Recurrent global states Modeling infinite computations as d-diagrams Vector clocks in d-diagrams Predicate detection on d-diagrams

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Backup Slides

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Computation Model A distributed computation consists of N sequential processes P1, P2,... PN A directed graph is used to represent the computation with the vertices corresponding to the events and the edges representing the dependencies. o Acyclic graphs can represent finite computations A consistent cut in the distributed graph represents a set of vertices such that if it contains a vertex e, it contains all its incoming neighbors as well o A consistent cut represents a valid global state The frontier of a consistent cut is the set of events who successors from the same process don't exist in the cut. o The frontier can be used to uniquely represent a consistent cut

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Trace Model: Partial Order Partial order: Lamports happened-before model [Lamport 78] –suitable for concurrent and distributed programs –encodes exponential number of total orders, captures bugs that may not be found with a total order 27

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Vector Clock for d-diagrams We prove that after sufficient unrolling, the smallest consistent cuts containing different iteration of all recurrent vertices can be obtained simply by shifts. This iteration is called the shift-diameter. We associate a p-timestamp with every recurrent vertex which allows us to generate the timestamp for any iteration of the vertex: PV(e) = (V(e^1), V(e^2),..., V(e^n); I(e)) where V(e^i) = vector timestamp of e^i (the i-th iteration of e); I(e) = V(e^(n+1)) - V(e^n) n = shift diameter Given p-timestamp, V(e^(n+j)) = V(e^n) + j * I(e)

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Detecting global predicates A predicate is a property defined on the states of the processes as well as channels We show that its sufficient to detect predicates on a finite part of the computation i.e. all possible consistent cuts that can be found in the full computation can be found in a finite subset obtained by rolling the d-diagram sufficient number of times. Hence o If a predicate is never true in finite part, then it'll never be true in the infinite computation o If a predicate becomes true on recurrent events, then it'll be true infinitely often during the computation. We also show that the number of unrollings required are less than N (number of processes)

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