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Protocol Verification with Merci Mark R. Tuttle and Amit Goel DTS SCL

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Introduction I love proof –Proof is the path to understanding why things work –But theorem provers are too hard for the masses (even me) I advocate model checking at Intel –It is the path to automated formal verification for the masses –But model checkers verify without explaining, and don’t scale But the world has changed –Decision procedures and SMT now automate some forms of proof –Is theorem proving now viable for nonspecialists in product groups? Slide 2

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Our result Amit wrote Merci: SMT-based proof checker from SCL –Systems modeled with guarded commands (like Murphi, TLA+) –Clean mapping to decision procedures of an SMT solver Mark validated a classical distributed algorithm –A novice: no prior exposure to Merci, little exposure to SMT –Model done in 3 days, proof done in 3 days, just 9 pages long –Model looks like ordinary code, invariants explain the algorithm Found little need to coach the prover about “obvious” things Slide 3

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Consensus Validity: –Each output was an input Agreement: –All outputs are equal Termination: –All nodes choose an output n1n1 n2n2 n3n3 010 111 nodes inputs outputs [Pease, Shostak, Lamport] message passing Slide 4

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A shocking result! Consensus is impossible in an asynchronous system if even one node can fail. –Asynchronous: no bound on node step time, msg delivery time –Failure: node just stops (crashes) A decade of papers –Different system models, different failure models –How fast? How few messages? How many failures Consensus is the “hardest problem” in concurrency! –but sometimes it can be solved… [Fischer, Lynch, Patterson] [Herlihy] Slide 5

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Synchronous model Computation is a sequence of rounds of message passing. nodes send messages nodes receive messages nodes change state round rround r+1 node Slide 6

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Crash failures At most t nodes can fail. n n is correct sends all messages n is silent sends no messages n crashes! sends some messages Slide 7

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Algorithm procedure consensus (node n) state ← { input } for each round r = 1, 2, …, t+1 do broadcast state to all nodes receive state 1, state 2, …, state k from other nodes state ← state 1 U state 2 U … U state k output ← min(state) Validity: each output was an input Termination: all nodes choose an output at end of round t+1 Agreement: ??? [Dolev, Strong] Slide 8

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Clean round: no nodes fail There is a clean round in t+1 rounds (at most t failures). Nodes have same state after a clean round. Nodes choose same output value min(state). Agreement! [Dwork, Moses] Clean round! Slide 9

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Merci A typed procedural language Guarded commands used to describe systems type node var array(node, bool) y = mk_array[node](false) var array(node, bool) critical =mk_array[node](false) var node turn transition unit req_critical (node n) require (!y[n]) { y[n] := true; } transition unit enter_critical (node n) require (y[n] && !critical[n] && turn=n) { critical[n] := true; } transition unit exit_critical (node n) require (critical[n]) {critical[n] := false; y[n] := false; nondet turn;} [Amit Goel]

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Merci A typed procedural language Guarded commands used to describe systems A goal description language for compositional reasoning def bool mutex = (node n1, node n2) (critical[n1] && critical[n2] => n1=n2) def bool aux = (node n) (critical[n] => turn=n) goal g0 = invariant mutex assuming aux goal g1 = invariant aux [Amit Goel]

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Merci A typed procedural language Guarded commands used to describe systems A goal description language for compositional reasoning A template system for extending the language template Set { type t // set type const bool mem (elem x, t s) const t add (elem x, t s) const t remove (elem x, t s) axiom mem_add = (elem x, elem y, t s) (mem (x, add (y, s)) = (x = y || mem (x, s))) axiom mem_remove = (elem x, elem y, t s) (mem (x, remove(y, s)) = (x !=y && mem(x, s))) } type node module Node= Set [Amit Goel]

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Crash failure model def bool is_crash_behavior (Nodes crashed, Nodes crashing, message_pattern deliver) = (node p) (p crashed => is_silent(p,deliver)) && (node p) (is_faulty(p,deliver) => p crashed || p crashing) && Nodes.disjoint(crashed,crashing) && Nodes.cardinality(crashed) + Nodes.cardinality(crashing) ≤ t faulty silent Slide 13

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Synchronous model for each node p initialize state of p for each round r for each p and q send msg from p to q for each p and q receive msg from p to q for each p update state of p phase init send recv comp program counter init[p] send[p][q] recv[p][q] comp[p] algorithm how? what? how? decide? decide! Slide 14

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phase ← send phase ← recv phase ← comp Synchronous model Transitions –initialize(p) –start_send –send(p,q) –start_recv –recv(p,q) –start_comp –comp(p) init[p] ← true send[p][q] ← true recv[p][q] ← true comp[p] ← true increment round send[q][p] ← false recv[p][q] ← false comp[p] ← fasle is_init_phase = phase = init init_phase_done = forall (node p) (init[p]) Slide 15

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transition start_sending () require ( is_init_phase && init_phase_done || is_comp_phase && comp_phase_done) { "send[p][q], recv[p][q], comp[p] <= false" "message[p][q] <= null_message" round := round + 1; phase := send; crashed := Nodes.union(crashed,crashing); nondet crashing; nondet deliver; assume is_crash_behavior(crashed,crash,deliver); } Slide 16

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transition send (node n, node m) require (is_send_phase) require (!send[n][m]) { messages[n][m] := (deliver [n][m] ? global_state[n] : null_message); send[n][m] := true; } initialize(p)8 lines start_send()16 linessend(p,q)9 lines start_recv()5 linesrecv(p,q)7 lines start_comp()5 linescomp(p)13 lines Transition size Slide 17

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Agreement proof Recall the agreement proof –A1: There is a clean round –A2: All states are equal at the end of a clean round –A3: All states remain equal after a clean round –A4: All nodes choose from their states the same output value Merci proof is short –A1: 7 lines –A2: 127 lines –A3: 12 lines –A4: 25 lines Merci proof is almost entirely at the algorithmic level Slide 18

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A1: There is a clean round def bool clean_round_by_round_t_plus_1 = round >= t+1 => !before_clean def bool faulty_grows_until_clean_round = before_clean => Nodes.cardinality(faulty) >= round goal clean1 = invariant faulty_grows_until_clean_round goal clean2 = invariant clean_round_by_round_t_plus_1 assuming faulty_grows_until_clean_round Slide 19

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A2: All states equal … def bool state_equality = (node n, node m) (noncrashed(n) && noncrashed(m) => state[n] = state[m]) def bool state_equality_in_clean = in_clean && send_phase_done && recv_phase_done => state_equality Proof –A2.1: If nonfaulty n has v, then n received v in a message –A2.2: That message was sent to everyone since round is clean –A2.3: If m received v in a message, then m has v –A2.4: So nonfaulty n and m have the same values Proof algorithmic and short: 48, 34, 15, and 30 lines long Slide 20

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Conclusion Classical fault-tolerant distributed algorithm proved w/Merci –Model looks like ordinary code, invariants explain the algorithm –Merci proof is 170 lines, Classical proof is 1+ page –Model and proof done in 6 days with no prior experience Yices made quantification hard –exists: usually have to produce the example by hand –forall: template instantiation wouldn’t find the right instantiation Yices counterexamples mostly useless –Get a context from first few lines, ignore the rest –“Is property false or is Yices failing to instantiate a forall template?” –BKM: Think about the algorithm itself, and ignore Yices output Slide 21

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