pwl rewal-slideshare

Ineluctable modality of the distributed

On Joseph Halpern’s work on knowledge in distributed

systems

Peter Alvaro UC Berkeley

choose-your-own-adventure talk

Last time at PWL…

•  The agreement problem(s) •  Impossibility results •  A “weakest” failure detector

Today: knowledge

It’s not just for byzantine stuff

I'm not a great fool, so I can clearly not choose the wine in front of you. But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.

Why you should care

A correct distributed program achieves (nontrivial) distributed property X. Some tricky questions before we start coding:

1.  Is X even attainable? 2.  Cheapest protocol that gets me X? 3.  How should I implement it?

A strong claim about distributed correctness properties

Uncertainty is what makes reasoning about distributed systems difficult. Uncertainty is the abundance of possibilities. Knowledge is the dual of possibility

A strong statement about protocols

How: Protocols just describe what actions to take based on local knowledge. Why: Protocols are just mechanisms to ensure that a group has shared knowledge of a fact.

A good paper about bridging the gap between properties and protocols

For example

•  Commit protocols – each agent knows the commit/abort

decision AND knows that all agents know the decision

•  Distributed garbage collection – an agent knows that no remote references

exist to a particular object, and that all other agents know

For example •  When the leader has received phase 2b messages for

value v and ballot bal from a majority of the acceptors, it knows that the value v has been chosen. [paxos]

•  a process takes a checkpoint when it knows that all processes on which it computationally depends took their checkpoints [An Efficient Protocol for Checkpointing Recovery in Distributed Systems, Kim and Park]

•  and therefore a cohort with a later viewstamp for some view knows everything known to a cohort with an earlier viewstamp for that view. [viewstamped replication]

•  Since each member of Si serves as an arbitrator, the requesting node knows that it is the only node that has been granted mutual exclusion [A sqrt(N) Algorithm for Mutual Exclusion in Decentralized Systems, Maekawa]

Warmup: RPC protocols

Hi!

Alice Bob


Hi!

Alice Bob

Issue: uncertainty! Uncertain environment è Uncertain outcomes


Alice Bob

Issue: uncertainty! Uncertain environment è Uncertain outcomes


Hi!

Retry Alice Bob


Hi!

Issues: infinite (sender) behavior & state, at-least-once delivery

Retry Alice Bob


Hi!

Retry with ACKS

Hi!

Alice Bob


Hi!

Retry with ACKS

Hi! Hi!

Alice Bob

Hi!


Hi!

Hi yourself

Retry with ACKS

Hi!

Issues: at-least once delivery

Hi!

Alice Bob

Hi!


Hi!

Hi yourself

Retry with ACKS

Hi!


Hi!

Alice Bob


Retry with ACKS


Alice Bob

Hi!

a good paper about principled distributed GC


Hi!

Issues: infinite receiver state

Receiver buffers, dedups

Alice Bob


Issues: infinite receiver state

Hi!

Receiver buffers, dedups

Alice Bob


Hi!

ACK-ACKing

Hi!

Alice Bob


Hi!

Hi yourself

ACK-ACKing

Hi!

Issue: uncertainty

Alice Bob


Hi!

Hi yourself

ACK-ACKing

Hi!

Alice Bob


ACK-ACKing

Hi!

Alice Bob

Ahoy


ACK-ACKing Alice Bob


ACK-ACKing

Issue: uncertainty

Alice Bob


Issues: infinite hot potato

Alice Bob

what does this remind me of?

Refresher: the two generals problem

Logic time

(propositional) logic

ϕ ϕ if ϕ is atomic ϕ ∧ ψ true if both ϕ and ψ are true ¬ϕ true if ϕ is false Sweet duality: ϕ ∨ ψ = ¬(¬ϕ ∧ ¬ψ) ϕ ⇒ ψ= ¬(ϕ ∧ ¬ψ)

q ⇒ p p = “the write is stable” q = “the write is acknowledged”

modality, duality

∃xϕ === ¬∀x ¬ϕ ¯ϕ === ¬£¬ϕ

Symbol Temporal Deon/c Epistemic

¯ Some8mes Is permi:ed Is possible

£ Always Is obligatory Is known

Knowledge is the dual of possibility

Epistemic modal logic

ϕ = “the write is stable” Kaliceϕ = “alice knows ϕ” KaliceKbobϕ = “alice knows bob knows ϕ” KaliceKbobKcarolϕ = “alice knows bob knows carol knows ϕ” […]

Epistemic modal logic

ϕ = “the write is stable” Eϕ = “everyone* knows ϕ” EEϕ = “everyone knows everyone knows ϕ” […]

A driver will not feel safe going when he sees a green light unless he knows that everyone else knows and follows the rules.

Common knowledge

ϕ = “the write is stable” Eϕ = “everyone* knows ϕ” EEϕ = “everyone knows everyone knows ϕ” […] Eiϕ = “(everyone knows * i) ϕ” Cϕ = E∞ϕ = “it is common knowledge that ϕ”

Distributed knowledge

ϕ = “the write is stable” Dϕ = “ϕ is implicitly known by the group” Sϕ = “someone knows ϕ”

Protocols climb the hierarchy Cϕ […]

Ek+1ϕ

[…] Eϕ Sϕ Dϕ ϕ


Ek+1ϕ


Deadlock detection ϕ is distributed knowledge

Someone knows ϕ


Ek+1ϕ


Reliable broadcast Someone knows ϕ

ϕ is distributed knowledge

Everyone knows ϕ


E3ϕ

E2ϕ Eϕ Sϕ Dϕ ϕ

Uniform Reliable broadcast

Someone knows ϕ


Everyone knows ϕ

Everyone knows everyone knows ϕ


E3ϕ

E2ϕ Eϕ Sϕ Dϕ ϕ

Someone knows ϕ


Everyone knows ϕ

Everyone knows everyone knows ϕ

Some crazy BFT protocol

(Everyone knows)k ϕ


E3ϕ

E2ϕ Eϕ Sϕ Dϕ ϕ

Knowledge Highway

E10ϕ 10 E100ϕ 100

Cϕ ∞

Applications of knowledge



Applications: impossibility

“in a system in which communication is not guaranteed, common knowledge of initially-undetermined facts is not attainable in any run of any protocol.” Corollary: the 2 generals problem is unsolvable

Let’s use knowledge to prove it!

But first… lots of formalism to get through L

Road map for the proof:

1.  Semantics of modal logic 2.  Distributed system model 3.  A quick and easy lemma 4.  Big theorem: Common knowledge is not

attainable via protocol 5.  Lemma 2: if the generals attack, they have

common knowledge of the attack. 6.  Corollary: 2 generals is unsolvable

Semantics

Semantics: structures

Formulae are well-formed, meaningless strings of symbols Structures give meaning to formulae

(in the very narrow sense of making them all either true or false)

S |= ϕ

Semantics: propositional structures

Propositional formula:

S |= p ∧ q

Need: 1.  a map S from variable names to T/F 2.  rules; e.g. S |= ϕ ∧ ψ iff S |= ϕ and S |= ψ

Semantics: first-order structures

First-order formula:

S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me)

Need: 1.  S assigns “records” to dog, big and likes. 2.  Rules; e.g. S |= ∀xφ iff for all d ∈ |S|, S[x := d] |= φ

Semantics: first-order structures

•  First-order logic:

S |= ∀x, dog(x) ⇒ big(x) ∧ likes(x, me)

dog

Rex

Fido

Rover

big

Rex

Fido

me

likes

Rex me

Fido me

Rover me

me me

couple good papers about using FO logic to program distributed systems

Semantics – modal logic

S |= (£¬p) ∧ (q ⇒ ¯r) Need: a structure that can interpret the propositional formulae under different modalities Kripke structure: (W, π, R) •  W is a set of worlds •  For each element of W, π is a propositional structure •  R is an accessibility relation among elements of W

S1 S3


Temporal logic S |= (£¬p) ∧ (q ⇒ ¯r)

q r

r q

S1 S3

S2

Kripke structure: (W, π, R)


Epistemic logic S |= r ∧ ¬Kir ∧ Ki(Kjr or Kj¬r) ∧ Kjr ∧ ¬Kj¬Kir

q r

r q

S1 S3

S2 i j

Kripke structure: (W, π, Ri)

a model of distributed systems

(r,t)

p1 p2 p3 p4 Idealized time

} h(p4,r,t)

A run r ∈ R

Knowledge-based interpretations

Knowledge interpretation: I = (R, π, {v1,v2,[..]}) Knowledge point: (I, r, t) R – a set of runs π – assigns a truth assignment to propositions for each point in R vi – A view function for R for some agent i (determined by h)

Kripke structure: (W, π, R)

choose-your-own-adventure

•  If you’d like to gloss over the proof and skip to other applications of knowledge, turn to page 62

•  If you’d like to dive into the weeds, turn to page 54.

Truth in a knowledge interpretation

(I,r,t) |= Cφ iff (I,r’,t’) |= Ekφ for all k Fixed point axiom: Cφ = E(φ ∧ Cφ) Induction rule: From φ ⇒ E(φ ∧ ψ) infer φ ⇒ Cψ

communication is not guaranteed

NG1: For all runs r and times t, there exists a run r’ extending (r,t) such that […] no messages are received in r’ at or after time t. NG2: If in run r processor pi does not receive any messages in the interval (t’,t), then there is a run r’ extending (r,t’) such that […] h(pi,r,t’’) = h(pi,r’,t’’) for all t’’ < t, and no processor pj != pi receives a message in r’ in the interval (t’,t).

Lemma 1

If, in two different runs (r and r’) of the same protocol, some h(p, r, t) = h(p, r’, t), then

(I, r, t) |= Cφ iff (I, r’, t) |= Cφ Sorry, no proof today!

Common knowledge is not attainable in a system in which communication is not guaranteed

Take runs r and r- in R, with the same initial configuration, s.t. no messages are received in r- up till time t. Then (I,r,t) |= Cφ iff (I,r-,t) |= Cφ. Proof (by induction on d(r)*): •  Base case: d(r)=0. h(p1,r,t) = h(p1,r-,t). By Lemma

1, (I,r,t) |= Cφ iff (I,r-,t) |= Cφ.

* d(r) is the number of messages received in run r.


Inductive case: d(r) = k+1. Let: •  t’ < t -- the latest time a message is received in r before t. •  pj -- a processor that received a message at t’ •  pi –a processor (!= pj)

By NG2, there is a run r’ extending (r,t’) s.t. h(pi,r,t’’)=h(pi,r’,t’’) for all t’’ <= t, and all processors (besides pi) receive no messages in the interval (t’, t). By construction, d(r’) <= k, so by the IH (I,r’,t) |= Cφ iff (I,r-,t) |= Cφ. But since h(pi,r,t) = h(pi,r’,t), by Lemma 1 (I,r’,t) |= Cφ iff (I,r,t) |= Cφ. So (I,r,t) |= Cφ iff (I,r-,t) |= Cφ. QED


Review: we showed that common knowledge cannot be gained (or lost) by exchanging messages.

Corollary: the 2 generals will never attack. But we still need to prove one more lemma: Any correct protocol for coordinated attack has the property that whenever the generals attack, it is common knowledge that they are attacking.

Lemma 2: coordinated attack requires common knowledge

Let ψ = the generals are attacking Assume the generals (A and B) attack at (r*, t*) – we show that (I,r*,t*) |= Cψ. Pick an arbitrary point (r,t). We show ψ ⇒ Eψ is valid in R. •  If (I,r,t) |= ψ, then the generals attack at (r,t). Consider (r’,t’), in

which A has the same history at (r,t). Since the protocol is deterministic (assumption), A must also attack in (r’,t’); since the protocol is correct, B does also, and so (I,r’,t’) |= ψ. It follows that (I,r,t) |= Eψ, so ψ ⇒ Eψ is valid in R.

•  If (I,r,t) |= ¬ψ, then trivially ψ ⇒ Eψ is valid in R. By the induction rule, ψ ⇒ Cψ is valid in R

Coup de grace

ψ = the generals are attacking 1.  By assumption, Cψ does not hold if no

messages are exchanged. 2.  By theorem 1, Cψ will never hold. 3.  By lemma 2, the generals cannot attack

unless Cψ.

Phew. but…?

Common knowledge is a prerequisite for agreement. Common knowledge is not attainable via protocol.

Halpern: These results may seem paradoxical.

Reality check

Fragile assumptions on which the proofs rest: •  Deterministic protocol •  Simultaneous agreement is necessary •  “Communication not guaranteed” •  Lack of useful a priori common knowledge

Bootstrapping common knowledge

•  The ``weakest failure detector’’ •  Spanner’s global clock •  Sequence wraparound

lower bounds for protocols [Hadzilacos, PODS’87]: A knowledge-theoretic analysis of atomic commitment protocols 1.  All of the variants of 2pc ((de-)centralized,

linear/nested, etc) are identical from a knowledge perspective

2.  All 2PC variants attain the minimum level of knowledge needed to commit

3.  3PC attains the minimum needed to commit without blocking

4.  Lower bound for messages: nested 2PC.

A good paper about automatically choosing cheap coordination mechanisms

protocol implementation / synthesis

•  Halpern and Fagin: knowledge-based programming [PODC’95] case of

K(Msg) and (KE(AckedMsg)) do deliver(Msg) K(Msg) and !KE(AckedMsg) do relay(Msg)

end

•  Matteo interlandi [Datalog2.0’11]: Knowlog: knowledge-enriched Dedalus

log(Tx_id,"abort")@next :-‐ Dvote(Vote,Tx_id),Vote=="no", par8cipants(X),transac8on(Tx_id,State),State=="vote-‐req".

A good paper about Dedalus

Monotonicity and knowledge

Monotonic: the more you know, the more you know.

Cϕ […]

E3ϕ

E2ϕ Eϕ Sϕ Dϕ ϕ

A good paper about monotonicity and distributed consistency

Remember

•  Knowledge is the dual of possibility •  Local knowledge dictates protocol

behavior •  The purpose of protocols is obtaining a

particular level of distributed knowledge •  Deep connections between semantic

structures and system behavior •  Common knowledge is unattainable via

protocol (but there is still hope)


E3ϕ

E2ϕ Eϕ Sϕ Dϕ ϕ

Knowledge Highway

E10ϕ 10 E100ϕ 100

Cϕ ∞

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