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Common knowledge: application to distributed systems

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Caesar Ogole, Jan Gerard Gerrits, Harrie de Groot, Julius Kidubuka & Stijn Colen ... The Kripke Model M associated with a distributed system is. M= S, R1 .Rm where: ... – PowerPoint PPT presentation

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Title: Common knowledge: application to distributed systems


1
Common knowledge application to distributed
systems
  • Caesar Ogole, Jan Gerard Gerrits, Harrie de
    Groot, Julius Kidubuka Stijn Colen

2
Common Knowledge in Distributed Systems
  • Looking back to the definition
  • The Kripke Model M associated with a distributed
    system is
  • MltS, R1 ..Rmgt
  • where
  • S( S1 ..Sm Si is a local state of
    processor i)
  • p S?P?(t, f),
  • Ri (s, t), Si ti for i1....m

3
Some limiting properties of M
  • M does not contain any information about the
    actual state transformations (that the system
    executes or is subject to).
  • The actual process is determined by
  • The structure of the process
  • The way they are programmed
  • The protocols by which they communicate

4
Introducing the notion of a run of system
  • Epistemic logic is limited in the sense that it
    cannot express anything about the way in which a
    process comes about.
  • However, it is possible to describe processor
    knowledge using the concept of a run
  • A run in M is defined as
  • s(1), s(2) ?
  • (? is not to be confused with
    )
  • Our main interest in a run
  • Behaviour of some common knowledge during a run
    (given M)

5
Some prior knowledge
  • Consider the figure below
  • 1 Proposition
  • If we let s be a state in the Kripke Model M, and
    K the upward cone of s, then
  • (i) (M, s)Cf if (M, t)f for all t Ks
  • (ii) if Cf holds in s (i.e. (M, s) Cf) then Cf
    holds in the world of ks

6
Proof
  • (i) (M, s) Cf ? (M, t)f for all t with s ?gt t
    ? (M, t) f for all t Ks
  • (ii)(proof (or hint) to be given)
  • Next some more concepts

7
Definition (2.2.3)Strongly Connected
Let M ltS, p, R1, , Rmgt and ? be defined
as before.
Then M is called strongly connected if for all
s, t ? S it holds that s ? t. Meaning Every
state is reachable from every other state in 0 or
more steps
8
Model
R1
s0
s1
si ? S
9
Model
R1
s0
s1
Ri
si ? S ti ? S
ti
10
Connected
R1
s0
s1
Ri
si ? S ti ? S
ti
S ? t
11
Strongly connected
R1
s0
s1
Ri
si ? S ti ? S
ti
S ? t
12
Proposition (2.2.3.1)Connected Distributed
Systems
The Kripke model associated with a distributed
system, is strongly connected, if m gt 1.
R1
(0,0)
(0,1)
All states are reachable within 2 steps, because
of the strongly connected relations.
R2
R2
R1
(1,0)
(1,1)
13
Proof s ? t
Prove for any s,t ? S in the Kripke model of the
distributed system that s ? t holds. s
(s1,s2,,sm) , t (t1,t2,,tm)  s
(s1,s2,,sm)?(s1,t2,,tm)?(t1,t2,,tm) ?t
R1 Ri i ?1 Thus s ? t
14
Example Model with multiple dimensions
si lt0,1,1,0,0,1,1,0gt si1
lt1,1,1,0,0,1,1,0gt ti lt1,1,1,0,0,0,1,0gt Every
state is reachable within 2 steps
15
Theorem (2.2.4) General Result
Let M be a strongly connected Kripke model.
Suppose that for some state s and a formula f it
holds that (M,s) ? Cf. Then M ? Cf
16
Proof
  • IF (M,s) ? Cf THEN M ? Cf
  • because
  • f is true for all states in Ks
  • In a strongly connected system all s ? Ks

17
Corollary
  • Let M be a Kripke model associated with a
    distributed system with processors 1, , m, (m gt
    1)
  • (M, s) ? Cp s ? S
  • M ? Cp
  • Common knowledge is constant through every run of
    M (Julius)

because a Kripke model of a distributed system is
strongly connected
18
Example 1
Given the following distributed
system Processors A, B, C Local states 0, 1
(let P p, q)
Describe the Kripke Model M for this system,
along with a truth assignment such that
  1. M ? Cp
  2. There is a global state such that (M, s) ? Eq,
    but not M ? Eq

19
Possible Worlds
(0,1,1)
(1,1,1)
(1,0,1)
(0,0,1)
(0,1,0)
(1,1,0)
(0,0,0)
(1,0,0)
20
Description of the model
M ltS, p, RA, RB, RCgt S (x, y, z) x, y, z ?
0,1 where s (x1, y1, z1) and t (x2,
y2, z2) RA (s, t) ? RA ? x1 x2 RB (s, t) ? RB
? y1 y2 RC (s, t) ? RC ? z1 z2 p ?s ? S
p(s)(p) t p(s)(q) f ? s (1,1,1)
21
Questions
1. M ? Cp P is defined true everywhere, so we
have M ? Cp. 2. There is a global state such
that (M, s) ? Eq, but not M ? Eq If we choose s
(0,0,0), we have (M, s) ? Eq. Since q is false
in (1,1,1), we have M ? Eq
22
Example 2
  • Show that for any Kripke model M it holds that M
    ? f ? M ? Cf
  • Answer Suppose M ? f.Then in all s ? S, (M, s)
    ? f.But then f is true in all Rc-successors of
    each world let s and t ? S such that (s,t) ? Rc.
    Since f is true in all states of S, we have (M,
    t) ? f, and thus (M, s) ? Cf.

23
Counter example
  • Counter example of M ? f ? Cf
  • In first example (cube). (M (0,0,0)) ? q ?
    Cqand thus M ? q ? Cq.

24
Example Increasing common knowledge
  • Model M ltS, p, R1, R2, RE, RC gt obtained asS
    a, b p(x)(p) t iff x a and R1 R2
    (a, a), (b,b). In run a ? b its the case that
    the common knowledge about p increases
  • We have (M, a) ? Cp while (M, b) ? Cp

R1R2
R1R2
a p
b p
25
Some comments
  • We would expect common knowledge in distributed
    systems to increase by communication
  • Why not?
  • Hence the Kripke model loses the property of
    being strongly connected

26
Plausible solution
  • Consider Kripke models M ltS, p, R1,..,
    Rmgtwhere S is a subset of S1,S2,,Sm rather than
    (S S1 Sm )
  • The task at hand is to prove that C-knowledge is
    constant, hence

27
Definition 2.2.11
  • A run s(1) ? s(2) ? .
  • is called non-simultaneous if for every
  • transition s(k) ? s(k1) there exists
  • a processor 1 i m with si(k) si(k1)

28
Theorem 2.2.12
  • In non-simultaneous runs common knowledge is
    constant

29
Proof of Theorem 2.2.12
  • Suppose s ? s' for s (s1, s2, , sm) and s'
    (s1', s2, , sm) with si si', and
    consequently (s, s')?Ri , and suppose (M, s') ?
    Cf.
  • Now it holds that
  • (M, s') ? Cf ? (M, s') ? ECf ? (M, s') ? KiCf

30
.
  • Since Ri is an equivalence relation, then it
    holds that
  • (s, s')?Ri ? (s', s)?Ri
  • Using the definition of the semantics of the
    Ki-operator, we have
  • (M, s) ? Cf

31
.
  • From above, any C-knowledge present in s' is also
    present in s and vice versa as well
  • Hence, C-knowledge is constant at the
    non-simultaneous transition s ? s'
  • Then by induction, C-knowledge is also constant
    in a non-simultaneous run.

32
Co-ordinated Attack Problem
  • Two separated generals co-ordinating an attack
  • Cf (fattack at time x!) necessary
  • Messengers may be captured by enemy

General B
General A
Communication
Hostile army
33
Attaining Cf
  • f, Messenger f
  • KBf, messenger KBf
  • KAKBf, messenger KAKBf
  • Ad infinitum
  • Cf is never attained (in finite time)
  • Even without actual deletion or delay (common
    knowledge about deletion or delay is enough)
  • Each message adds only one level of knowledge

34
Proof by induction no finite amount of messages
is enough
  • 0 messages KBf
  • Inductive step, k messages insufficient Cf
  • If k1 suffice
  • k1s sender attacks without confirmation
  • k1 was apparently irrelevant
  • k should have sufficed
  • which contradicts the inductive hypothesis

35
Non-guaranteed communication
  • NG1 for all r and t, r exists extending (r,t)
  • r has same history and internal clock as r
  • r receives no messages on or after t
  • NG2 if in r, pi does not receive messages in
    (t, t)
  • r exists extending (r, t), with h(pi, r, t)
    h(pi, r, t) for all t lt t
  • no other processor pj receives message in r in
    t, t)

36
Consequence of NG1 NG2
  • If Cf can be attained by communication, Cf can be
    attained without communication
  • Since no k messages are enough, either is
    impossible in the current problem
  • Proof by induction follows

37
C without guaranteed communication (1)
  • Theorem
  • r run in R
  • d(r) amount of messages in r up to time t
  • r same run in R, no messages up to time t
  • (I, r, t) Cf ? (I, r, t) Cf
  • d(r) 0
  • h(p1, r, t) h(p1, r, t)
  • (I, r, t) Cf ? (I, r, t) Cf

38
C without guaranteed communication (2)
  • Assume hypothesis holds for all runs r with
    d(r) k
  • Assume d(r) k 1
  • t lt t is latest time of message reception in r
    before t
  • pj receives message at t in r
  • There is a run r for which h(pi, r, t)h(pi,
    r, t) for all t t
  • Other processor pk receives no messages in t,
    t)

39
C without guaranteed communication (3)
  • d(r) lt k
  • Inductive hypothesis, when d(r) k (I, r, t)
    Cf ? (I, r, t) Cf
  • Since h(pi, r, t) h(pi, r, t)
  • (I, r, t) Cf ? (I, r, t) Cf
  • Therefore (I, r, t) Cf ? (I, r, t) Cf

40
Possible solution
  • Problem
  • t gt n gt b gt a OR t gt n gt a gt b
  • Attack, I will attack once I am sure we both
    will.
  • Solution
  • t gt b gt n gt a OR t gt a gt n gt b
  • Attack, please ack, I will not re-ack.

41
Discussion
  • Does TCP protocol solve the problem?
  • Are there real-life equivalents of this problem?
  • With less strict requirements?
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