[PPT] - Intro duction 1 Goals of the lecture: W eak Conjunctive PowerPoint Presentation

SLIDE 1 Intro duction 1 Goals

f

the lecture: W eak Conjunctive Predicates

Logic

fo r global p redicates

W

eak conjunctive algo rithm References: Garg and W aldec k er 94 c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 2 Intro duction 2 Logic fo r Global Predicates

Three

syntactic catego ries in the logic

b
l,

lin and form.

Syntax:

fo rm ::= A: lin j E: lin lin ::= 3 lin j lin ^ lin j :lin j b

l

b

l

::= a p redicate

ver

a global state

bool

: b

lean

exp ression dened

n

a single global state

f

the system.

Example:

if the global state has (x = 3; y = 6), then the bool (x

y

) is true.

l

in is a temp

ral

fo rmula dened

ver

a global sequence.

bool

is true in g if it is true in the last state

f

g .

3

lin: there exists a p rex

f

g such that lin is true fo r the p rex.

2

and _: duals

f

3 and ^.

A

form is dened fo r a run and it is simply a lin qualied with universal (A :)

r

existential (E :) quantier. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 3 Intro duction 3 Semantics g j = bool i g m j = bool , g m = last state in g g j = :l in i :g j = l in g j = 3l in i 9i : g i j = l in g j = l in 1 , ! l in 2 i 9i < j : g i j = l in 1 ^ g j j = l in 2 r j =A:l in i 8g : g 2 l inear (r ) : g j = l in r j =E:l in i 9g : g 2 l inear (r ) : g j = l in

str
ng

p redicate: fo rmulas sta rting with A

we

ak p redicate: fo rmulas sta rting with E . c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 4 Intro duction 4 W eak Conjunctive Predicates A w eak conjunctive p redicate (W CP) is true fo r a given run if and

nly

if there exists a global sequence consistent with that run in which all conjuncts a re true in some global state.

useful

fo r bad

r

undesirable p redicates

Example:

the classical mutual exclusion p roblem.

detect

erro rs that ma y b e hidden in some run due to race conditions. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 5 Intro duction 5 Imp

rtance
f

W eak Conjunctive Predicates

Sucient

fo r detection

f

any b

lean

exp ression

f

lo cal p redicates.

Example

x; y and z a re in three dierent p ro cesses. Then, E:3ev en(x) ^ ((y < 0) _ (z > 6))

E:3(ev

en(x) ^ (y < 0))_ E :3(ev en(x) ^ (z > 6))

the

global p redicate is satised b y

nly

a nite numb er

f

p

ssible

global states.

Example,

E : 3(x = y ), x and y a re in dierent p ro cesses. (x = y ) is not a lo c al p redicate Assume that x and y can

nly

tak e values f0; 1g.

A

: 2bool can b e easily detected, why ? c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 6 Intro duction 6 Conditions fo r W eak Conjunctive Predicates

LP

i : a lo cal p redicate in the p ro cess P i

LP

i (s): the p redicate LP i is true in the state s.

s

2 r [i]: s

ccurs

in the sequence r [i]. can assume k

n

b ecause LP i ^ LP j is just another lo cal p redicate if LP i and LP j b elong to the same p ro cess. Theorem 1 E : 3(LP 1 ^ LP 2 ^ : : : LP k ) is true for a run r i for al l 1

i
m

9s i 2 r [i] such that LP i is true in state s i , and s i and s j ar e inc

mp

ar able for i 6= j . d d d d d t t t d t A consisten t cut

s

1 s 2 s k c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 7 Intro duction 7 Detection

f

W eak Conjunctive Predicates: Centralized Algo rithm

One

p ro cess serves as a check er.

Other

p ro cesses : non-check er p ro cesses. Each non-check er p ro cess maintains its lo cal l cmv ector (last causal message vecto r).

F
r

P j , lcmvecto r[i] (i 6= j ) is the message id

f

the most recent message from P i (to anyb

dy)

which has a causal relationship to P j .

lcmvecto

r[j] fo r the p ro cess P j is the next message id that P j will use. e e e u u u e u Lo cal Snapshot

@

@ @ @ R @ @ @ @ R @ @ @ @ R U . . . . . . . . . c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 8 Intro duction 8 Algo rithm: Non-check er p ro cesses v ar lcmvecto r: arra y [1..n]

f

integer; init 8i : i 6= id :lcmvecto r[i] = 0; lcmvecto r[id] = 1; rstag: b

lean

init true; lo cal p red: Bo

lean

Exp ression; 2 F

r

sending do send (p rog, lcmvecto r, : : : ); lcmvecto r[id]++ ; rstag:=true; 2 Up

n

receive (p rog, msg lcmvecto r, : : : ) do 8i :lcmvecto r[i]:=m ax (lcm vecto r[i ], msg lcmvecto r[i]); 2 Up

n

(lo cal p red = true)^ rstag do rstag := false; send (dbg, lcmvecto r) to the check er p ro cess; c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 9 Intro duction 9 Optimizatio n Sucient to send the lcmvecto r

nce

after each message is sent irresp ective

f

the numb er

f

messages received.

l
cal

(s) denote that the lo cal p redicate is true in state s.

f

ir st(s): the lo cal p redicate is true fo r the rst time since the most recently sent message.

w

cp(s 1 ; s 2 ; :::; s m ): if s 1 ; s 2 ; :::s m a re the states in dierent p ro cesses making the w cp true. Theorem 2 9s 1 ; :::; s m : w cp(s 1 ; s 2 ; :::s m ) , h9s 1 ; ::; s m : w cp(s 1 ; s 2 ; :::; s m ) ^ 8i : 1

i
m

: f ir st(s i )i

@

@ R @ @ R

P

j s i s j s i s j P i c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 10 Intro duction 10 Complexi t y

Space

complexit y: the a rra y l cmv ector and is O (n).

message

complexit y is O (m s ) where m s is the numb er

f

p rogram messages sent.

In

addition, p rogram messages have to include time vecto rs.

Time

complexit y

detection
f

lo cal p redicates

maintain

time vecto rs (O (n)=message ). c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 11 Intro duction 11 Check er Pro cess

Incoming

debug messages from p ro cesses a re enqueued in the app rop riate queue.

assume

that the check er p ro cess gets its message from any p ro cess in FIF O. Lemma 1 If the lcmve ctor at the he ad

f
ne

queue is less than the lcmve ctor at the he ad

f

any

ther

queue, then the smal ler lcmve ctor may b e eliminate d fr

m

further c

nsider

ation in che cking to se e if the WCP is satise d.

Non-checker Process 1 Non-checker Non-checker Process 2 Process N queue 1 queue N Process Checker Checker

c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 12 Intro duction 12 F

rmal

Description v ar q 1 : : : q m : queue

f

lcmvecto r; changed, new changed: set

f

f1,2,...,mg 2 Up

n

recv(elem) from P k do insert(q k , elem); if (head(q k ) = elem) then b egin changed := f k g; while (changed 6= ) b egin new changed := fg; for i in changed, and j in f1,2,...,m gdo if (:empty (q i ) ^ :empty (q j )) then b egin c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 13 Intro duction 13 F

rmal

Description [Contd.] if head(q i ) < head(q j ) then new changed:=new changed [ fig; if head(q j ) < head(q i ) then new changed:=new changed [ fjg; end; /* if */ changed := new changed; for i in changed do deletehead(q i ); end;/* while */ if 8i : :empty (q i ) then found:=true; end; /* if */ chang ed: the set

f

indices fo r which the head

f

the queues have b een up dated. Theorem 3 The ab

ve

algorithm r e quir es at most O (m 2 p) c

mp

arisons. c Vija y K. Ga rg Distributed Systems F all 94

SLIDE 14 Intro duction 14 Lo w er Bounds Theorem 4 A ny algorithm which determines whether ther e exists a set

f

inc

mp

ar able ve ctors

f

size m in m chains

f

size at most p, makes at le ast pm(m

1)=2

c

mp

arisons. Pro

f:

Case 1 : p = 1 Case 2 : p > 1 c Vija y K. Ga rg Distributed Systems F all 94