[PPT] - Conjunctive Predicates 1 Goals of the lecture: Conjunctive PowerPoint Presentation

SLIDE 1 Conjunctive Predicates 1 Goals

f

the lecture: Conjunctive Predicates

Direct

dep endency algo rithm

T
k

en based decentralized algo rithm

Channel

Predicates Reference: Chapter 5. c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 2 Conjunctive Predicates 2 Algo rithm fo r application p ro cess P i

Assume

fully connected net w

rk
Mattern's

vecto r clo ck

Notation:
(i;

k ): the k th state

n

p ro cess P i (o r simply k ) c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 3 Conjunctive Predicates 3 Monito r Pro cesses fo r W CP

Monito

r p ro cesses resp

nsible

fo r sea rching fo r a W CP cut.

The

tok en sto res a candidate cut.

The

tok en also sto res info rmation to determine whether the candidate cut is consistent. c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 4 Conjunctive Predicates 4 Info rmal Description

A

tok en is sent to a p ro cess P i

nly

when the current cut is not consistent. Sp ecically , when current state from P i happ ened b efo re some

ther

state in the candidate cut.

Once

the monito r p ro cess fo r P i has eliminated the current state,

receive

a new state from the applicatio n p ro cess

check

fo r consistency conditions again.

This

p ro cess is rep eated until

all

states a re eliminate d from some p ro cess P i

r
the

W CP is detected. c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 5 Conjunctive Predicates 5 T

k

en

A

monito r p ro cess is active

nly

if it has the tok en.

tok

en consists

f

t w

vecto

rs G and col

r

.

G

is a global state vecto r rep resents the candidate global cut

G[i]

= k indicates that state (i; k ) is pa rt

f

the current cut.

W

e maintain the inva riant that G[i] = k implies that any global cut C with (i; s) 2 C and s < k cannot satisfy the W CP.

c
lor

, indicates which states have b een eliminate d.

If

col

r

[i] = r ed then state (i; G[i]) has b een eliminated and can never satisfy the global p redicate.

If

col

r

[i] = g r een, then there is no state in G such that (i; G[i]) happ ened b efo re that state. c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 6 Conjunctive Predicates 6 Monito r Pro cess Algo rithm v ar candidate:arra y[1..n]

f

in teger;

n

receiving the tok en (G,color) while (color[i] = red) do receiv e candidate from application pro cess P i if (candidate.v clo c k[i] > G[i]) then G[i] := candidate.v clo c k[i]; color[i]:=green; endwhile for j 6= i: if (candidate.v clo c k[j] > G[j]) then G[j] := candidate.v clo c k[j]; color[j]:=red; endif endfor if (9 j: color[j] = red) then send tok en to P j else detect := true; Figure 1: Monitor Pro cess Algorithm c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 7 Conjunctive Predicates 7 Co rrectness

f

W CP Detection Algo rithm The algo rithm co rrectly detects the rst cut that satises a W CP. Lemma 1 F

r

any i, 1. G[i] 6= ^ col

r

[i] = r ed ) 9j : j 6= i : (i; G[i]) ! (j; G[j ]); 2. col

r

[i] = g r een ) 8k : (i; G[i]) 6! (k ; G[k ]); 3. (col

r

[i] = g r een)^(col

r

[j ] = g r een) ) (i; G[i])k(j; G[j ]). 4. If (col

r

[i] = r ed), then ther e is no glob al cut satisfying the WCP which includes (i; G[i]). c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 8 Conjunctive Predicates 8 Analysis

f

Single-T

k

en W CP Algo rithm

time

complexit y: the total computation time fo r all p ro cesses is O (n 2 m)

Every

time a state is eliminate d, O (n) w

rk

is p erfo rmed

There

a re at most mn states.

Message

complexit y: the total numb er

f

messages O (mn).

the

tok en is sent at most mn times.

each

monito r receives at most m messages from its applicati

n

p ro- cess.

Communication

bit complexit y: O (n 2 m).

size
f

b

th

the tok en and the candidate messages is O (n).

space

complexit y: O (mn) space is required b y the algo rithm fo r every p ro cess.

the

buer fo r holding messages c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 9 Conjunctive Predicates 9 Channel Predicates

A

channel p redicate: any b

lean

function

f

the accumula- tion

f

send and receive events

n

that channel.

Only

uni-directional channels s; t: states at dierent p ro cesses. s:send[t:p]: string

f

all messages sent at

r

b efo re state s from s:p to t:p. t:r eceiv ed[s:p]: string

f

all messages received at

r

b efo re state t from t:p to s:p. The channel p redicate can then b e written as: c j (s:send[t:p]; t:r eceiv ed[s:p])

r

in sho rt notation as: c j (S; R )

c

j (s:send[t:p]; t:r eceiv e[s:p])

Requirements

fo r monotonicit y c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 10 Conjunctive Predicates 10 Examples Example 1 Empty channels: len(S)=len(R): This sa ys that if a channel p redicate is false, then it cannot b e made true b y sending mo re messages without receiving mo re messages. Example 2 Nonempty channels: (ns > nr ): (ns

nr

> nk ) /* at least k messages in the channel */ c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 11 Conjunctive Predicates 11 GCP-cuts C : global cuts that satisfy a GCP with monotone channel p redicates

C
D

i 8i : C [i]

D

[i]. W e sho w that the concept

f

rst cut that satises a GCP is w ell-dened. Theorem 2 If C ; D 2 C , then their gr e atest lower b

und

is also in C . Pr

f:

c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 12 Conjunctive Predicates 12 Example: no rst cut in general p redicate: There a re an

dd

numb er

f

messages in the channel. true

nly

at p

ints

C [1] and D [1] fo r P 1 , and C [2] and D [2] fo r P 2 . the GCP is true in the cut C and D but not in their greatest lo w er b

und.

C[1] D[1] D[2] C[2]

Figure 2: consisten t cuts satisfying a GCP is not a lattice. c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 13 Conjunctive Predicates 13 Non-check er p ro cess algo rithm initial l y 8j : j 6= i :lcmvecto r[j ] = 0; lcmvecto r[i] = 1; rsag = true;incsend = increcv = ;; F

r

sending m do send (lcmvecto r, m); lcmvecto r[i]++ ; rstag:=true; incsend:= incsend

m;

Up

n

receive (msg lcmvecto r, m) do lcmvecto r:=max (lcm vecto r, msg lcmvecto r); rstag:=true; increcv:= increcv

m;

Up

n

(lo cal p red = true)^ rstag do send (lcmvecto r,incsend,increcv) to check er ; rstag := false; incsend:=increcv:=;; c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 14 Conjunctive Predicates 14 Data Structures

f

the Check er Pro cess

p

er-p ro cess data

cut:a

rra y[1..n]

f

struct v:vecto r

f

integer; colo r:red, green

The

colo r

f

a state is either red

r

green. green: the current state is concurrent with the current states from all

ther

green p ro cesses. red: the current state cannot b e pa rt

f

a GCP cut

A

FIF O queue

f

successive lo cal snapshots from this p ro cess.

q:a

rra y[1..n]

f

queues

f

struct

v:vecto

r

f

integer;

incsend:a

rra y[1..n]

f

sequences

f

messages;

increcv:a

rra y[1..n]

f

sets

f

messages; c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 15 Conjunctive Predicates 15 P er-Channel Data three data structures fo r each channel: 1. A p ending-send list: messages sent but not y et received S[i,j]: sequence

f

messageinfo; 2. A p ending-receive list:

rdered

list

f

message sequence num- b ers. R[i,j]: sets

f

messageinfo ; 3. A CP-state ag. V alue

f

channel p redicates

T

(true)

nly

if the channel p redicate fo r that channel is true fo r the current cut

F

(false)

nly

if the channel p redicate fo r that channel is false fo r the current cut. The CP-state ag can tak e the value X (unk

wn)

at any time. cp[i,j]:X,F,T c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 16 Conjunctive Predicates 16 F

rmal

description S[1..n,1..n], R[1..n,1..n] : sequence

f

message; cp[1..n,1..n] : fX, F, Tg; cut : arra y[1..n]

f

struct f v : v ector

f

in teger; color : fred, greeng; incsend, increcv : sequence

f

messages g initially cut[i].v = 0; cut[i].color = red; S[i,j], R[i,j] = ;; rep eat while (9 i : (cut[i].color = red) ^ (q[i] 6= ;)) cut[i] := receiv e(q[i]); pain t-state(i); up date-c hannels(i); endwhile if (9 i,j : cp[i,j] = X ^ cut[i].color = green ^ cut[j].color = green) then cp[i,j] := c hanp(S[i,j]); if (cp[i,j] = F) then if (send-mono(i,j)) cut[j].color := red; else cut[i].color := red; /* receiv e-mono(i,j) */ un til (8 i : cut[i].color = green) ^ (8 i,j : cp[i,j] = T) detect := true; c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 17 Conjunctive Predicates 17 Up date Channels up date-c hannels(i) for (j : cut[i].incsend[j] 6= ;) do S := S[i,j]; R := R[i,j]; S[i,j] := S

(cut[i].incsend[j]
R

); R[i,j] := R

cut[i].incsend[j];

if (: send-mono(i,j) _ cp[i,j] = T ) cp[i,j] := X; for (j : cut[i].increcv[j] 6= ;) do S := S[j,i]; R := R[j,i]; R[j,i] := R

(cut[i].increcv[j]
S

); S[j,i] := S

cut[i].increcv[j];

if (: recv-mono(j,i) _ cp[j,i] = T ) cp[j,i] := X; c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 18 Conjunctive Predicates 18 Overhead analysis

Time

complexit y:

any

state is compa red to at most n

ther

states.

There

a re mn states in all. Therefo re, mn 2 compa risons

at

most t w

evaluations
f

the p redicate p er message.

at

most 2mn message send and receive events.

each

p redicate evaluation tak es at most c time units, The total time sp ent is 2mnc.

Space

complexit y

n

queues each with at most m elements. assume that comp

nent
f

each vecto r and every message: a constant numb er

f

bits.

Therefo

re, fo r each queue: O (mn).

Summing

up all incremental channel histo ries, w e get O (m).

T
tal

space required b y the check er p ro cess is O (mn 2 ). c Vija y K. Ga rg Distributed Systems Sp ring 96

SLIDE 19 Conjunctive Predicates 19

Message

Complexit y Every p ro cess sends at most m messages to the check er p ro cess. Using same assumptions (space complexit y): O (mn) bits sent b y each p ro cess. c Vija y K. Ga rg Distributed Systems Sp ring 96