Conjunctive Predicates 1 Goals of the lecture: Conjunctive Predicates � Direct dep endency algo rithm � T ok en based decentralized algo rithm � Channel Predicates Reference: Chapter 5. � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 2 Algo rithm fo r application p ro cess P i � Assume fully connected net w o rk � Mattern's vecto r clo ck � Notation: � ( i; k ) : the k th state on p ro cess P (o r simply k ) i � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 3 Monito r Pro cesses fo r W CP � Monito r p ro cesses resp onsible 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. � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 4 Info rmal Description � A tok en is sent to a p ro cess P only when the current cut i is not consistent. Sp eci�cally , when current state from P i happ ened b efo re some other state in the candidate cut. � Once the monito r p ro cess fo r P has eliminated the current i 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 o r i � the W CP is detected. � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 5 T ok en � A monito r p ro cess is active only if it has the tok en. � tok en consists of t w o vecto rs G and col or . � G is a global state vecto r rep resents the candidate global cut � G [ i ] = k indicates that state ( i; k ) is pa rt of 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 olor , indicates which states have b een eliminate d. � If col or [ i ] = r ed then state ( i; G [ i ]) has b een eliminated and can never satisfy the global p redicate. � If col or [ i ] = g r een , then there is no state in G such that ( i; G [ i ]) happ ened b efo re that state. � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 6 Monito r Pro cess Algo rithm v ar candidate:arra y[1..n] of in teger; on 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 � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 7 Co rrectness of W CP Detection Algo rithm The algo rithm co rrectly detects the �rst cut that satis�es a W CP. Lemma 1 F or any i , 1. G [ i ] 6 = 0 ^ col or [ i ] = r ed ) 9 j : j 6 = i : ( i; G [ i ]) ! ( j; G [ j ]) ; 2. col or [ i ] = g r een ) 8 k : ( i; G [ i ]) 6! ( k ; G [ k ]) ; 3. ( col or [ i ] = g r een ) ^ ( col or [ j ] = g r een ) ) ( i; G [ i ]) k ( j; G [ j ]) . 4. If ( col or [ i ] = r ed ) , then ther e is no glob al cut satisfying the WCP which includes ( i; G [ i ]) . � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 8 Analysis of Single-T ok en W CP Algo rithm � time complexit y: the total computation time fo r all p ro cesses 2 is O ( n m ) � Every time a state is eliminate d, O ( n ) w o rk is p erfo rmed � There a re at most mn states. � Message complexit y: the total numb er of messages O ( mn ) . � the tok en is sent at most mn times. � each monito r receives at most m messages from its applicati on p ro- cess. 2 � Communication bit complexit y: O ( n m ) . � size of b oth 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 bu�er fo r holding messages � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 9 Channel Predicates � A channel p redicate: any b o olean function of the accumula- tion of send and receive events on that channel. � Only uni-directional channels s; t : states at di�erent p ro cesses. s:send [ t:p ] : string of all messages sent at o r b efo re state s from s:p to t:p . t:r eceiv ed [ s:p ] : string of all messages received at o r b efo re state t from t:p to s:p . The channel p redicate can then b e written as: c ( s:send [ t:p ] ; t:r eceiv ed [ s:p ]) j o r in sho rt notation as: c ( S; R ) � c ( s:send [ t:p ] ; t:r eceiv e [ s:p ]) j j � Requirements fo r monotonicit y � Vija c y K. Ga rg Distributed Systems Sp ring 96
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 */ � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 11 GCP-cuts C : global cuts that satisfy a GCP with monotone channel p red- icates � C � D i� 8 i : C [ i ] � D [ i ] . W e sho w that the concept of �rst cut that satis�es a GCP is w ell-de�ned. Theorem 2 If C ; D 2 C , then their gr e atest lower b ound is also in C . Pr o of: � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 12 Example: no �rst cut in general p redicate: There a re an o dd numb er of messages in the channel. true only at p oints C [1] and D [1] fo r P , and C [2] and D [2] fo r 1 P . 2 the GCP is true in the cut C and D but not in their greatest lo w er b ound. Figure 2: consisten t cuts satisfying a GCP is not a lattice. C[1] D[1] � Vija c y K. Ga rg Distributed Systems Sp ring 96 C[2] D[2]
Conjunctive Predicates 13 Non-check er p ro cess algo rithm initial l y 8 j : j 6 = i :lcmvecto r[ j ] = 0; lcmvecto r[ i ] = 1; �rs�ag = true;incsend = increcv = ; ; F or sending m do send (lcmvecto r, m); lcmvecto r[ i ]++ ; �rst�ag:=true; incsend:= incsend � m; Up on receive (msg lcmvecto r, m) do lcmvecto r:=max (lcm vecto r, msg lcmvecto r); �rst�ag:=true; increcv:= increcv � m; Up on (lo cal p red = true) ^ �rst�ag do send (lcmvecto r,incsend,increcv) to check er ; �rst�ag := false; incsend:=increcv:= ; ; � Vija c y K. Ga rg Distributed Systems Sp ring 96
Conjunctive Predicates 14 Data Structures of the Check er Pro cess - p er-p ro cess data � cut:a rra y[1..n] of struct v:vecto r of integer; colo r:red, green � The colo r of a state is either red o r green. green: the current state is concurrent with the current states from all other green p ro cesses. red: the current state cannot b e pa rt of a GCP cut � A FIF O queue of successive lo cal snapshots from this p ro cess. � q:a rra y[1..n] of queues of struct � v:vecto r of integer; � incsend:a rra y[1..n] of sequences of messages; � increcv:a rra y[1..n] of sets of messages; � Vija c y K. Ga rg Distributed Systems Sp ring 96
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 of messageinfo; 2. A p ending-receive list: o rdered list of message sequence num- b ers. R[i,j]: sets of messageinfo ; 3. A CP-state �ag. V alue of channel p redicates � T (true) only if the channel p redicate fo r that channel is true fo r the current cut � F (false) only 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 o wn) at any time. cp[i,j]:X,F,T � Vija c y K. Ga rg Distributed Systems Sp ring 96
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