The Weakest Failure Detector for Message Passing Set-Agreement

Abstract

Abstract. In the set-agreement problem, n processes seek to agree on at most n−1 different values. This paper determines the weakest failure detector to solve this problem in a message-passing system where processes may fail by crashing. This failure detector, called the Loneliness detector and denoted L, outputs one of two values, "true" or "false" such that:(1) there is at least one process where L outputs always "false", and (2) if only one process is correct, L eventually outputs "true" at this process.

Anonymous asynchronous systems: the case of failure detectors

“…It consequently sets sigma i to Π (line 14). Otherwise, p i computes a non-trivial value for sigma i from the set candidates (lines [15][16][17][18]. To that end, rank( ) is defined as the position of the index in the queue alive i (line 16).…”

“…When the set Π is the value of sigma i (line 14, the safety property is trivially satisfied. Hence, let us consider two processes p i1 and p i2 , such that the values of sigma i1 and sigma i2 have been computed at any time instants τ 1 and τ 2 , respectively (at lines lines [15][16][17][18].…”

“…Process p i then enters a waiting loop (lines [9][10][11][12][13][14][15][16][17][18][19], that (as we will see in the proof) it eventually exits at line 10 or 13 after having assigned a value to est2 i . The exit at line 10 is to prevent p i from blocking forever in phase 2 when processes have already progressed to phase 3 of the current round r i .…”

Chasing the Weakest Failure Detector for k-Set Agreement in Message-Passing Systems

“…The implementation of the operation propose() is made up of two sequential phases: a read phase followed by write phase. broadcast REQ_W(r, pst i , est i ); (11) repeat Q i ← qr i (12) until (∀j ∈ Q i : RSP_W(r, pst i , lre j , pos j , est j ) received from p j ) end repeat; (13) let rcv i = { lre j , pos j , est j : RSP_W(r, pst i , lre j , pos j , est j ) received}; (14) if ( [7] Succinct description of the algorithm: the read phase When it invokes propose(r, v), a process p i first broadcasts a read-request message (line 01) to (a) obtain information on values proposed in previous rounds (if any) and (b) learn if other processes have started higher rounds.…”

“…The Loneliness failure detector (denoted L) has been proposed in [12] where it is proved that it is the weakest failure detector for solving (n − 1)-set agreement in the asynchronous message-passing model with t = n − 1. Such a failure detector provides each process p with a boolean (that p can only read) such that the boolean of at least one process remains always false and, if all but one process crash, the boolean of the remaining process becomes and remains true forever.…”