Distributed Algorithms: Asynch R/W SM Computability Eli Gafni, UCLA Summer Course, CRI, Haifa U, Israel
Computational Models • Serial Van-Neuman – Turing Machines, RAM – PRAM • Interleaving – ND Turing Machine - exits a paths – Asynch Distributed - for all paths
Message-Passing Interleaving • N processors each with own program of the type “upon receiving message X in state Z do Y”. • Y may involve sending messages to certain other processors. • Special message type ``start’’ that can be received (or acted upon) only as first message.
MP Cont’ed • ``Configuration’’ - messages in the ``ether’’ distained to processors + states of processors. • ``next Configuration’’ - choose a messge from the ``ether’’ and deliver it to its destination processor. Operate the ``Upon’’ procedure. Change processor’s state, and place new messages in the ``ether’’. • Initial configuartion - Start message in ``ether’’ to each processor. Processors in intial state.
Full-Information • History of a processor determines its state. • History contains more info than ``state. ’’ • Processors’ programs are ``common knowledge. ’’ • W. l. o. g when interested in computability rather than efficiency - message is the history of the sender.
Computability (Problems/Tasks) • Each processor eventually halts with ``output’’. • Some n-tuple outputs are valid some are not. • To prevent ``default’’ solutions the output tuples are parametrized by the set of processors that ``started. ’’
Fail-Free Model • Every message eventually delivered, and all processors eventually respond. • All problems are solvable: – Receive/send message to/from all – Determine the set of ``starters’’ – Apply default output to that set of starters
Faults • Communication: ``channel’’ goes down. • Processor: Fail-Stop, Byzantine. • Many other faults possible: Discuss.
Communication • Synchronous: Proceed by ``rounds’’. Every message sent at the beginning of a round is received by the end of the round. • Assume 2 processors/ 2 channels and one of the channels may fail-stop to deliver
Consensus • Proc’s vote ``attack/retreat. ’’ • If both vote same and no communication failure both eventually decide their vote. • Else they decide both either ``attack’’ or ``retreat. ’’
2 procs synch cons w/ single channel failure in a round: Impossible. • Cannot do it with no communication • Cannot do it in 1 round: – A_1 donot receive must decide A since its view is compatible with A_2 no fault run. – Similarly R_2. – Say w. l. o. g R_2/A_1 no fault is R – Fail the message to A_1 to get contradiction.
Impossibility Cont’ed • In general A_1 has to decide A comm with A_2 whether it receive or does not receive the last message. • It must have commited to A at the end of the previous round, so does A_2, so the last round is not necessary, Contradiction.
N>2 • No alg for even N when N-1 channels may fail - one of the 2 procs emulate N-1 procs and ine emulates the remaining one. • Alg for less then N-1 channels: – – Send input to all, receive Send all that received to all, receive Until have heard input of all: decide. (prove liveness…)
From Synch to ``Asynch’’ • What if the faults are less N-1 in each round, but the set of faults may ``jump around’’? • Correctness of the prev alg does not depend on the faults being static.
Proc fail-stop failure • What if synch, no comm failure, but proc fail-stop. • What if t at most can fail-stop? • In t+1 rounds there is a ``clean’’ round that the view of all inputs is shared by all.
What if single failure that ``Jumps’’ • In each round some or all messages from a SINGLE proc may not be delivered. • If N=2 obviously cannot do cons since can emulate comm failure. • What if N=3? ---- suspense.
SWMR Async Shared Memory • n procs p_0, …, p_n and n cells C_1, …, C_n. • Proc p_k writes to C_k and can read all cells one at a time. • Configuration: each proc at a state that enable writing its cell, or reading some cell. • A proc is chosen it reads or writes, changes state, and another is chosen, etc.
Properties of SM • If all procs write and then read all cells in arbitrary order, then each p_k returns the set of procs S_k it has seen write: – What is the property of the sets that makes them realization of SM execution? – At least one sees all, continue inductively – Fat Immediate Snapshots
Immediate Snapshots • p_k in S_k (proc reads itself) • p_k subseteq p_j or vice versa • If p_j in S_k then S_j subseteq S_k • Can you implement IS in SWMR SM? • Atomic Snap is I without the last property • IS subseteq AS
How do you Take AS in the middle of Comp rather than as a task? • Use sequence numbers with each new value written, double sacn until success • AS as a model - the read operation returns the whole memory rather than a single cell.
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