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Implementing Sequentially Consistent Programs on Processor Consistent Platforms Lisa Higham and Jalal Kawash University of Calgary, Canada American University of Sharjah, UAE

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Outline Memory consistency models –Sequential consistency –P-RAM –Coherence –PC-G Compiling from SC to PC-G –Good news –Bad news –Proof

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Multi-Processor’s Computation PPPP..…… O o1o1 o1o1 o1o1 o1o1 o2o2 o2o2 o2o2 o2o2 o3o3 o3o3 o3o3 o3o3 ……..…

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Sequential Consistency PPPP..…… Switch Shared Memory

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Sequential Consistency: Example PPP xy r(y) 4 r(x) 2 w(x, 2) w(y, 1) w(x, 3) w(y, 4) w(x, 3) w(y, 4) r(y) 4 w(x, 2) w(y, 1) r(x) 2 0000003421

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Sequential Consistency: Definition A computation is sequentially consistent iff a valid total order on O such that (O, ) (O, ) prog

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P-RAM Copies of Memory P P P P FIFO Channels xyzxyz xyzxyz

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P-RAM: Example w(x, 1) x 1 w(y, 2) y 2 P P P P PPPP r(y) 2 r(x) 0 r(x) 1 r(y) 0 w(y, 2)w(x, 1) 000021 000021 000021 000021 r(x) 1 r(y) 0 r(y) 2 r(x) 0 xy xy xy xy

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P-RAM Definition A computation is P-RAM iff for each process p, a valid total order such that (O|p O|w, ) (O|p O|w, ) Lp ∩ prog ∩ Lp

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Coherence x z y PPPP …...

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x 2y 3 Coherence: Example 00 x y P PPP w(x, 2) r(y) 0r(x) 0 w(y, 3) 23 w(x, 2) r(y) 0 r(x) 0 w(y, 3) P

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Coherence: Definition A computation is Coherent iff for each variable x, a valid total order such that (O| x, ) (O| x, ) progx x

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PC-G: P-RAM and Coherence P-RAM Coherence PC-G

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P-CG Definition A computation is P-CG iff for each process p, a valid total order such that –(O|p O|w, ) (O|p O|w, ) – processes q, and variable x: (O|w ∩ O|x, ) = (O|w ∩ O|x, ) Lp ∩ prog ∩ Lp Lq

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P-CG vs. SC Algorithms are designed for SC machines Some of them work directly when run on P-CG (e.g. Peterson 2) Most of the SC algorithms do not work on P-CG machines (e.g. test&set and Bakery algorithm)

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Can we transform an SC algorithm to an equivalent P-CG algorithm? Can we find a compiler that transforms any SC algorithm to an equivalent P-CG algorithm?

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Program Transformation and Interpretation Program P Program α(P) Transformation α Interpretation D C= {Computations of P on SC machines} D= {Computations of α(P) on M machines} E = {Interpretations of D on SC machines} C E Execute P on SC Execute α(P) on PC-G

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Program Implementation Program P Program α(P) Transformation α C E Interpretation D C= {SC Computations of P} D= {P-CG Computations of α(P)} E = {Interpretations of D} If program P, α implements P, then α is a compiler from SC to PC-G Execute P on SC Execute α(P) on PC-G

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Transformation Function α m: a new multi-writer variable Instructionα(Instruction) write(s y, val)write(m, id(y)); write(s y, val); write(m, id(y)) read(s y )

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Results Claim 1: – implements Lamport’s Bakery algorithm for 2 processes on PC-G Claim 2: – is a compiler from SC to PC-G for any program provided: Only 2 processes Only single-writer variables

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Transformation Example PP w(x, 1) r(y) w(y, 4) w(y, 2) r(x) Program Under SC: if r(y) returns 4, then r(x) returns 1 Possible PC-G Views PP w(x, 1) w(y, 4) r(y) 4 w(y, 2) w(y, 4) w(y, 2) r(x) 0 w(x, 1) Under PC-G: r(y) returns 4 and r(x) may return 0

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Transformation Example Program (Program) PP w(x, 1) r(x) w(y, 4) r(y) w(y, 2) w 4 (m, P) w 1 (m, P) w 2 (m, P) w 1 (m, P) w 2 (m, P) w 3 (m, P)

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Transformation Example P P r(x) r(x) must return 1 w(x, 1)w(y, 4) w(y, 2) View for w 1 (m, P) w 2 (m, P) w 3 (m, P) w 4 (m, P) w 1 (m, P) w 2 (m, P) If r(y) returns 4 w(x, 1)w(y, 4) w(y, 2) w 1 (m, P) w 2 (m, P) w 3 (m, P) w 4 (m, P) w 1 (m, P) w 2 (m, P) r(y)

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m m m m m m Proof Sketch PP w1w1 w w2w2..… m1m1 m5m5 m2m2 m3m3 m4m4 m6m6 Program PP PC-G Views … m1m1 m5m5 m2m2 m3m3 m4m4 m6m6 w w1w1 w2w2 System View … m1m1 m5m5 m2m2 m3m3 m4m4 m6m6 w w1w1 w2w2 ….. m2m2 m3m3 m4m4 m6m6 m1m1 m5m5

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Proof Sketch System view – Contains all reads and writes by both processes –Maintains program order –Is valid

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Summary Compiler: Only one additional variable Only writes to that variable Provided: Two processors Single writer variables

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Impossibilities For more than 2 processors, there is no compiler from SC to PC-G that: Only adds write instructions (with any number of variables) nor Uses only one additional variable (with any number of reads and writes)

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Pros and Cons Restricted –2 processes –Only single writer variables Valuable –ME Lamport’s Bakery algorithm –Wait-free test&set had no known solutions in weak memory consistency

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Wait-Free Test&set Define test&set if (s i = you and s j ≠ rst) the return 1 repeat s i choose case s j is: you, rst: s i me me : s i you choose: s i random (me, you) end case until (s i ≠ s j ) if (s i = me) then return 0 else return 1 m i Define reset s i rst

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Conclusions α works for any two-process program with single-writer variables α works for particular programs with > 2 processes (randomized wait-free n-process test&set) If there is a transformation that work for other cases, it must be more complicated: –Cannot be write-adding –Must use more than one additional variable

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Thank You?Thank You?

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m m m m m m Proof Sketch PP w1w1 w w2w2..… m1m1 m5m5 m2m2 m3m3 m4m4 m6m6 Program PP PC-G Views … m1m1 m5m5 m2m2 m3m3 m4m4 m6m6.. m2m2 m3m3 m4m4 m6m6 m1m1 m5m5 w w1w1 w2w2 System View … m1m1 m5m5 m2m2 m3m3 m4m4 m6m6 w w1w1 w2w2 …

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