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CIS 720. Correctness of Concurrent Programs. i := 1 max := A[1] do i < n i = i + 1 if (max < A[i]) max = A[i] od. Concurrency rule.

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## CIS 720

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**CIS 720**Correctness of Concurrent Programs**i := 1**max := A[1] do i < n i = i + 1 if (max < A[i]) max = A[i] od**Concurrency rule**for all i, { Pi } Si { Qi } { P1 /\ …. /\ Pn} co S1 // …. // Snoc { Q1 /\ …. /\ Qn}**x = 0; y = 1; x = 0;**co co x = 1 x = x + 1 // // y = y + 1 x = x + 2 oc oc**Interference freedom**• Let abe a statement and C be an assertion. • NI(a, C) iff { pre(a) /\ C } a { C } • Execution of a does not invalidates (or interferes) with C.**{ Pi } Si { Qi }, for all I, are interference free if**• For all i for all assertions C in proof outline of Si, for all actions a in Sj, i != j NI(a, C) holds**Concurrency rule**{ Pi } Si { Qi } are interference free { P1 /\ …. /\ Pn} co S1 // …. // Snoc { Q1 /\ …. /\ Qn}**x= 0;**co x = x + 1 // x = x + 2 od**Bank transactions**co // Transfer(x,y,am): Auditor: ac[x] = ac[x] – am; total = 0; i = 0; ac[y] = ac[y] + am do i < n total = total + ac[i] i = i + 1 od oc**Bank transactions**co // Transfer(x,y,am): Auditor: if < !auditon ac[x] = ac[x] – am; auditon = 1; total = 0; i = 0; ac[y] = ac[y] + am > [] do i < n <auditon && ( (x < i && y < i) ||(x > I && y > i)) total = total + ac[i] ac[x] = ac[x] – am; ac[y] = ac[y] + am> i = i + 1 fi od auditon = 0 oc**Techniques to avoid interference**• Disjoint variables • If the write set of each process is disjoint from the read and write set of other processes, then the proof outlines will be interference free.**Avoiding interference**• Weakened assertions x= 0; co x = x + 1 // x = x + 2 od**Avoiding interference**• Global Invariants: Assume that every assertion in the proof outline of Pi is of the form I /\ L, where L refers to local variables of Pi or to variables that only Pi writes. - Then the proof outlines will be interference free.**Concurrency rule**for all i, { I} Si { I } { I} co S1 // …. // Sn { I}**Avoiding interference**• Synchronization

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