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@ -27,7 +27,7 @@ Explanation of the Linux-Kernel Memory Consistency Model
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19. AND THEN THERE WAS ALPHA
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20. THE HAPPENS-BEFORE RELATION: hb
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21. THE PROPAGATES-BEFORE RELATION: pb
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22. RCU RELATIONS: rcu-link, gp, rscs, rcu-fence, and rb
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22. RCU RELATIONS: rcu-link, rcu-gp, rcu-rscsi, rcu-fence, and rb
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23. LOCKING
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24. ODDS AND ENDS
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@ -1430,8 +1430,8 @@ they execute means that it cannot have cycles. This requirement is
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the content of the LKMM's "propagation" axiom.
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RCU RELATIONS: rcu-link, gp, rscs, rcu-fence, and rb
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----------------------------------------------------
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RCU RELATIONS: rcu-link, rcu-gp, rcu-rscsi, rcu-fence, and rb
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-------------------------------------------------------------
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RCU (Read-Copy-Update) is a powerful synchronization mechanism. It
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rests on two concepts: grace periods and read-side critical sections.
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@ -1446,17 +1446,19 @@ As far as memory models are concerned, RCU's main feature is its
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Grace-Period Guarantee, which states that a critical section can never
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span a full grace period. In more detail, the Guarantee says:
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If a critical section starts before a grace period then it
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must end before the grace period does. In addition, every
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store that propagates to the critical section's CPU before the
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end of the critical section must propagate to every CPU before
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the end of the grace period.
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For any critical section C and any grace period G, at least
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one of the following statements must hold:
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If a critical section ends after a grace period ends then it
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must start after the grace period does. In addition, every
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store that propagates to the grace period's CPU before the
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start of the grace period must propagate to every CPU before
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the start of the critical section.
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(1) C ends before G does, and in addition, every store that
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propagates to C's CPU before the end of C must propagate to
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every CPU before G ends.
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(2) G starts before C does, and in addition, every store that
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propagates to G's CPU before the start of G must propagate
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to every CPU before C starts.
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In particular, it is not possible for a critical section to both start
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before and end after a grace period.
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Here is a simple example of RCU in action:
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@ -1483,10 +1485,11 @@ The Grace Period Guarantee tells us that when this code runs, it will
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never end with r1 = 1 and r2 = 0. The reasoning is as follows. r1 = 1
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means that P0's store to x propagated to P1 before P1 called
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synchronize_rcu(), so P0's critical section must have started before
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P1's grace period. On the other hand, r2 = 0 means that P0's store to
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y, which occurs before the end of the critical section, did not
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propagate to P1 before the end of the grace period, violating the
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Guarantee.
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P1's grace period, contrary to part (2) of the Guarantee. On the
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other hand, r2 = 0 means that P0's store to y, which occurs before the
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end of the critical section, did not propagate to P1 before the end of
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the grace period, contrary to part (1). Together the results violate
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the Guarantee.
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In the kernel's implementations of RCU, the requirements for stores
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to propagate to every CPU are fulfilled by placing strong fences at
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@ -1504,11 +1507,11 @@ before" or "ends after" a grace period? Some aspects of the meaning
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are pretty obvious, as in the example above, but the details aren't
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entirely clear. The LKMM formalizes this notion by means of the
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rcu-link relation. rcu-link encompasses a very general notion of
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"before": Among other things, X ->rcu-link Z includes cases where X
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happens-before or is equal to some event Y which is equal to or comes
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before Z in the coherence order. When Y = Z this says that X ->rfe Z
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implies X ->rcu-link Z. In addition, when Y = X it says that X ->fr Z
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and X ->co Z each imply X ->rcu-link Z.
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"before": If E and F are RCU fence events (i.e., rcu_read_lock(),
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rcu_read_unlock(), or synchronize_rcu()) then among other things,
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E ->rcu-link F includes cases where E is po-before some memory-access
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event X, F is po-after some memory-access event Y, and we have any of
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X ->rfe Y, X ->co Y, or X ->fr Y.
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The formal definition of the rcu-link relation is more than a little
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obscure, and we won't give it here. It is closely related to the pb
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@ -1516,171 +1519,173 @@ relation, and the details don't matter unless you want to comb through
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a somewhat lengthy formal proof. Pretty much all you need to know
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about rcu-link is the information in the preceding paragraph.
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The LKMM also defines the gp and rscs relations. They bring grace
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periods and read-side critical sections into the picture, in the
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The LKMM also defines the rcu-gp and rcu-rscsi relations. They bring
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grace periods and read-side critical sections into the picture, in the
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following way:
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E ->gp F means there is a synchronize_rcu() fence event S such
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that E ->po S and either S ->po F or S = F. In simple terms,
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there is a grace period po-between E and F.
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E ->rcu-gp F means that E and F are in fact the same event,
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and that event is a synchronize_rcu() fence (i.e., a grace
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period).
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E ->rscs F means there is a critical section delimited by an
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rcu_read_lock() fence L and an rcu_read_unlock() fence U, such
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that E ->po U and either L ->po F or L = F. You can think of
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this as saying that E and F are in the same critical section
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(in fact, it also allows E to be po-before the start of the
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critical section and F to be po-after the end).
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E ->rcu-rscsi F means that E and F are the rcu_read_unlock()
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and rcu_read_lock() fence events delimiting some read-side
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critical section. (The 'i' at the end of the name emphasizes
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that this relation is "inverted": It links the end of the
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critical section to the start.)
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If we think of the rcu-link relation as standing for an extended
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"before", then X ->gp Y ->rcu-link Z says that X executes before a
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grace period which ends before Z executes. (In fact it covers more
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than this, because it also includes cases where X executes before a
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grace period and some store propagates to Z's CPU before Z executes
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but doesn't propagate to some other CPU until after the grace period
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ends.) Similarly, X ->rscs Y ->rcu-link Z says that X is part of (or
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before the start of) a critical section which starts before Z
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executes.
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"before", then X ->rcu-gp Y ->rcu-link Z roughly says that X is a
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grace period which ends before Z begins. (In fact it covers more than
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this, because it also includes cases where some store propagates to
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Z's CPU before Z begins but doesn't propagate to some other CPU until
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after X ends.) Similarly, X ->rcu-rscsi Y ->rcu-link Z says that X is
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the end of a critical section which starts before Z begins.
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The LKMM goes on to define the rcu-fence relation as a sequence of gp
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and rscs links separated by rcu-link links, in which the number of gp
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links is >= the number of rscs links. For example:
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The LKMM goes on to define the rcu-fence relation as a sequence of
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rcu-gp and rcu-rscsi links separated by rcu-link links, in which the
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number of rcu-gp links is >= the number of rcu-rscsi links. For
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example:
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X ->gp Y ->rcu-link Z ->rscs T ->rcu-link U ->gp V
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X ->rcu-gp Y ->rcu-link Z ->rcu-rscsi T ->rcu-link U ->rcu-gp V
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would imply that X ->rcu-fence V, because this sequence contains two
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gp links and only one rscs link. (It also implies that X ->rcu-fence T
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and Z ->rcu-fence V.) On the other hand:
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rcu-gp links and one rcu-rscsi link. (It also implies that
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X ->rcu-fence T and Z ->rcu-fence V.) On the other hand:
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X ->rscs Y ->rcu-link Z ->rscs T ->rcu-link U ->gp V
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X ->rcu-rscsi Y ->rcu-link Z ->rcu-rscsi T ->rcu-link U ->rcu-gp V
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does not imply X ->rcu-fence V, because the sequence contains only
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one gp link but two rscs links.
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one rcu-gp link but two rcu-rscsi links.
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The rcu-fence relation is important because the Grace Period Guarantee
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means that rcu-fence acts kind of like a strong fence. In particular,
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if W is a write and we have W ->rcu-fence Z, the Guarantee says that W
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will propagate to every CPU before Z executes.
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E ->rcu-fence F implies not only that E begins before F ends, but also
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that any write po-before E will propagate to every CPU before any
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instruction po-after F can execute. (However, it does not imply that
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E must execute before F; in fact, each synchronize_rcu() fence event
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is linked to itself by rcu-fence as a degenerate case.)
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To prove this in full generality requires some intellectual effort.
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We'll consider just a very simple case:
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W ->gp X ->rcu-link Y ->rscs Z.
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G ->rcu-gp W ->rcu-link Z ->rcu-rscsi F.
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This formula means that there is a grace period G and a critical
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section C such that:
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This formula means that G and W are the same event (a grace period),
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and there are events X, Y and a read-side critical section C such that:
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1. W is po-before G;
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1. G = W is po-before or equal to X;
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2. X is equal to or po-after G;
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2. X comes "before" Y in some sense (including rfe, co and fr);
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3. X comes "before" Y in some sense;
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2. Y is po-before Z;
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4. Y is po-before the end of C;
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4. Z is the rcu_read_unlock() event marking the end of C;
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5. Z is equal to or po-after the start of C.
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5. F is the rcu_read_lock() event marking the start of C.
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From 2 - 4 we deduce that the grace period G ends before the critical
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section C. Then the second part of the Grace Period Guarantee says
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not only that G starts before C does, but also that W (which executes
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on G's CPU before G starts) must propagate to every CPU before C
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starts. In particular, W propagates to every CPU before Z executes
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(or finishes executing, in the case where Z is equal to the
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rcu_read_lock() fence event which starts C.) This sort of reasoning
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can be expanded to handle all the situations covered by rcu-fence.
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From 1 - 4 we deduce that the grace period G ends before the critical
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section C. Then part (2) of the Grace Period Guarantee says not only
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that G starts before C does, but also that any write which executes on
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G's CPU before G starts must propagate to every CPU before C starts.
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In particular, the write propagates to every CPU before F finishes
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executing and hence before any instruction po-after F can execute.
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This sort of reasoning can be extended to handle all the situations
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covered by rcu-fence.
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Finally, the LKMM defines the RCU-before (rb) relation in terms of
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rcu-fence. This is done in essentially the same way as the pb
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relation was defined in terms of strong-fence. We will omit the
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details; the end result is that E ->rb F implies E must execute before
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F, just as E ->pb F does (and for much the same reasons).
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details; the end result is that E ->rb F implies E must execute
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before F, just as E ->pb F does (and for much the same reasons).
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Putting this all together, the LKMM expresses the Grace Period
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Guarantee by requiring that the rb relation does not contain a cycle.
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Equivalently, this "rcu" axiom requires that there are no events E and
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F with E ->rcu-link F ->rcu-fence E. Or to put it a third way, the
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axiom requires that there are no cycles consisting of gp and rscs
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alternating with rcu-link, where the number of gp links is >= the
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number of rscs links.
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Equivalently, this "rcu" axiom requires that there are no events E
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and F with E ->rcu-link F ->rcu-fence E. Or to put it a third way,
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the axiom requires that there are no cycles consisting of rcu-gp and
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rcu-rscsi alternating with rcu-link, where the number of rcu-gp links
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is >= the number of rcu-rscsi links.
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Justifying the axiom isn't easy, but it is in fact a valid
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formalization of the Grace Period Guarantee. We won't attempt to go
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through the detailed argument, but the following analysis gives a
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taste of what is involved. Suppose we have a violation of the first
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part of the Guarantee: A critical section starts before a grace
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period, and some store propagates to the critical section's CPU before
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the end of the critical section but doesn't propagate to some other
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CPU until after the end of the grace period.
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taste of what is involved. Suppose both parts of the Guarantee are
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violated: A critical section starts before a grace period, and some
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store propagates to the critical section's CPU before the end of the
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critical section but doesn't propagate to some other CPU until after
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the end of the grace period.
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Putting symbols to these ideas, let L and U be the rcu_read_lock() and
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rcu_read_unlock() fence events delimiting the critical section in
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question, and let S be the synchronize_rcu() fence event for the grace
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period. Saying that the critical section starts before S means there
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are events E and F where E is po-after L (which marks the start of the
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critical section), E is "before" F in the sense of the rcu-link
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relation, and F is po-before the grace period S:
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are events Q and R where Q is po-after L (which marks the start of the
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critical section), Q is "before" R in the sense used by the rcu-link
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relation, and R is po-before the grace period S. Thus we have:
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L ->po E ->rcu-link F ->po S.
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L ->rcu-link S.
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Let W be the store mentioned above, let Z come before the end of the
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Let W be the store mentioned above, let Y come before the end of the
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critical section and witness that W propagates to the critical
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section's CPU by reading from W, and let Y on some arbitrary CPU be a
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witness that W has not propagated to that CPU, where Y happens after
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section's CPU by reading from W, and let Z on some arbitrary CPU be a
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witness that W has not propagated to that CPU, where Z happens after
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some event X which is po-after S. Symbolically, this amounts to:
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S ->po X ->hb* Y ->fr W ->rf Z ->po U.
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S ->po X ->hb* Z ->fr W ->rf Y ->po U.
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The fr link from Y to W indicates that W has not propagated to Y's CPU
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at the time that Y executes. From this, it can be shown (see the
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discussion of the rcu-link relation earlier) that X and Z are related
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by rcu-link, yielding:
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The fr link from Z to W indicates that W has not propagated to Z's CPU
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at the time that Z executes. From this, it can be shown (see the
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discussion of the rcu-link relation earlier) that S and U are related
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by rcu-link:
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S ->po X ->rcu-link Z ->po U.
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S ->rcu-link U.
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The formulas say that S is po-between F and X, hence F ->gp X. They
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also say that Z comes before the end of the critical section and E
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comes after its start, hence Z ->rscs E. From all this we obtain:
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Since S is a grace period we have S ->rcu-gp S, and since L and U are
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the start and end of the critical section C we have U ->rcu-rscsi L.
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From this we obtain:
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F ->gp X ->rcu-link Z ->rscs E ->rcu-link F,
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S ->rcu-gp S ->rcu-link U ->rcu-rscsi L ->rcu-link S,
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a forbidden cycle. Thus the "rcu" axiom rules out this violation of
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the Grace Period Guarantee.
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For something a little more down-to-earth, let's see how the axiom
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works out in practice. Consider the RCU code example from above, this
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time with statement labels added to the memory access instructions:
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time with statement labels added:
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int x, y;
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P0()
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{
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rcu_read_lock();
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W: WRITE_ONCE(x, 1);
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X: WRITE_ONCE(y, 1);
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rcu_read_unlock();
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L: rcu_read_lock();
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X: WRITE_ONCE(x, 1);
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Y: WRITE_ONCE(y, 1);
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U: rcu_read_unlock();
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}
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P1()
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{
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int r1, r2;
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Y: r1 = READ_ONCE(x);
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synchronize_rcu();
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Z: r2 = READ_ONCE(y);
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Z: r1 = READ_ONCE(x);
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S: synchronize_rcu();
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W: r2 = READ_ONCE(y);
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}
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If r2 = 0 at the end then P0's store at X overwrites the value that
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P1's load at Z reads from, so we have Z ->fre X and thus Z ->rcu-link X.
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In addition, there is a synchronize_rcu() between Y and Z, so therefore
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we have Y ->gp Z.
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If r2 = 0 at the end then P0's store at Y overwrites the value that
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P1's load at W reads from, so we have W ->fre Y. Since S ->po W and
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also Y ->po U, we get S ->rcu-link U. In addition, S ->rcu-gp S
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because S is a grace period.
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If r1 = 1 at the end then P1's load at Y reads from P0's store at W,
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so we have W ->rcu-link Y. In addition, W and X are in the same critical
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section, so therefore we have X ->rscs W.
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If r1 = 1 at the end then P1's load at Z reads from P0's store at X,
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so we have X ->rfe Z. Together with L ->po X and Z ->po S, this
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yields L ->rcu-link S. And since L and U are the start and end of a
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critical section, we have U ->rcu-rscsi L.
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Then X ->rscs W ->rcu-link Y ->gp Z ->rcu-link X is a forbidden cycle,
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violating the "rcu" axiom. Hence the outcome is not allowed by the
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LKMM, as we would expect.
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Then U ->rcu-rscsi L ->rcu-link S ->rcu-gp S ->rcu-link U is a
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forbidden cycle, violating the "rcu" axiom. Hence the outcome is not
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allowed by the LKMM, as we would expect.
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For contrast, let's see what can happen in a more complicated example:
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@ -1690,51 +1695,52 @@ For contrast, let's see what can happen in a more complicated example:
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{
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int r0;
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rcu_read_lock();
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W: r0 = READ_ONCE(x);
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X: WRITE_ONCE(y, 1);
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rcu_read_unlock();
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L0: rcu_read_lock();
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r0 = READ_ONCE(x);
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WRITE_ONCE(y, 1);
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U0: rcu_read_unlock();
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}
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P1()
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{
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int r1;
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Y: r1 = READ_ONCE(y);
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synchronize_rcu();
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Z: WRITE_ONCE(z, 1);
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r1 = READ_ONCE(y);
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S1: synchronize_rcu();
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WRITE_ONCE(z, 1);
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}
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P2()
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{
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int r2;
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rcu_read_lock();
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U: r2 = READ_ONCE(z);
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V: WRITE_ONCE(x, 1);
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rcu_read_unlock();
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L2: rcu_read_lock();
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r2 = READ_ONCE(z);
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WRITE_ONCE(x, 1);
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U2: rcu_read_unlock();
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}
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If r0 = r1 = r2 = 1 at the end, then similar reasoning to before shows
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that W ->rscs X ->rcu-link Y ->gp Z ->rcu-link U ->rscs V ->rcu-link W.
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However this cycle is not forbidden, because the sequence of relations
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contains fewer instances of gp (one) than of rscs (two). Consequently
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the outcome is allowed by the LKMM. The following instruction timing
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diagram shows how it might actually occur:
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that U0 ->rcu-rscsi L0 ->rcu-link S1 ->rcu-gp S1 ->rcu-link U2 ->rcu-rscsi
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L2 ->rcu-link U0. However this cycle is not forbidden, because the
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sequence of relations contains fewer instances of rcu-gp (one) than of
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rcu-rscsi (two). Consequently the outcome is allowed by the LKMM.
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|
The following instruction timing diagram shows how it might actually
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|
occur:
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|
P0 P1 P2
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|
-------------------- -------------------- --------------------
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|
rcu_read_lock()
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|
X: WRITE_ONCE(y, 1)
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Y: r1 = READ_ONCE(y)
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WRITE_ONCE(y, 1)
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|
r1 = READ_ONCE(y)
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|
synchronize_rcu() starts
|
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|
|
. rcu_read_lock()
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|
. V: WRITE_ONCE(x, 1)
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|
W: r0 = READ_ONCE(x) .
|
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|
. WRITE_ONCE(x, 1)
|
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|
r0 = READ_ONCE(x) .
|
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|
|
rcu_read_unlock() .
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|
|
synchronize_rcu() ends
|
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|
Z: WRITE_ONCE(z, 1)
|
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|
U: r2 = READ_ONCE(z)
|
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|
WRITE_ONCE(z, 1)
|
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|
|
r2 = READ_ONCE(z)
|
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|
|
rcu_read_unlock()
|
|
|
|
|
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|
|
This requires P0 and P2 to execute their loads and stores out of
|
|
|
|
@ -1744,6 +1750,15 @@ section in P0 both starts before P1's grace period does and ends
|
|
|
|
|
before it does, and the critical section in P2 both starts after P1's
|
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|
|
grace period does and ends after it does.
|
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|
|
Addendum: The LKMM now supports SRCU (Sleepable Read-Copy-Update) in
|
|
|
|
|
addition to normal RCU. The ideas involved are much the same as
|
|
|
|
|
above, with new relations srcu-gp and srcu-rscsi added to represent
|
|
|
|
|
SRCU grace periods and read-side critical sections. There is a
|
|
|
|
|
restriction on the srcu-gp and srcu-rscsi links that can appear in an
|
|
|
|
|
rcu-fence sequence (the srcu-rscsi links must be paired with srcu-gp
|
|
|
|
|
links having the same SRCU domain with proper nesting); the details
|
|
|
|
|
are relatively unimportant.
|
|
|
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|
LOCKING
|
|
|
|
|
-------
|
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|
|