[PATCH tools/lkmm 2/2] tools/memory-model: Redefine rb in terms of rcu-fence

Paul E. McKenney
2018-03-23T15:49:32Z

From: Alan Stern <stern@rowland.harvard.edu>

This patch reorganizes the definition of rb in the Linux Kernel Memory
Consistency Model. The relation is now expressed in terms of
rcu-fence, which consists of a sequence of gp and rscs links separated
by rcu-link links, in which the number of occurrences of gp is >= the
number of occurrences of rscs.

Arguments similar to those published in
http://diy.inria.fr/linux/long.pdf show that rcu-fence behaves like an
inter-CPU strong fence. Furthermore, the definition of rb in terms of
rcu-fence is highly analogous to the definition of pb in terms of
strong-fence, which can help explain why rcu-path expresses a form of
temporal ordering.

This change should not affect the semantics of the memory model, just
its internal organization.

Signed-off-by: Alan Stern <stern@rowland.harvard.edu>
Reviewed-by: Andrea Parri <parri.andrea@gmail.com>
Signed-off-by: Paul E. McKenney <paulmck@linux.vnet.ibm.com>
---
tools/memory-model/Documentation/explanation.txt | 168 +++++++++++++++--------
tools/memory-model/linux-kernel.cat | 33 +++--
2 files changed, 129 insertions(+), 72 deletions(-)

diff --git a/tools/memory-model/Documentation/explanation.txt b/tools/memory-model/Documentation/explanation.txt
index 1a387d703212..1b09f3175a1f 100644
--- a/tools/memory-model/Documentation/explanation.txt
+++ b/tools/memory-model/Documentation/explanation.txt
@@ -27,7 +27,7 @@ Explanation of the Linux-Kernel Memory Consistency Model
19. AND THEN THERE WAS ALPHA
20. THE HAPPENS-BEFORE RELATION: hb
21. THE PROPAGATES-BEFORE RELATION: pb
- 22. RCU RELATIONS: rcu-link, gp-link, rscs-link, and rb
+ 22. RCU RELATIONS: rcu-link, gp, rscs, rcu-fence, and rb
23. ODDS AND ENDS

@@ -1451,8 +1451,8 @@ they execute means that it cannot have cycles. This requirement is
the content of the LKMM's "propagation" axiom.

-RCU RELATIONS: rcu-link, gp-link, rscs-link, and rb
----------------------------------------------------
+RCU RELATIONS: rcu-link, gp, rscs, rcu-fence, and rb
+----------------------------------------------------

RCU (Read-Copy-Update) is a powerful synchronization mechanism. It
rests on two concepts: grace periods and read-side critical sections.
@@ -1537,49 +1537,100 @@ relation, and the details don't matter unless you want to comb through
a somewhat lengthy formal proof. Pretty much all you need to know
about rcu-link is the information in the preceding paragraph.

-The LKMM goes on to define the gp-link and rscs-link relations. They
-bring grace periods and read-side critical sections into the picture,
-in the following way:
+The LKMM also defines the gp and rscs relations. They bring grace
+periods and read-side critical sections into the picture, in the
+following way:

- E ->gp-link F means there is a synchronize_rcu() fence event S
- and an event X such that E ->po S, either S ->po X or S = X,
- and X ->rcu-link F. In other words, E and F are linked by a
- grace period followed by an instance of rcu-link.
+ E ->gp F means there is a synchronize_rcu() fence event S such
+ that E ->po S and either S ->po F or S = F. In simple terms,
+ there is a grace period po-between E and F.

- E ->rscs-link F means there is a critical section delimited by
- an rcu_read_lock() fence L and an rcu_read_unlock() fence U,
- and an event X such that E ->po U, either L ->po X or L = X,
- and X ->rcu-link F. Roughly speaking, this says that some
- event in the same critical section as E is linked by rcu-link
- to F.
+ E ->rscs F means there is a critical section delimited by an
+ rcu_read_lock() fence L and an rcu_read_unlock() fence U, such
+ that E ->po U and either L ->po F or L = F. You can think of
+ this as saying that E and F are in the same critical section
+ (in fact, it also allows E to be po-before the start of the
+ critical section and F to be po-after the end).

If we think of the rcu-link relation as standing for an extended
-"before", then E ->gp-link F says that E executes before a grace
-period which ends before F executes. (In fact it covers more than
-this, because it also includes cases where E executes before a grace
-period and some store propagates to F's CPU before F executes and
-doesn't propagate to some other CPU until after the grace period
-ends.) Similarly, E ->rscs-link F says that E is part of (or before
-the start of) a critical section which starts before F executes.
+"before", then X ->gp Y ->rcu-link Z says that X executes before a
+grace period which ends before Z executes. (In fact it covers more
+than this, because it also includes cases where X executes before a
+grace period and some store propagates to Z's CPU before Z executes
+but doesn't propagate to some other CPU until after the grace period
+ends.) Similarly, X ->rscs Y ->rcu-link Z says that X is part of (or
+before the start of) a critical section which starts before Z
+executes.
+
+The LKMM goes on to define the rcu-fence relation as a sequence of gp
+and rscs links separated by rcu-link links, in which the number of gp
+links is >= the number of rscs links. For example:
+
+ X ->gp Y ->rcu-link Z ->rscs T ->rcu-link U ->gp V
+
+would imply that X ->rcu-fence V, because this sequence contains two
+gp links and only one rscs link. (It also implies that X ->rcu-fence T
+and Z ->rcu-fence V.) On the other hand:
+
+ X ->rscs Y ->rcu-link Z ->rscs T ->rcu-link U ->gp V
+
+does not imply X ->rcu-fence V, because the sequence contains only
+one gp link but two rscs links.
+
+The rcu-fence relation is important because the Grace Period Guarantee
+means that rcu-fence acts kind of like a strong fence. In particular,
+if W is a write and we have W ->rcu-fence Z, the Guarantee says that W
+will propagate to every CPU before Z executes.
+
+To prove this in full generality requires some intellectual effort.
+We'll consider just a very simple case:
+
+ W ->gp X ->rcu-link Y ->rscs Z.
+
+This formula means that there is a grace period G and a critical
+section C such that:
+
+ 1. W is po-before G;
+
+ 2. X is equal to or po-after G;
+
+ 3. X comes "before" Y in some sense;
+
+ 4. Y is po-before the end of C;
+
+ 5. Z is equal to or po-after the start of C.
+
+From 2 - 4 we deduce that the grace period G ends before the critical
+section C. Then the second part of the Grace Period Guarantee says
+not only that G starts before C does, but also that W (which executes
+on G's CPU before G starts) must propagate to every CPU before C
+starts. In particular, W propagates to every CPU before Z executes
+(or finishes executing, in the case where Z is equal to the
+rcu_read_lock() fence event which starts C.) This sort of reasoning
+can be expanded to handle all the situations covered by rcu-fence.
+
+Finally, the LKMM defines the RCU-before (rb) relation in terms of
+rcu-fence. This is done in essentially the same way as the pb
+relation was defined in terms of strong-fence. We will omit the
+details; the end result is that E ->rb F implies E must execute before
+F, just as E ->pb F does (and for much the same reasons).

Putting this all together, the LKMM expresses the Grace Period
-Guarantee by requiring that there are no cycles consisting of gp-link
-and rscs-link links in which the number of gp-link instances is >= the
-number of rscs-link instances. It does this by defining the rb
-relation to link events E and F whenever it is possible to pass from E
-to F by a sequence of gp-link and rscs-link links with at least as
-many of the former as the latter. The LKMM's "rcu" axiom then says
-that there are no events E with E ->rb E.
-
-Justifying this axiom takes some intellectual effort, but it is in
-fact a valid formalization of the Grace Period Guarantee. We won't
-attempt to go through the detailed argument, but the following
-analysis gives a taste of what is involved. Suppose we have a
-violation of the first part of the Guarantee: A critical section
-starts before a grace period, and some store propagates to the
-critical section's CPU before the end of the critical section but
-doesn't propagate to some other CPU until after the end of the grace
-period.
+Guarantee by requiring that the rb relation does not contain a cycle.
+Equivalently, this "rcu" axiom requires that there are no events E and
+F with E ->rcu-link F ->rcu-fence E. Or to put it a third way, the
+axiom requires that there are no cycles consisting of gp and rscs
+alternating with rcu-link, where the number of gp links is >= the
+number of rscs links.
+
+Justifying the axiom isn't easy, but it is in fact a valid
+formalization of the Grace Period Guarantee. We won't attempt to go
+through the detailed argument, but the following analysis gives a
+taste of what is involved. Suppose we have a violation of the first
+part of the Guarantee: A critical section starts before a grace
+period, and some store propagates to the critical section's CPU before
+the end of the critical section but doesn't propagate to some other
+CPU until after the end of the grace period.

Putting symbols to these ideas, let L and U be the rcu_read_lock() and
rcu_read_unlock() fence events delimiting the critical section in
@@ -1606,11 +1657,14 @@ by rcu-link, yielding:

S ->po X ->rcu-link Z ->po U.

-The formulas say that S is po-between F and X, hence F ->gp-link Z
-via X. They also say that Z comes before the end of the critical
-section and E comes after its start, hence Z ->rscs-link F via E. But
-now we have a forbidden cycle: F ->gp-link Z ->rscs-link F. Thus the
-"rcu" axiom rules out this violation of the Grace Period Guarantee.
+The formulas say that S is po-between F and X, hence F ->gp X. They
+also say that Z comes before the end of the critical section and E
+comes after its start, hence Z ->rscs E. From all this we obtain:
+
+ F ->gp X ->rcu-link Z ->rscs E ->rcu-link F,
+
+a forbidden cycle. Thus the "rcu" axiom rules out this violation of
+the Grace Period Guarantee.

For something a little more down-to-earth, let's see how the axiom
works out in practice. Consider the RCU code example from above, this
@@ -1639,15 +1693,15 @@ time with statement labels added to the memory access instructions:
If r2 = 0 at the end then P0's store at X overwrites the value that
P1's load at Z reads from, so we have Z ->fre X and thus Z ->rcu-link X.
In addition, there is a synchronize_rcu() between Y and Z, so therefore
-we have Y ->gp-link X.
+we have Y ->gp Z.

If r1 = 1 at the end then P1's load at Y reads from P0's store at W,
so we have W ->rcu-link Y. In addition, W and X are in the same critical
-section, so therefore we have X ->rscs-link Y.
+section, so therefore we have X ->rscs W.

-This gives us a cycle, Y ->gp-link X ->rscs-link Y, with one gp-link
-and one rscs-link, violating the "rcu" axiom. Hence the outcome is
-not allowed by the LKMM, as we would expect.
+Then X ->rscs W ->rcu-link Y ->gp Z ->rcu-link X is a forbidden cycle,
+violating the "rcu" axiom. Hence the outcome is not allowed by the
+LKMM, as we would expect.

For contrast, let's see what can happen in a more complicated example:

@@ -1683,15 +1737,11 @@ For contrast, let's see what can happen in a more complicated example:
}

If r0 = r1 = r2 = 1 at the end, then similar reasoning to before shows
-that W ->rscs-link Y via X, Y ->gp-link U via Z, and U ->rscs-link W
-via V. And just as before, this gives a cycle:
-
- W ->rscs-link Y ->gp-link U ->rscs-link W.
-
-However, this cycle has fewer gp-link instances than rscs-link
-instances, and consequently the outcome is not forbidden by the LKMM.
-The following instruction timing diagram shows how it might actually
-occur:
+that W ->rscs X ->rcu-link Y ->gp Z ->rcu-link U ->rscs V ->rcu-link W.
+However this cycle is not forbidden, because the sequence of relations
+contains fewer instances of gp (one) than of rscs (two). Consequently
+the outcome is allowed by the LKMM. The following instruction timing
+diagram shows how it might actually occur:

P0 P1 P2
-------------------- -------------------- --------------------
diff --git a/tools/memory-model/linux-kernel.cat b/tools/memory-model/linux-kernel.cat
index cdf682859d4e..1e5c4653dd12 100644
--- a/tools/memory-model/linux-kernel.cat
+++ b/tools/memory-model/linux-kernel.cat
@@ -102,20 +102,27 @@ let rscs = po ; crit^-1 ; po?
*)
let rcu-link = hb* ; pb* ; prop

-(* Chains that affect the RCU grace-period guarantee *)
-let gp-link = gp ; rcu-link
-let rscs-link = rscs ; rcu-link
-
(*
- * A cycle containing at least as many grace periods as RCU read-side
- * critical sections is forbidden.
+ * Any sequence containing at least as many grace periods as RCU read-side
+ * critical sections (joined by rcu-link) acts as a generalized strong fence.
*)
-let rec rb =
- gp-link |
- (gp-link ; rscs-link) |
- (rscs-link ; gp-link) |
- (rb ; rb) |
- (gp-link ; rb ; rscs-link) |
- (rscs-link ; rb ; gp-link)
+let rec rcu-fence = gp |
+ (gp ; rcu-link ; rscs) |
+ (rscs ; rcu-link ; gp) |
+ (gp ; rcu-link ; rcu-fence ; rcu-link ; rscs) |
+ (rscs ; rcu-link ; rcu-fence ; rcu-link ; gp) |
+ (rcu-fence ; rcu-link ; rcu-fence)
+
+(* rb orders instructions just as pb does *)
+let rb = prop ; rcu-fence ; hb* ; pb*

irreflexive rb as rcu
+
+(*
+ * The happens-before, propagation, and rcu constraints are all
+ * expressions of temporal ordering. They could be replaced by
+ * a single constraint on an "executes-before" relation, xb:
+ *
+ * let xb = hb | pb | rb
+ * acyclic xb as executes-before
+ *)
--
2.5.2