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Explanation of the Linux-Kernel Memory Consistency Model
:Author: Alan Stern <>
:Created: October 2017
.. Contents
9. DEPENDENCY RELATIONS: data, addr, and ctrl
10. THE READS-FROM RELATION: rf, rfi, and rfe
12. THE FROM-READS RELATION: fr, fri, and fre
22. RCU RELATIONS: rcu-link, rcu-gp, rcu-rscsi, rcu-fence, and rb
The Linux-kernel memory consistency model (LKMM) is rather complex and
obscure. This is particularly evident if you read through the
linux-kernel.bell and files that make up the formal
version of the model; they are extremely terse and their meanings are
far from clear.
This document describes the ideas underlying the LKMM, but excluding
the modeling of bare C (or plain) shared memory accesses. It is meant
for people who want to understand how the model was designed. It does
not go into the details of the code in the .bell and .cat files;
rather, it explains in English what the code expresses symbolically.
Sections 2 (BACKGROUND) through 5 (ORDERING AND CYCLES) are aimed
toward beginners; they explain what memory consistency models are and
the basic notions shared by all such models. People already familiar
with these concepts can skim or skip over them. Sections 6 (EVENTS)
through 12 (THE FROM_READS RELATION) describe the fundamental
relations used in many models. Starting in Section 13 (AN OPERATIONAL
MODEL), the workings of the LKMM itself are covered.
Warning: The code examples in this document are not written in the
proper format for litmus tests. They don't include a header line, the
initializations are not enclosed in braces, the global variables are
not passed by pointers, and they don't have an "exists" clause at the
end. Converting them to the right format is left as an exercise for
the reader.
A memory consistency model (or just memory model, for short) is
something which predicts, given a piece of computer code running on a
particular kind of system, what values may be obtained by the code's
load instructions. The LKMM makes these predictions for code running
as part of the Linux kernel.
In practice, people tend to use memory models the other way around.
That is, given a piece of code and a collection of values specified
for the loads, the model will predict whether it is possible for the
code to run in such a way that the loads will indeed obtain the
specified values. Of course, this is just another way of expressing
the same idea.
For code running on a uniprocessor system, the predictions are easy:
Each load instruction must obtain the value written by the most recent
store instruction accessing the same location (we ignore complicating
factors such as DMA and mixed-size accesses.) But on multiprocessor
systems, with multiple CPUs making concurrent accesses to shared
memory locations, things aren't so simple.
Different architectures have differing memory models, and the Linux
kernel supports a variety of architectures. The LKMM has to be fairly
permissive, in the sense that any behavior allowed by one of these
architectures also has to be allowed by the LKMM.
Here is a simple example to illustrate the basic concepts. Consider
some code running as part of a device driver for an input device. The
driver might contain an interrupt handler which collects data from the
device, stores it in a buffer, and sets a flag to indicate the buffer
is full. Running concurrently on a different CPU might be a part of
the driver code being executed by a process in the midst of a read(2)
system call. This code tests the flag to see whether the buffer is
ready, and if it is, copies the data back to userspace. The buffer
and the flag are memory locations shared between the two CPUs.
We can abstract out the important pieces of the driver code as follows
(the reason for using WRITE_ONCE() and READ_ONCE() instead of simple
assignment statements is discussed later):
int buf = 0, flag = 0;
WRITE_ONCE(buf, 1);
WRITE_ONCE(flag, 1);
int r1;
int r2 = 0;
r1 = READ_ONCE(flag);
if (r1)
r2 = READ_ONCE(buf);
Here the P0() function represents the interrupt handler running on one
CPU and P1() represents the read() routine running on another. The
value 1 stored in buf represents input data collected from the device.
Thus, P0 stores the data in buf and then sets flag. Meanwhile, P1
reads flag into the private variable r1, and if it is set, reads the
data from buf into a second private variable r2 for copying to
userspace. (Presumably if flag is not set then the driver will wait a
while and try again.)
This pattern of memory accesses, where one CPU stores values to two
shared memory locations and another CPU loads from those locations in
the opposite order, is widely known as the "Message Passing" or MP
pattern. It is typical of memory access patterns in the kernel.
Please note that this example code is a simplified abstraction. Real
buffers are usually larger than a single integer, real device drivers
usually use sleep and wakeup mechanisms rather than polling for I/O
completion, and real code generally doesn't bother to copy values into
private variables before using them. All that is beside the point;
the idea here is simply to illustrate the overall pattern of memory
accesses by the CPUs.
A memory model will predict what values P1 might obtain for its loads
from flag and buf, or equivalently, what values r1 and r2 might end up
with after the code has finished running.
Some predictions are trivial. For instance, no sane memory model would
predict that r1 = 42 or r2 = -7, because neither of those values ever
gets stored in flag or buf.
Some nontrivial predictions are nonetheless quite simple. For
instance, P1 might run entirely before P0 begins, in which case r1 and
r2 will both be 0 at the end. Or P0 might run entirely before P1
begins, in which case r1 and r2 will both be 1.
The interesting predictions concern what might happen when the two
routines run concurrently. One possibility is that P1 runs after P0's
store to buf but before the store to flag. In this case, r1 and r2
will again both be 0. (If P1 had been designed to read buf
unconditionally then we would instead have r1 = 0 and r2 = 1.)
However, the most interesting possibility is where r1 = 1 and r2 = 0.
If this were to occur it would mean the driver contains a bug, because
incorrect data would get sent to the user: 0 instead of 1. As it
happens, the LKMM does predict this outcome can occur, and the example
driver code shown above is indeed buggy.
The first widely cited memory model, and the simplest to understand,
is Sequential Consistency. According to this model, systems behave as
if each CPU executed its instructions in order but with unspecified
timing. In other words, the instructions from the various CPUs get
interleaved in a nondeterministic way, always according to some single
global order that agrees with the order of the instructions in the
program source for each CPU. The model says that the value obtained
by each load is simply the value written by the most recently executed
store to the same memory location, from any CPU.
For the MP example code shown above, Sequential Consistency predicts
that the undesired result r1 = 1, r2 = 0 cannot occur. The reasoning
goes like this:
Since r1 = 1, P0 must store 1 to flag before P1 loads 1 from
it, as loads can obtain values only from earlier stores.
P1 loads from flag before loading from buf, since CPUs execute
their instructions in order.
P1 must load 0 from buf before P0 stores 1 to it; otherwise r2
would be 1 since a load obtains its value from the most recent
store to the same address.
P0 stores 1 to buf before storing 1 to flag, since it executes
its instructions in order.
Since an instruction (in this case, P1's store to flag) cannot
execute before itself, the specified outcome is impossible.
However, real computer hardware almost never follows the Sequential
Consistency memory model; doing so would rule out too many valuable
performance optimizations. On ARM and PowerPC architectures, for
instance, the MP example code really does sometimes yield r1 = 1 and
r2 = 0.
x86 and SPARC follow yet a different memory model: TSO (Total Store
Ordering). This model predicts that the undesired outcome for the MP
pattern cannot occur, but in other respects it differs from Sequential
Consistency. One example is the Store Buffer (SB) pattern, in which
each CPU stores to its own shared location and then loads from the
other CPU's location:
int x = 0, y = 0;
int r0;
r0 = READ_ONCE(y);
int r1;
r1 = READ_ONCE(x);
Sequential Consistency predicts that the outcome r0 = 0, r1 = 0 is
impossible. (Exercise: Figure out the reasoning.) But TSO allows
this outcome to occur, and in fact it does sometimes occur on x86 and
SPARC systems.
The LKMM was inspired by the memory models followed by PowerPC, ARM,
x86, Alpha, and other architectures. However, it is different in
detail from each of them.
Memory models are all about ordering. Often this is temporal ordering
(i.e., the order in which certain events occur) but it doesn't have to
be; consider for example the order of instructions in a program's
source code. We saw above that Sequential Consistency makes an
important assumption that CPUs execute instructions in the same order
as those instructions occur in the code, and there are many other
instances of ordering playing central roles in memory models.
The counterpart to ordering is a cycle. Ordering rules out cycles:
It's not possible to have X ordered before Y, Y ordered before Z, and
Z ordered before X, because this would mean that X is ordered before
itself. The analysis of the MP example under Sequential Consistency
involved just such an impossible cycle:
W: P0 stores 1 to flag executes before
X: P1 loads 1 from flag executes before
Y: P1 loads 0 from buf executes before
Z: P0 stores 1 to buf executes before
W: P0 stores 1 to flag.
In short, if a memory model requires certain accesses to be ordered,
and a certain outcome for the loads in a piece of code can happen only
if those accesses would form a cycle, then the memory model predicts
that outcome cannot occur.
The LKMM is defined largely in terms of cycles, as we will see.
The LKMM does not work directly with the C statements that make up
kernel source code. Instead it considers the effects of those
statements in a more abstract form, namely, events. The model
includes three types of events:
Read events correspond to loads from shared memory, such as
calls to READ_ONCE(), smp_load_acquire(), or
Write events correspond to stores to shared memory, such as
calls to WRITE_ONCE(), smp_store_release(), or atomic_set().
Fence events correspond to memory barriers (also known as
fences), such as calls to smp_rmb() or rcu_read_lock().
These categories are not exclusive; a read or write event can also be
a fence. This happens with functions like smp_load_acquire() or
spin_lock(). However, no single event can be both a read and a write.
Atomic read-modify-write accesses, such as atomic_inc() or xchg(),
correspond to a pair of events: a read followed by a write. (The
write event is omitted for executions where it doesn't occur, such as
a cmpxchg() where the comparison fails.)
Other parts of the code, those which do not involve interaction with
shared memory, do not give rise to events. Thus, arithmetic and
logical computations, control-flow instructions, or accesses to
private memory or CPU registers are not of central interest to the
memory model. They only affect the model's predictions indirectly.
For example, an arithmetic computation might determine the value that
gets stored to a shared memory location (or in the case of an array
index, the address where the value gets stored), but the memory model
is concerned only with the store itself -- its value and its address
-- not the computation leading up to it.
Events in the LKMM can be linked by various relations, which we will
describe in the following sections. The memory model requires certain
of these relations to be orderings, that is, it requires them not to
have any cycles.
The most important relation between events is program order (po). You
can think of it as the order in which statements occur in the source
code after branches are taken into account and loops have been
unrolled. A better description might be the order in which
instructions are presented to a CPU's execution unit. Thus, we say
that X is po-before Y (written as "X ->po Y" in formulas) if X occurs
before Y in the instruction stream.
This is inherently a single-CPU relation; two instructions executing
on different CPUs are never linked by po. Also, it is by definition
an ordering so it cannot have any cycles.
po-loc is a sub-relation of po. It links two memory accesses when the
first comes before the second in program order and they access the
same memory location (the "-loc" suffix).
Although this may seem straightforward, there is one subtle aspect to
program order we need to explain. The LKMM was inspired by low-level
architectural memory models which describe the behavior of machine
code, and it retains their outlook to a considerable extent. The
read, write, and fence events used by the model are close in spirit to
individual machine instructions. Nevertheless, the LKMM describes
kernel code written in C, and the mapping from C to machine code can
be extremely complex.
Optimizing compilers have great freedom in the way they translate
source code to object code. They are allowed to apply transformations
that add memory accesses, eliminate accesses, combine them, split them
into pieces, or move them around. The use of READ_ONCE(), WRITE_ONCE(),
or one of the other atomic or synchronization primitives prevents a
large number of compiler optimizations. In particular, it is guaranteed
that the compiler will not remove such accesses from the generated code
(unless it can prove the accesses will never be executed), it will not
change the order in which they occur in the code (within limits imposed
by the C standard), and it will not introduce extraneous accesses.
The MP and SB examples above used READ_ONCE() and WRITE_ONCE() rather
than ordinary memory accesses. Thanks to this usage, we can be certain
that in the MP example, the compiler won't reorder P0's write event to
buf and P0's write event to flag, and similarly for the other shared
memory accesses in the examples.
Since private variables are not shared between CPUs, they can be
accessed normally without READ_ONCE() or WRITE_ONCE(). In fact, they
need not even be stored in normal memory at all -- in principle a
private variable could be stored in a CPU register (hence the convention
that these variables have names starting with the letter 'r').
The protections provided by READ_ONCE(), WRITE_ONCE(), and others are
not perfect; and under some circumstances it is possible for the
compiler to undermine the memory model. Here is an example. Suppose
both branches of an "if" statement store the same value to the same
r1 = READ_ONCE(x);
if (r1) {
... /* do something */
} else {
... /* do something else */
For this code, the LKMM predicts that the load from x will always be
executed before either of the stores to y. However, a compiler could
lift the stores out of the conditional, transforming the code into
something resembling:
r1 = READ_ONCE(x);
if (r1) {
... /* do something */
} else {
... /* do something else */
Given this version of the code, the LKMM would predict that the load
from x could be executed after the store to y. Thus, the memory
model's original prediction could be invalidated by the compiler.
Another issue arises from the fact that in C, arguments to many
operators and function calls can be evaluated in any order. For
r1 = f(5) + g(6);
The object code might call f(5) either before or after g(6); the
memory model cannot assume there is a fixed program order relation
between them. (In fact, if the functions are inlined then the
compiler might even interleave their object code.)
DEPENDENCY RELATIONS: data, addr, and ctrl
We say that two events are linked by a dependency relation when the
execution of the second event depends in some way on a value obtained
from memory by the first. The first event must be a read, and the
value it obtains must somehow affect what the second event does.
There are three kinds of dependencies: data, address (addr), and
control (ctrl).
A read and a write event are linked by a data dependency if the value
obtained by the read affects the value stored by the write. As a very
simple example:
int x, y;
r1 = READ_ONCE(x);
WRITE_ONCE(y, r1 + 5);
The value stored by the WRITE_ONCE obviously depends on the value
loaded by the READ_ONCE. Such dependencies can wind through
arbitrarily complicated computations, and a write can depend on the
values of multiple reads.
A read event and another memory access event are linked by an address
dependency if the value obtained by the read affects the location
accessed by the other event. The second event can be either a read or
a write. Here's another simple example:
int a[20];
int i;
r1 = READ_ONCE(i);
r2 = READ_ONCE(a[r1]);
Here the location accessed by the second READ_ONCE() depends on the
index value loaded by the first. Pointer indirection also gives rise
to address dependencies, since the address of a location accessed
through a pointer will depend on the value read earlier from that
Finally, a read event and another memory access event are linked by a
control dependency if the value obtained by the read affects whether
the second event is executed at all. Simple example:
int x, y;
r1 = READ_ONCE(x);
if (r1)
WRITE_ONCE(y, 1984);
Execution of the WRITE_ONCE() is controlled by a conditional expression
which depends on the value obtained by the READ_ONCE(); hence there is
a control dependency from the load to the store.
It should be pretty obvious that events can only depend on reads that
come earlier in program order. Symbolically, if we have R ->data X,
R ->addr X, or R ->ctrl X (where R is a read event), then we must also
have R ->po X. It wouldn't make sense for a computation to depend
somehow on a value that doesn't get loaded from shared memory until
later in the code!
THE READS-FROM RELATION: rf, rfi, and rfe
The reads-from relation (rf) links a write event to a read event when
the value loaded by the read is the value that was stored by the
write. In colloquial terms, the load "reads from" the store. We
write W ->rf R to indicate that the load R reads from the store W. We
further distinguish the cases where the load and the store occur on
the same CPU (internal reads-from, or rfi) and where they occur on
different CPUs (external reads-from, or rfe).
For our purposes, a memory location's initial value is treated as
though it had been written there by an imaginary initial store that
executes on a separate CPU before the program runs.
Usage of the rf relation implicitly assumes that loads will always
read from a single store. It doesn't apply properly in the presence
of load-tearing, where a load obtains some of its bits from one store
and some of them from another store. Fortunately, use of READ_ONCE()
and WRITE_ONCE() will prevent load-tearing; it's not possible to have:
int x = 0;
WRITE_ONCE(x, 0x1234);
int r1;
r1 = READ_ONCE(x);
and end up with r1 = 0x1200 (partly from x's initial value and partly
from the value stored by P0).
On the other hand, load-tearing is unavoidable when mixed-size
accesses are used. Consider this example:
union {
u32 w;
u16 h[2];
} x;
WRITE_ONCE(x.h[0], 0x1234);
WRITE_ONCE(x.h[1], 0x5678);
int r1;
r1 = READ_ONCE(x.w);
If r1 = 0x56781234 (little-endian!) at the end, then P1 must have read
from both of P0's stores. It is possible to handle mixed-size and
unaligned accesses in a memory model, but the LKMM currently does not
attempt to do so. It requires all accesses to be properly aligned and
of the location's actual size.
Cache coherence is a general principle requiring that in a
multi-processor system, the CPUs must share a consistent view of the
memory contents. Specifically, it requires that for each location in
shared memory, the stores to that location must form a single global
ordering which all the CPUs agree on (the coherence order), and this
ordering must be consistent with the program order for accesses to
that location.
To put it another way, for any variable x, the coherence order (co) of
the stores to x is simply the order in which the stores overwrite one
another. The imaginary store which establishes x's initial value
comes first in the coherence order; the store which directly
overwrites the initial value comes second; the store which overwrites
that value comes third, and so on.
You can think of the coherence order as being the order in which the
stores reach x's location in memory (or if you prefer a more
hardware-centric view, the order in which the stores get written to
x's cache line). We write W ->co W' if W comes before W' in the
coherence order, that is, if the value stored by W gets overwritten,
directly or indirectly, by the value stored by W'.
Coherence order is required to be consistent with program order. This
requirement takes the form of four coherency rules:
Write-write coherence: If W ->po-loc W' (i.e., W comes before
W' in program order and they access the same location), where W
and W' are two stores, then W ->co W'.
Write-read coherence: If W ->po-loc R, where W is a store and R
is a load, then R must read from W or from some other store
which comes after W in the coherence order.
Read-write coherence: If R ->po-loc W, where R is a load and W
is a store, then the store which R reads from must come before
W in the coherence order.
Read-read coherence: If R ->po-loc R', where R and R' are two
loads, then either they read from the same store or else the
store read by R comes before the store read by R' in the
coherence order.
This is sometimes referred to as sequential consistency per variable,
because it means that the accesses to any single memory location obey
the rules of the Sequential Consistency memory model. (According to
Wikipedia, sequential consistency per variable and cache coherence
mean the same thing except that cache coherence includes an extra
requirement that every store eventually becomes visible to every CPU.)
Any reasonable memory model will include cache coherence. Indeed, our
expectation of cache coherence is so deeply ingrained that violations
of its requirements look more like hardware bugs than programming
int x;
WRITE_ONCE(x, 17);
WRITE_ONCE(x, 23);
If the final value stored in x after this code ran was 17, you would
think your computer was broken. It would be a violation of the
write-write coherence rule: Since the store of 23 comes later in
program order, it must also come later in x's coherence order and
thus must overwrite the store of 17.
int x = 0;
int r1;
r1 = READ_ONCE(x);
WRITE_ONCE(x, 666);
If r1 = 666 at the end, this would violate the read-write coherence
rule: The READ_ONCE() load comes before the WRITE_ONCE() store in
program order, so it must not read from that store but rather from one
coming earlier in the coherence order (in this case, x's initial
int x = 0;
int r1, r2;
r1 = READ_ONCE(x);
r2 = READ_ONCE(x);
If r1 = 5 (reading from P0's store) and r2 = 0 (reading from the
imaginary store which establishes x's initial value) at the end, this
would violate the read-read coherence rule: The r1 load comes before
the r2 load in program order, so it must not read from a store that
comes later in the coherence order.
(As a minor curiosity, if this code had used normal loads instead of
READ_ONCE() in P1, on Itanium it sometimes could end up with r1 = 5
and r2 = 0! This results from parallel execution of the operations
encoded in Itanium's Very-Long-Instruction-Word format, and it is yet
another motivation for using READ_ONCE() when accessing shared memory
Just like the po relation, co is inherently an ordering -- it is not
possible for a store to directly or indirectly overwrite itself! And
just like with the rf relation, we distinguish between stores that
occur on the same CPU (internal coherence order, or coi) and stores
that occur on different CPUs (external coherence order, or coe).
On the other hand, stores to different memory locations are never
related by co, just as instructions on different CPUs are never
related by po. Coherence order is strictly per-location, or if you
prefer, each location has its own independent coherence order.
THE FROM-READS RELATION: fr, fri, and fre
The from-reads relation (fr) can be a little difficult for people to
grok. It describes the situation where a load reads a value that gets
overwritten by a store. In other words, we have R ->fr W when the
value that R reads is overwritten (directly or indirectly) by W, or
equivalently, when R reads from a store which comes earlier than W in
the coherence order.
For example:
int x = 0;
int r1;
r1 = READ_ONCE(x);
The value loaded from x will be 0 (assuming cache coherence!), and it
gets overwritten by the value 2. Thus there is an fr link from the
READ_ONCE() to the WRITE_ONCE(). If the code contained any later
stores to x, there would also be fr links from the READ_ONCE() to
As with rf, rfi, and rfe, we subdivide the fr relation into fri (when
the load and the store are on the same CPU) and fre (when they are on
different CPUs).
Note that the fr relation is determined entirely by the rf and co
relations; it is not independent. Given a read event R and a write
event W for the same location, we will have R ->fr W if and only if
the write which R reads from is co-before W. In symbols,
(R ->fr W) := (there exists W' with W' ->rf R and W' ->co W).
The LKMM is based on various operational memory models, meaning that
the models arise from an abstract view of how a computer system
operates. Here are the main ideas, as incorporated into the LKMM.
The system as a whole is divided into the CPUs and a memory subsystem.
The CPUs are responsible for executing instructions (not necessarily
in program order), and they communicate with the memory subsystem.
For the most part, executing an instruction requires a CPU to perform
only internal operations. However, loads, stores, and fences involve
When CPU C executes a store instruction, it tells the memory subsystem
to store a certain value at a certain location. The memory subsystem
propagates the store to all the other CPUs as well as to RAM. (As a
special case, we say that the store propagates to its own CPU at the
time it is executed.) The memory subsystem also determines where the
store falls in the location's coherence order. In particular, it must
arrange for the store to be co-later than (i.e., to overwrite) any
other store to the same location which has already propagated to CPU C.
When a CPU executes a load instruction R, it first checks to see
whether there are any as-yet unexecuted store instructions, for the
same location, that come before R in program order. If there are, it
uses the value of the po-latest such store as the value obtained by R,
and we say that the store's value is forwarded to R. Otherwise, the
CPU asks the memory subsystem for the value to load and we say that R
is satisfied from memory. The memory subsystem hands back the value
of the co-latest store to the location in question which has already
propagated to that CPU.
(In fact, the picture needs to be a little more complicated than this.
CPUs have local caches, and propagating a store to a CPU really means
propagating it to the CPU's local cache. A local cache can take some
time to process the stores that it receives, and a store can't be used
to satisfy one of the CPU's loads until it has been processed. On
most architectures, the local caches process stores in
First-In-First-Out order, and consequently the processing delay
doesn't matter for the memory model. But on Alpha, the local caches
have a partitioned design that results in non-FIFO behavior. We will
discuss this in more detail later.)
Note that load instructions may be executed speculatively and may be
restarted under certain circumstances. The memory model ignores these
premature executions; we simply say that the load executes at the
final time it is forwarded or satisfied.
Executing a fence (or memory barrier) instruction doesn't require a
CPU to do anything special other than informing the memory subsystem
about the fence. However, fences do constrain the way CPUs and the
memory subsystem handle other instructions, in two respects.
First, a fence forces the CPU to execute various instructions in
program order. Exactly which instructions are ordered depends on the
type of fence:
Strong fences, including smp_mb() and synchronize_rcu(), force
the CPU to execute all po-earlier instructions before any
po-later instructions;
smp_rmb() forces the CPU to execute all po-earlier loads
before any po-later loads;
smp_wmb() forces the CPU to execute all po-earlier stores
before any po-later stores;
Acquire fences, such as smp_load_acquire(), force the CPU to
execute the load associated with the fence (e.g., the load
part of an smp_load_acquire()) before any po-later
Release fences, such as smp_store_release(), force the CPU to
execute all po-earlier instructions before the store
associated with the fence (e.g., the store part of an
Second, some types of fence affect the way the memory subsystem
propagates stores. When a fence instruction is executed on CPU C:
For each other CPU C', smp_wmb() forces all po-earlier stores
on C to propagate to C' before any po-later stores do.
For each other CPU C', any store which propagates to C before
a release fence is executed (including all po-earlier
stores executed on C) is forced to propagate to C' before the
store associated with the release fence does.
Any store which propagates to C before a strong fence is
executed (including all po-earlier stores on C) is forced to
propagate to all other CPUs before any instructions po-after
the strong fence are executed on C.
The propagation ordering enforced by release fences and strong fences
affects stores from other CPUs that propagate to CPU C before the
fence is executed, as well as stores that are executed on C before the
fence. We describe this property by saying that release fences and
strong fences are A-cumulative. By contrast, smp_wmb() fences are not
A-cumulative; they only affect the propagation of stores that are
executed on C before the fence (i.e., those which precede the fence in
program order).
rcu_read_lock(), rcu_read_unlock(), and synchronize_rcu() fences have
other properties which we discuss later.
The fences which affect propagation order (i.e., strong, release, and
smp_wmb() fences) are collectively referred to as cumul-fences, even
though smp_wmb() isn't A-cumulative. The cumul-fence relation is
defined to link memory access events E and F whenever:
E and F are both stores on the same CPU and an smp_wmb() fence
event occurs between them in program order; or
F is a release fence and some X comes before F in program order,
where either X = E or else E ->rf X; or
A strong fence event occurs between some X and F in program
order, where either X = E or else E ->rf X.
The operational model requires that whenever W and W' are both stores
and W ->cumul-fence W', then W must propagate to any given CPU
before W' does. However, for different CPUs C and C', it does not
require W to propagate to C before W' propagates to C'.
The LKMM is derived from the restrictions imposed by the design
outlined above. These restrictions involve the necessity of
maintaining cache coherence and the fact that a CPU can't operate on a
value before it knows what that value is, among other things.
The formal version of the LKMM is defined by five requirements, or
Sequential consistency per variable: This requires that the
system obey the four coherency rules.
Atomicity: This requires that atomic read-modify-write
operations really are atomic, that is, no other stores can
sneak into the middle of such an update.
Happens-before: This requires that certain instructions are
executed in a specific order.
Propagation: This requires that certain stores propagate to
CPUs and to RAM in a specific order.
Rcu: This requires that RCU read-side critical sections and
grace periods obey the rules of RCU, in particular, the
Grace-Period Guarantee.
The first and second are quite common; they can be found in many
memory models (such as those for C11/C++11). The "happens-before" and
"propagation" axioms have analogs in other memory models as well. The
"rcu" axiom is specific to the LKMM.
Each of these axioms is discussed below.
According to the principle of cache coherence, the stores to any fixed
shared location in memory form a global ordering. We can imagine
inserting the loads from that location into this ordering, by placing
each load between the store that it reads from and the following
store. This leaves the relative positions of loads that read from the
same store unspecified; let's say they are inserted in program order,
first for CPU 0, then CPU 1, etc.
You can check that the four coherency rules imply that the rf, co, fr,
and po-loc relations agree with this global ordering; in other words,
whenever we have X ->rf Y or X ->co Y or X ->fr Y or X ->po-loc Y, the
X event comes before the Y event in the global ordering. The LKMM's
"coherence" axiom expresses this by requiring the union of these
relations not to have any cycles. This means it must not be possible
to find events
X0 -> X1 -> X2 -> ... -> Xn -> X0,
where each of the links is either rf, co, fr, or po-loc. This has to
hold if the accesses to the fixed memory location can be ordered as
cache coherence demands.
Although it is not obvious, it can be shown that the converse is also
true: This LKMM axiom implies that the four coherency rules are
What does it mean to say that a read-modify-write (rmw) update, such
as atomic_inc(&x), is atomic? It means that the memory location (x in
this case) does not get altered between the read and the write events
making up the atomic operation. In particular, if two CPUs perform
atomic_inc(&x) concurrently, it must be guaranteed that the final
value of x will be the initial value plus two. We should never have
the following sequence of events:
CPU 0 loads x obtaining 13;
CPU 1 loads x obtaining 13;
CPU 0 stores 14 to x;
CPU 1 stores 14 to x;
where the final value of x is wrong (14 rather than 15).
In this example, CPU 0's increment effectively gets lost because it
occurs in between CPU 1's load and store. To put it another way, the
problem is that the position of CPU 0's store in x's coherence order
is between the store that CPU 1 reads from and the store that CPU 1
The same analysis applies to all atomic update operations. Therefore,
to enforce atomicity the LKMM requires that atomic updates follow this
rule: Whenever R and W are the read and write events composing an
atomic read-modify-write and W' is the write event which R reads from,
there must not be any stores coming between W' and W in the coherence
order. Equivalently,
(R ->rmw W) implies (there is no X with R ->fr X and X ->co W),
where the rmw relation links the read and write events making up each
atomic update. This is what the LKMM's "atomic" axiom says.
There are many situations where a CPU is obligated to execute two
instructions in program order. We amalgamate them into the ppo (for
"preserved program order") relation, which links the po-earlier
instruction to the po-later instruction and is thus a sub-relation of
The operational model already includes a description of one such
situation: Fences are a source of ppo links. Suppose X and Y are
memory accesses with X ->po Y; then the CPU must execute X before Y if
any of the following hold:
A strong (smp_mb() or synchronize_rcu()) fence occurs between
X and Y;
X and Y are both stores and an smp_wmb() fence occurs between
X and Y are both loads and an smp_rmb() fence occurs between
X is also an acquire fence, such as smp_load_acquire();
Y is also a release fence, such as smp_store_release().
Another possibility, not mentioned earlier but discussed in the next
section, is:
X and Y are both loads, X ->addr Y (i.e., there is an address
dependency from X to Y), and X is a READ_ONCE() or an atomic
Dependencies can also cause instructions to be executed in program
order. This is uncontroversial when the second instruction is a
store; either a data, address, or control dependency from a load R to
a store W will force the CPU to execute R before W. This is very
simply because the CPU cannot tell the memory subsystem about W's
store before it knows what value should be stored (in the case of a
data dependency), what location it should be stored into (in the case
of an address dependency), or whether the store should actually take
place (in the case of a control dependency).
Dependencies to load instructions are more problematic. To begin with,
there is no such thing as a data dependency to a load. Next, a CPU
has no reason to respect a control dependency to a load, because it
can always satisfy the second load speculatively before the first, and
then ignore the result if it turns out that the second load shouldn't
be executed after all. And lastly, the real difficulties begin when
we consider address dependencies to loads.
To be fair about it, all Linux-supported architectures do execute
loads in program order if there is an address dependency between them.
After all, a CPU cannot ask the memory subsystem to load a value from
a particular location before it knows what that location is. However,
the split-cache design used by Alpha can cause it to behave in a way
that looks as if the loads were executed out of order (see the next
section for more details). The kernel includes a workaround for this
problem when the loads come from READ_ONCE(), and therefore the LKMM
includes address dependencies to loads in the ppo relation.
On the other hand, dependencies can indirectly affect the ordering of
two loads. This happens when there is a dependency from a load to a
store and a second, po-later load reads from that store:
R ->dep W ->rfi R',
where the dep link can be either an address or a data dependency. In
this situation we know it is possible for the CPU to execute R' before
W, because it can forward the value that W will store to R'. But it
cannot execute R' before R, because it cannot forward the value before
it knows what that value is, or that W and R' do access the same
location. However, if there is merely a control dependency between R
and W then the CPU can speculatively forward W to R' before executing
R; if the speculation turns out to be wrong then the CPU merely has to
restart or abandon R'.
(In theory, a CPU might forward a store to a load when it runs across
an address dependency like this:
r1 = READ_ONCE(ptr);
WRITE_ONCE(*r1, 17);
r2 = READ_ONCE(*r1);
because it could tell that the store and the second load access the
same location even before it knows what the location's address is.
However, none of the architectures supported by the Linux kernel do
Two memory accesses of the same location must always be executed in
program order if the second access is a store. Thus, if we have
R ->po-loc W
(the po-loc link says that R comes before W in program order and they
access the same location), the CPU is obliged to execute W after R.
If it executed W first then the memory subsystem would respond to R's
read request with the value stored by W (or an even later store), in
violation of the read-write coherence rule. Similarly, if we had
W ->po-loc W'
and the CPU executed W' before W, then the memory subsystem would put
W' before W in the coherence order. It would effectively cause W to
overwrite W', in violation of the write-write coherence rule.
(Interestingly, an early ARMv8 memory model, now obsolete, proposed
allowing out-of-order writes like this to occur. The model avoided
violating the write-write coherence rule by requiring the CPU not to
send the W write to the memory subsystem at all!)
As mentioned above, the Alpha architecture is unique in that it does
not appear to respect address dependencies to loads. This means that
code such as the following:
int x = 0;
int y = -1;
int *ptr = &y;
WRITE_ONCE(ptr, &x);
int *r1;
int r2;
r1 = ptr;
r2 = READ_ONCE(*r1);
can malfunction on Alpha systems (notice that P1 uses an ordinary load
to read ptr instead of READ_ONCE()). It is quite possible that r1 = &x
and r2 = 0 at the end, in spite of the address dependency.
At first glance this doesn't seem to make sense. We know that the
smp_wmb() forces P0's store to x to propagate to P1 before the store
to ptr does. And since P1 can't execute its second load
until it knows what location to load from, i.e., after executing its
first load, the value x = 1 must have propagated to P1 before the
second load executed. So why doesn't r2 end up equal to 1?
The answer lies in the Alpha's split local caches. Although the two
stores do reach P1's local cache in the proper order, it can happen
that the first store is processed by a busy part of the cache while
the second store is processed by an idle part. As a result, the x = 1
value may not become available for P1's CPU to read until after the
ptr = &x value does, leading to the undesirable result above. The
final effect is that even though the two loads really are executed in
program order, it appears that they aren't.
This could not have happened if the local cache had processed the
incoming stores in FIFO order. By contrast, other architectures
maintain at least the appearance of FIFO order.
In practice, this difficulty is solved by inserting a special fence
between P1's two loads when the kernel is compiled for the Alpha
architecture. In fact, as of version 4.15, the kernel automatically
adds this fence (called smp_read_barrier_depends() and defined as
nothing at all on non-Alpha builds) after every READ_ONCE() and atomic
load. The effect of the fence is to cause the CPU not to execute any
po-later instructions until after the local cache has finished
processing all the stores it has already received. Thus, if the code
was changed to:
int *r1;
int r2;
r1 = READ_ONCE(ptr);
r2 = READ_ONCE(*r1);
then we would never get r1 = &x and r2 = 0. By the time P1 executed
its second load, the x = 1 store would already be fully processed by
the local cache and available for satisfying the read request. Thus
we have yet another reason why shared data should always be read with
READ_ONCE() or another synchronization primitive rather than accessed
The LKMM requires that smp_rmb(), acquire fences, and strong fences
share this property with smp_read_barrier_depends(): They do not allow
the CPU to execute any po-later instructions (or po-later loads in the
case of smp_rmb()) until all outstanding stores have been processed by
the local cache. In the case of a strong fence, the CPU first has to
wait for all of its po-earlier stores to propagate to every other CPU
in the system; then it has to wait for the local cache to process all
the stores received as of that time -- not just the stores received
when the strong fence began.
And of course, none of this matters for any architecture other than
The happens-before relation (hb) links memory accesses that have to
execute in a certain order. hb includes the ppo relation and two
others, one of which is rfe.
W ->rfe R implies that W and R are on different CPUs. It also means
that W's store must have propagated to R's CPU before R executed;
otherwise R could not have read the value stored by W. Therefore W
must have executed before R, and so we have W ->hb R.
The equivalent fact need not hold if W ->rfi R (i.e., W and R are on
the same CPU). As we have already seen, the operational model allows
W's value to be forwarded to R in such cases, meaning that R may well
execute before W does.
It's important to understand that neither coe nor fre is included in
hb, despite their similarities to rfe. For example, suppose we have
W ->coe W'. This means that W and W' are stores to the same location,
they execute on different CPUs, and W comes before W' in the coherence
order (i.e., W' overwrites W). Nevertheless, it is possible for W' to
execute before W, because the decision as to which store overwrites
the other is made later by the memory subsystem. When the stores are
nearly simultaneous, either one can come out on top. Similarly,
R ->fre W means that W overwrites the value which R reads, but it
doesn't mean that W has to execute after R. All that's necessary is
for the memory subsystem not to propagate W to R's CPU until after R
has executed, which is possible if W executes shortly before R.
The third relation included in hb is like ppo, in that it only links
events that are on the same CPU. However it is more difficult to
explain, because it arises only indirectly from the requirement of
cache coherence. The relation is called prop, and it links two events
on CPU C in situations where a store from some other CPU comes after
the first event in the coherence order and propagates to C before the
second event executes.
This is best explained with some examples. The simplest case looks
like this:
int x;
int r1;
r1 = READ_ONCE(x);
If r1 = 8 at the end then P0's accesses must have executed in program
order. We can deduce this from the operational model; if P0's load
had executed before its store then the value of the store would have
been forwarded to the load, so r1 would have ended up equal to 1, not
8. In this case there is a prop link from P0's write event to its read
event, because P1's store came after P0's store in x's coherence
order, and P1's store propagated to P0 before P0's load executed.
An equally simple case involves two loads of the same location that
read from different stores:
int x = 0;
int r1, r2;
r1 = READ_ONCE(x);
r2 = READ_ONCE(x);
If r1 = 0 and r2 = 9 at the end then P0's accesses must have executed
in program order. If the second load had executed before the first
then the x = 9 store must have been propagated to P0 before the first
load executed, and so r1 would have been 9 rather than 0. In this
case there is a prop link from P0's first read event to its second,
because P1's store overwrote the value read by P0's first load, and
P1's store propagated to P0 before P0's second load executed.
Less trivial examples of prop all involve fences. Unlike the simple
examples above, they can require that some instructions are executed
out of program order. This next one should look familiar:
int buf = 0, flag = 0;
WRITE_ONCE(buf, 1);
WRITE_ONCE(flag, 1);
int r1;
int r2;
r1 = READ_ONCE(flag);
r2 = READ_ONCE(buf);
This is the MP pattern again, with an smp_wmb() fence between the two
stores. If r1 = 1 and r2 = 0 at the end then there is a prop link
from P1's second load to its first (backwards!). The reason is
similar to the previous examples: The value P1 loads from buf gets
overwritten by P0's store to buf, the fence guarantees that the store
to buf will propagate to P1 before the store to flag does, and the
store to flag propagates to P1 before P1 reads flag.
The prop link says that in order to obtain the r1 = 1, r2 = 0 result,
P1 must execute its second load before the first. Indeed, if the load
from flag were executed first, then the buf = 1 store would already
have propagated to P1 by the time P1's load from buf executed, so r2
would have been 1 at the end, not 0. (The reasoning holds even for
Alpha, although the details are more complicated and we will not go
into them.)
But what if we put an smp_rmb() fence between P1's loads? The fence
would force the two loads to be executed in program order, and it
would generate a cycle in the hb relation: The fence would create a ppo
link (hence an hb link) from the first load to the second, and the
prop relation would give an hb link from the second load to the first.
Since an instruction can't execute before itself, we are forced to
conclude that if an smp_rmb() fence is added, the r1 = 1, r2 = 0
outcome is impossible -- as it should be.
The formal definition of the prop relation involves a coe or fre link,
followed by an arbitrary number of cumul-fence links, ending with an
rfe link. You can concoct more exotic examples, containing more than
one fence, although this quickly leads to diminishing returns in terms
of complexity. For instance, here's an example containing a coe link
followed by two cumul-fences and an rfe link, utilizing the fact that
release fences are A-cumulative:
int x, y, z;
int r0;
r0 = READ_ONCE(z);
int r2;
r2 = READ_ONCE(y);
smp_store_release(&z, 1);
If x = 2, r0 = 1, and r2 = 1 after this code runs then there is a prop
link from P0's store to its load. This is because P0's store gets
overwritten by P1's store since x = 2 at the end (a coe link), the
smp_wmb() ensures that P1's store to x propagates to P2 before the
store to y does (the first cumul-fence), the store to y propagates to P2
before P2's load and store execute, P2's smp_store_release()
guarantees that the stores to x and y both propagate to P0 before the
store to z does (the second cumul-fence), and P0's load executes after the
store to z has propagated to P0 (an rfe link).
In summary, the fact that the hb relation links memory access events
in the order they execute means that it must not have cycles. This
requirement is the content of the LKMM's "happens-before" axiom.
The LKMM defines yet another relation connected to times of
instruction execution, but it is not included in hb. It relies on the
particular properties of strong fences, which we cover in the next
The propagates-before (pb) relation capitalizes on the special
features of strong fences. It links two events E and F whenever some
store is coherence-later than E and propagates to every CPU and to RAM
before F executes. The formal definition requires that E be linked to
F via a coe or fre link, an arbitrary number of cumul-fences, an
optional rfe link, a strong fence, and an arbitrary number of hb
links. Let's see how this definition works out.
Consider first the case where E is a store (implying that the sequence
of links begins with coe). Then there are events W, X, Y, and Z such
E ->coe W ->cumul-fence* X ->rfe? Y ->strong-fence Z ->hb* F,
where the * suffix indicates an arbitrary number of links of the
specified type, and the ? suffix indicates the link is optional (Y may
be equal to X). Because of the cumul-fence links, we know that W will
propagate to Y's CPU before X does, hence before Y executes and hence
before the strong fence executes. Because this fence is strong, we
know that W will propagate to every CPU and to RAM before Z executes.
And because of the hb links, we know that Z will execute before F.
Thus W, which comes later than E in the coherence order, will
propagate to every CPU and to RAM before F executes.
The case where E is a load is exactly the same, except that the first
link in the sequence is fre instead of coe.
The existence of a pb link from E to F implies that E must execute
before F. To see why, suppose that F executed first. Then W would
have propagated to E's CPU before E executed. If E was a store, the
memory subsystem would then be forced to make E come after W in the
coherence order, contradicting the fact that E ->coe W. If E was a
load, the memory subsystem would then be forced to satisfy E's read
request with the value stored by W or an even later store,
contradicting the fact that E ->fre W.
A good example illustrating how pb works is the SB pattern with strong
int x = 0, y = 0;
int r0;
r0 = READ_ONCE(y);
int r1;
r1 = READ_ONCE(x);
If r0 = 0 at the end then there is a pb link from P0's load to P1's
load: an fre link from P0's load to P1's store (which overwrites the
value read by P0), and a strong fence between P1's store and its load.
In this example, the sequences of cumul-fence and hb links are empty.
Note that this pb link is not included in hb as an instance of prop,
because it does not start and end on the same CPU.
Similarly, if r1 = 0 at the end then there is a pb link from P1's load
to P0's. This means that if both r1 and r2 were 0 there would be a
cycle in pb, which is not possible since an instruction cannot execute
before itself. Thus, adding smp_mb() fences to the SB pattern
prevents the r0 = 0, r1 = 0 outcome.
In summary, the fact that the pb relation links events in the order
they execute means that it cannot have cycles. This requirement is
the content of the LKMM's "propagation" axiom.
RCU RELATIONS: rcu-link, rcu-gp, rcu-rscsi, 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.
A grace period is the span of time occupied by a call to
synchronize_rcu(). A read-side critical section (or just critical
section, for short) is a region of code delimited by rcu_read_lock()
at the start and rcu_read_unlock() at the end. Critical sections can
be nested, although we won't make use of this fact.
As far as memory models are concerned, RCU's main feature is its
Grace-Period Guarantee, which states that a critical section can never
span a full grace period. In more detail, the Guarantee says:
For any critical section C and any grace period G, at least
one of the following statements must hold:
(1) C ends before G does, and in addition, every store that
propagates to C's CPU before the end of C must propagate to
every CPU before G ends.
(2) G starts before C does, and in addition, every store that
propagates to G's CPU before the start of G must propagate
to every CPU before C starts.
In particular, it is not possible for a critical section to both start
before and end after a grace period.
Here is a simple example of RCU in action:
int x, y;
int r1, r2;
r1 = READ_ONCE(x);
r2 = READ_ONCE(y);
The Grace Period Guarantee tells us that when this code runs, it will
never end with r1 = 1 and r2 = 0. The reasoning is as follows. r1 = 1
means that P0's store to x propagated to P1 before P1 called
synchronize_rcu(), so P0's critical section must have started before
P1's grace period, contrary to part (2) of the Guarantee. On the
other hand, r2 = 0 means that P0's store to y, which occurs before the
end of the critical section, did not propagate to P1 before the end of
the grace period, contrary to part (1). Together the results violate
the Guarantee.
In the kernel's implementations of RCU, the requirements for stores
to propagate to every CPU are fulfilled by placing strong fences at
suitable places in the RCU-related code. Thus, if a critical section
starts before a grace period does then the critical section's CPU will
execute an smp_mb() fence after the end of the critical section and
some time before the grace period's synchronize_rcu() call returns.
And if a critical section ends after a grace period does then the
synchronize_rcu() routine will execute an smp_mb() fence at its start
and some time before the critical section's opening rcu_read_lock()
What exactly do we mean by saying that a critical section "starts
before" or "ends after" a grace period? Some aspects of the meaning
are pretty obvious, as in the example above, but the details aren't
entirely clear. The LKMM formalizes this notion by means of the
rcu-link relation. rcu-link encompasses a very general notion of
"before": If E and F are RCU fence events (i.e., rcu_read_lock(),
rcu_read_unlock(), or synchronize_rcu()) then among other things,
E ->rcu-link F includes cases where E is po-before some memory-access
event X, F is po-after some memory-access event Y, and we have any of
X ->rfe Y, X ->co Y, or X ->fr Y.
The formal definition of the rcu-link relation is more than a little
obscure, and we won't give it here. It is closely related to the pb
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 also defines the rcu-gp and rcu-rscsi relations. They bring
grace periods and read-side critical sections into the picture, in the
following way:
E ->rcu-gp F means that E and F are in fact the same event,
and that event is a synchronize_rcu() fence (i.e., a grace
E ->rcu-rscsi F means that E and F are the rcu_read_unlock()
and rcu_read_lock() fence events delimiting some read-side
critical section. (The 'i' at the end of the name emphasizes
that this relation is "inverted": It links the end of the
critical section to the start.)
If we think of the rcu-link relation as standing for an extended
"before", then X ->rcu-gp Y ->rcu-link Z roughly says that X is a
grace period which ends before Z begins. (In fact it covers more than
this, because it also includes cases where some store propagates to
Z's CPU before Z begins but doesn't propagate to some other CPU until
after X ends.) Similarly, X ->rcu-rscsi Y ->rcu-link Z says that X is
the end of a critical section which starts before Z begins.
The LKMM goes on to define the rcu-fence relation as a sequence of
rcu-gp and rcu-rscsi links separated by rcu-link links, in which the
number of rcu-gp links is >= the number of rcu-rscsi links. For
X ->rcu-gp Y ->rcu-link Z ->rcu-rscsi T ->rcu-link U ->rcu-gp V
would imply that X ->rcu-fence V, because this sequence contains two
rcu-gp links and one rcu-rscsi link. (It also implies that
X ->rcu-fence T and Z ->rcu-fence V.) On the other hand:
X ->rcu-rscsi Y ->rcu-link Z ->rcu-rscsi T ->rcu-link U ->rcu-gp V
does not imply X ->rcu-fence V, because the sequence contains only
one rcu-gp link but two rcu-rscsi 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,
E ->rcu-fence F implies not only that E begins before F ends, but also
that any write po-before E will propagate to every CPU before any
instruction po-after F can execute. (However, it does not imply that
E must execute before F; in fact, each synchronize_rcu() fence event
is linked to itself by rcu-fence as a degenerate case.)
To prove this in full generality requires some intellectual effort.
We'll consider just a very simple case:
G ->rcu-gp W ->rcu-link Z ->rcu-rscsi F.
This formula means that G and W are the same event (a grace period),
and there are events X, Y and a read-side critical section C such that:
1. G = W is po-before or equal to X;
2. X comes "before" Y in some sense (including rfe, co and fr);
2. Y is po-before Z;
4. Z is the rcu_read_unlock() event marking the end of C;
5. F is the rcu_read_lock() event marking the start of C.
From 1 - 4 we deduce that the grace period G ends before the critical
section C. Then part (2) of the Grace Period Guarantee says not only
that G starts before C does, but also that any write which executes on
G's CPU before G starts must propagate to every CPU before C starts.
In particular, the write propagates to every CPU before F finishes
executing and hence before any instruction po-after F can execute.
This sort of reasoning can be extended 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 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 rcu-gp and
rcu-rscsi alternating with rcu-link, where the number of rcu-gp links
is >= the number of rcu-rscsi 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 both parts of the Guarantee are
violated: 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
question, and let S be the synchronize_rcu() fence event for the grace
period. Saying that the critical section starts before S means there
are events Q and R where Q is po-after L (which marks the start of the
critical section), Q is "before" R in the sense used by the rcu-link
relation, and R is po-before the grace period S. Thus we have:
L ->rcu-link S.
Let W be the store mentioned above, let Y come before the end of the
critical section and witness that W propagates to the critical
section's CPU by reading from W, and let Z on some arbitrary CPU be a
witness that W has not propagated to that CPU, where Z happens after
some event X which is po-after S. Symbolically, this amounts to:
S ->po X ->hb* Z ->fr W ->rf Y ->po U.
The fr link from Z to W indicates that W has not propagated to Z's CPU
at the time that Z executes. From this, it can be shown (see the
discussion of the rcu-link relation earlier) that S and U are related
by rcu-link:
S ->rcu-link U.
Since S is a grace period we have S ->rcu-gp S, and since L and U are
the start and end of the critical section C we have U ->rcu-rscsi L.
From this we obtain:
S ->rcu-gp S ->rcu-link U ->rcu-rscsi L ->rcu-link S,
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
time with statement labels added:
int x, y;
L: rcu_read_lock();
X: WRITE_ONCE(x, 1);
Y: WRITE_ONCE(y, 1);
U: rcu_read_unlock();
int r1, r2;
Z: r1 = READ_ONCE(x);
S: synchronize_rcu();
W: r2 = READ_ONCE(y);
If r2 = 0 at the end then P0's store at Y overwrites the value that
P1's load at W reads from, so we have W ->fre Y. Since S ->po W and
also Y ->po U, we get S ->rcu-link U. In addition, S ->rcu-gp S
because S is a grace period.
If r1 = 1 at the end then P1's load at Z reads from P0's store at X,
so we have X ->rfe Z. Together with L ->po X and Z ->po S, this
yields L ->rcu-link S. And since L and U are the start and end of a
critical section, we have U ->rcu-rscsi L.
Then U ->rcu-rscsi L ->rcu-link S ->rcu-gp S ->rcu-link U 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:
int x, y, z;
int r0;
L0: rcu_read_lock();
r0 = READ_ONCE(x);
U0: rcu_read_unlock();
int r1;
r1 = READ_ONCE(y);
S1: synchronize_rcu();
int r2;
L2: rcu_read_lock();
r2 = READ_ONCE(z);
U2: rcu_read_unlock();
If r0 = r1 = r2 = 1 at the end, then similar reasoning to before shows
that U0 ->rcu-rscsi L0 ->rcu-link S1 ->rcu-gp S1 ->rcu-link U2 ->rcu-rscsi
L2 ->rcu-link U0. However this cycle is not forbidden, because the
sequence of relations contains fewer instances of rcu-gp (one) than of
rcu-rscsi (two). Consequently the outcome is allowed by the LKMM.
The following instruction timing diagram shows how it might actually
P0 P1 P2
-------------------- -------------------- --------------------
r1 = READ_ONCE(y)
synchronize_rcu() starts
. rcu_read_lock()
. WRITE_ONCE(x, 1)
r0 = READ_ONCE(x) .
rcu_read_unlock() .
synchronize_rcu() ends
r2 = READ_ONCE(z)
This requires P0 and P2 to execute their loads and stores out of
program order, but of course they are allowed to do so. And as you
can see, the Grace Period Guarantee is not violated: The critical
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
grace period does and ends after it does.
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.
The LKMM includes locking. In fact, there is special code for locking
in the formal model, added in order to make tools run faster.
However, this special code is intended to be more or less equivalent
to concepts we have already covered. A spinlock_t variable is treated
the same as an int, and spin_lock(&s) is treated almost the same as:
while (cmpxchg_acquire(&s, 0, 1) != 0)
This waits until s is equal to 0 and then atomically sets it to 1,
and the read part of the cmpxchg operation acts as an acquire fence.
An alternate way to express the same thing would be:
r = xchg_acquire(&s, 1);
along with a requirement that at the end, r = 0. Similarly,
spin_trylock(&s) is treated almost the same as:
return !cmpxchg_acquire(&s, 0, 1);
which atomically sets s to 1 if it is currently equal to 0 and returns
true if it succeeds (the read part of the cmpxchg operation acts as an
acquire fence only if the operation is successful). spin_unlock(&s)
is treated almost the same as:
smp_store_release(&s, 0);
The "almost" qualifiers above need some explanation. In the LKMM, the
store-release in a spin_unlock() and the load-acquire which forms the
first half of the atomic rmw update in a spin_lock() or a successful
spin_trylock() -- we can call these things lock-releases and
lock-acquires -- have two properties beyond those of ordinary releases
and acquires.
First, when a lock-acquire reads from a lock-release, the LKMM
requires that every instruction po-before the lock-release must
execute before any instruction po-after the lock-acquire. This would
naturally hold if the release and acquire operations were on different
CPUs, but the LKMM says it holds even when they are on the same CPU.
For example:
int x, y;
spinlock_t s;
int r1, r2;
r1 = READ_ONCE(x);
r2 = READ_ONCE(y);
Here the second spin_lock() reads from the first spin_unlock(), and
therefore the load of x must execute before the load of y. Thus we
cannot have r1 = 1 and r2 = 0 at the end (this is an instance of the
MP pattern).
This requirement does not apply to ordinary release and acquire
fences, only to lock-related operations. For instance, suppose P0()
in the example had been written as:
int r1, r2, r3;
r1 = READ_ONCE(x);
smp_store_release(&s, 1);
r3 = smp_load_acquire(&s);
r2 = READ_ONCE(y);
Then the CPU would be allowed to forward the s = 1 value from the
smp_store_release() to the smp_load_acquire(), executing the
instructions in the following order:
r3 = smp_load_acquire(&s); // Obtains r3 = 1
r2 = READ_ONCE(y);
r1 = READ_ONCE(x);
smp_store_release(&s, 1); // Value is forwarded
and thus it could load y before x, obtaining r2 = 0 and r1 = 1.
Second, when a lock-acquire reads from a lock-release, and some other
stores W and W' occur po-before the lock-release and po-after the
lock-acquire respectively, the LKMM requires that W must propagate to
each CPU before W' does. For example, consider:
int x, y;
spinlock_t x;
int r1;
r1 = READ_ONCE(x);
int r2, r3;
r2 = READ_ONCE(y);
r3 = READ_ONCE(x);
If r1 = 1 at the end then the spin_lock() in P1 must have read from
the spin_unlock() in P0. Hence the store to x must propagate to P2
before the store to y does, so we cannot have r2 = 1 and r3 = 0.
These two special requirements for lock-release and lock-acquire do
not arise from the operational model. Nevertheless, kernel developers
have come to expect and rely on them because they do hold on all
architectures supported by the Linux kernel, albeit for various
differing reasons.
This section covers material that didn't quite fit anywhere in the
earlier sections.
The descriptions in this document don't always match the formal
version of the LKMM exactly. For example, the actual formal
definition of the prop relation makes the initial coe or fre part
optional, and it doesn't require the events linked by the relation to
be on the same CPU. These differences are very unimportant; indeed,
instances where the coe/fre part of prop is missing are of no interest
because all the other parts (fences and rfe) are already included in
hb anyway, and where the formal model adds prop into hb, it includes
an explicit requirement that the events being linked are on the same
Another minor difference has to do with events that are both memory
accesses and fences, such as those corresponding to smp_load_acquire()
calls. In the formal model, these events aren't actually both reads
and fences; rather, they are read events with an annotation marking
them as acquires. (Or write events annotated as releases, in the case
smp_store_release().) The final effect is the same.
Although we didn't mention it above, the instruction execution
ordering provided by the smp_rmb() fence doesn't apply to read events
that are part of a non-value-returning atomic update. For instance,
r1 = READ_ONCE(y);
it is not guaranteed that the load from y will execute after the
update to x. This is because the ARMv8 architecture allows
non-value-returning atomic operations effectively to be executed off
the CPU. Basically, the CPU tells the memory subsystem to increment
x, and then the increment is carried out by the memory hardware with
no further involvement from the CPU. Since the CPU doesn't ever read
the value of x, there is nothing for the smp_rmb() fence to act on.
The LKMM defines a few extra synchronization operations in terms of
things we have already covered. In particular, rcu_dereference() is
treated as READ_ONCE() and rcu_assign_pointer() is treated as
smp_store_release() -- which is basically how the Linux kernel treats
There are a few oddball fences which need special treatment:
smp_mb__before_atomic(), smp_mb__after_atomic(), and
smp_mb__after_spinlock(). The LKMM uses fence events with special
annotations for them; they act as strong fences just like smp_mb()
except for the sets of events that they order. Instead of ordering
all po-earlier events against all po-later events, as smp_mb() does,
they behave as follows:
smp_mb__before_atomic() orders all po-earlier events against
po-later atomic updates and the events following them;
smp_mb__after_atomic() orders po-earlier atomic updates and
the events preceding them against all po-later events;
smp_mb_after_spinlock() orders po-earlier lock acquisition
events and the events preceding them against all po-later
Interestingly, RCU and locking each introduce the possibility of
deadlock. When faced with code sequences such as:
what does the LKMM have to say? Answer: It says there are no allowed
executions at all, which makes sense. But this can also lead to
misleading results, because if a piece of code has multiple possible
executions, some of which deadlock, the model will report only on the
non-deadlocking executions. For example:
int x, y;
int r0;
r0 = READ_ONCE(y);
if (READ_ONCE(x) > 0) {
WRITE_ONCE(y, 36);
Is it possible to end up with r0 = 36 at the end? The LKMM will tell
you it is not, but the model won't mention that this is because P1
will self-deadlock in the executions where it stores 36 in y.