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 Explanation of the Linux-Kernel Memory Consistency Model
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 :Author: Alan Stern <stern@rowland.harvard.edu>
 :Created: October 2017
 
 .. Contents
 
   1. INTRODUCTION
   2. BACKGROUND
   3. A SIMPLE EXAMPLE
   4. A SELECTION OF MEMORY MODELS
   5. ORDERING AND CYCLES
   6. EVENTS
   7. THE PROGRAM ORDER RELATION: po AND po-loc
   8. A WARNING
   9. DEPENDENCY RELATIONS: data, addr, and ctrl
   10. THE READS-FROM RELATION: rf, rfi, and rfe
   11. CACHE COHERENCE AND THE COHERENCE ORDER RELATION: co, coi, and coe
   12. THE FROM-READS RELATION: fr, fri, and fre
   13. AN OPERATIONAL MODEL
   14. PROPAGATION ORDER RELATION: cumul-fence
   15. DERIVATION OF THE LKMM FROM THE OPERATIONAL MODEL
   16. SEQUENTIAL CONSISTENCY PER VARIABLE
   17. ATOMIC UPDATES: rmw
   18. THE PRESERVED PROGRAM ORDER RELATION: ppo
   19. AND THEN THERE WAS ALPHA
   20. THE HAPPENS-BEFORE RELATION: hb
   21. THE PROPAGATES-BEFORE RELATION: pb
   22. RCU RELATIONS: rcu-link, rcu-gp, rcu-rscsi, rcu-order, rcu-fence, and rb
   23. LOCKING
   24. PLAIN ACCESSES AND DATA RACES
   25. ODDS AND ENDS
 
 
 
 INTRODUCTION
 ------------
 
 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 linux-kernel.cat 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.  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.
 
 
 BACKGROUND
 ----------
 
 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.
 
 
 A SIMPLE EXAMPLE
 ----------------
 
 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;
 
         P0()
         {
                 WRITE_ONCE(buf, 1);
                 WRITE_ONCE(flag, 1);
         }
 
         P1()
         {
                 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.
 
 
 A SELECTION OF MEMORY MODELS
 ----------------------------
 
 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, P0'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;
 
         P0()
         {
                 int r0;
 
                 WRITE_ONCE(x, 1);
                 r0 = READ_ONCE(y);
         }
 
         P1()
         {
                 int r1;
 
                 WRITE_ONCE(y, 1);
                 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.
 
 
 ORDERING AND CYCLES
 -------------------
 
 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.
 
 
 EVENTS
 ------
 
 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
         rcu_dereference().
 
         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 PROGRAM ORDER RELATION: po AND po-loc
 -----------------------------------------
 
 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').
 
 
 A WARNING
 ---------
 
 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
 location:
 
         r1 = READ_ONCE(x);
         if (r1) {
                 WRITE_ONCE(y, 2);
                 ...  /* do something */
         } else {
                 WRITE_ONCE(y, 2);
                 ...  /* 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);
         WRITE_ONCE(y, 2);
         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
 example:
 
         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 function calls 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
 pointer.
 
 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!
 
 Here's a trick question: When is a dependency not a dependency?  Answer:
 When it is purely syntactic rather than semantic.  We say a dependency
 between two accesses is purely syntactic if the second access doesn't
 actually depend on the result of the first.  Here is a trivial example:
 
         r1 = READ_ONCE(x);
         WRITE_ONCE(y, r1 * 0);
 
 There appears to be a data dependency from the load of x to the store
 of y, since the value to be stored is computed from the value that was
 loaded.  But in fact, the value stored does not really depend on
 anything since it will always be 0.  Thus the data dependency is only
 syntactic (it appears to exist in the code) but not semantic (the
 second access will always be the same, regardless of the value of the
 first access).  Given code like this, a compiler could simply discard
 the value returned by the load from x, which would certainly destroy
 any dependency.  (The compiler is not permitted to eliminate entirely
 the load generated for a READ_ONCE() -- that's one of the nice
 properties of READ_ONCE() -- but it is allowed to ignore the load's
 value.)
 
 It's natural to object that no one in their right mind would write
 code like the above.  However, macro expansions can easily give rise
 to this sort of thing, in ways that often are not apparent to the
 programmer.
 
 Another mechanism that can lead to purely syntactic dependencies is
 related to the notion of "undefined behavior".  Certain program
 behaviors are called "undefined" in the C language specification,
 which means that when they occur there are no guarantees at all about
 the outcome.  Consider the following example:
 
         int a[1];
         int i;
 
         r1 = READ_ONCE(i);
         r2 = READ_ONCE(a[r1]);
 
 Access beyond the end or before the beginning of an array is one kind
 of undefined behavior.  Therefore the compiler doesn't have to worry
 about what will happen if r1 is nonzero, and it can assume that r1
 will always be zero regardless of the value actually loaded from i.
 (If the assumption turns out to be wrong the resulting behavior will
 be undefined anyway, so the compiler doesn't care!)  Thus the value
 from the load can be discarded, breaking the address dependency.
 
 The LKMM is unaware that purely syntactic dependencies are different
 from semantic dependencies and therefore mistakenly predicts that the
 accesses in the two examples above will be ordered.  This is another
 example of how the compiler can undermine the memory model.  Be warned.
 
 
 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 main 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;
 
         P0()
         {
                 WRITE_ONCE(x, 0x1234);
         }
 
         P1()
         {
                 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;
 
         P0()
         {
                 WRITE_ONCE(x.h[0], 0x1234);
                 WRITE_ONCE(x.h[1], 0x5678);
         }
 
         P1()
         {
                 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 AND THE COHERENCE ORDER RELATION: co, coi, and coe
 ------------------------------------------------------------------
 
 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
 errors:
 
         int x;
 
         P0()
         {
                 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;
 
         P0()
         {
                 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
 value).
 
         int x = 0;
 
         P0()
         {
                 WRITE_ONCE(x, 5);
         }
 
         P1()
         {
                 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
 locations.)
 
 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;
 
         P0()
         {
                 int r1;
 
                 r1 = READ_ONCE(x);
                 WRITE_ONCE(x, 2);
         }
 
 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
 them.
 
 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).
 
 
 AN OPERATIONAL MODEL
 --------------------
 
 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
 more.
 
 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
         instructions;
 
         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
         smp_store_release()).
 
 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.
 
 
 PROPAGATION ORDER RELATION: cumul-fence
 ---------------------------------------
 
 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'.
 
 
 DERIVATION OF THE LKMM FROM THE OPERATIONAL MODEL
 -------------------------------------------------
 
 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 six requirements, or
 axioms:
 
         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.
 
         Plain-coherence: This requires that plain memory accesses
         (those not using READ_ONCE(), WRITE_ONCE(), etc.) must obey
         the operational model's rules regarding cache coherence.
 
 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" and "plain-coherence" axioms are specific to the LKMM.
 
 Each of these axioms is discussed below.
 
 
 SEQUENTIAL CONSISTENCY PER VARIABLE
 -----------------------------------
 
 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
 obeyed.
 
 
 ATOMIC UPDATES: rmw
 -------------------
 
 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
 performs.
 
 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.
 
 
 THE PRESERVED PROGRAM ORDER RELATION: ppo
 -----------------------------------------
 
 There are many situations where a CPU is obliged 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
 po.
 
 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
         them;
 
         X and Y are both loads and an smp_rmb() fence occurs between
         them;
 
         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
         access.
 
 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
 this.)
 
 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!)
 
 
 AND THEN THERE WAS ALPHA
 ------------------------
 
 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;
 
         P0()
         {
                 WRITE_ONCE(x, 1);
                 smp_wmb();
                 WRITE_ONCE(ptr, &x);
         }
 
         P1()
         {
                 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 after every READ_ONCE() and atomic load on Alpha.  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:
 
         P1()
         {
                 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
 directly.
 
 The LKMM requires that smp_rmb(), acquire fences, and strong fences
 share this property: 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
 Alpha.
 
 
 THE HAPPENS-BEFORE RELATION: hb
 -------------------------------
 
 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;
 
         P0()
         {
                 int r1;
 
                 WRITE_ONCE(x, 1);
                 r1 = READ_ONCE(x);
         }
 
         P1()
         {
                 WRITE_ONCE(x, 8);
         }
 
 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;
 
         P0()
         {
                 int r1, r2;
 
                 r1 = READ_ONCE(x);
                 r2 = READ_ONCE(x);
         }
 
         P1()
         {
                 WRITE_ONCE(x, 9);
         }
 
 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;
 
         P0()
         {
                 WRITE_ONCE(buf, 1);
                 smp_wmb();
                 WRITE_ONCE(flag, 1);
         }
 
         P1()
         {
                 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;
 
         P0()
         {
                 int r0;
 
                 WRITE_ONCE(x, 1);
                 r0 = READ_ONCE(z);
         }
 
         P1()
         {
                 WRITE_ONCE(x, 2);
                 smp_wmb();
                 WRITE_ONCE(y, 1);
         }
 
         P2()
         {
                 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
 section.
 
 
 THE PROPAGATES-BEFORE RELATION: pb
 ----------------------------------
 
 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
 that:
 
         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
 fences:
 
         int x = 0, y = 0;
 
         P0()
         {
                 int r0;
 
                 WRITE_ONCE(x, 1);
                 smp_mb();
                 r0 = READ_ONCE(y);
         }
 
         P1()
         {
                 int r1;
 
                 WRITE_ONCE(y, 1);
                 smp_mb();
                 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-order, 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;
 
         P0()
         {
                 rcu_read_lock();
                 WRITE_ONCE(x, 1);
                 WRITE_ONCE(y, 1);
                 rcu_read_unlock();
         }
 
         P1()
         {
                 int r1, r2;
 
                 r1 = READ_ONCE(x);
                 synchronize_rcu();
                 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()
 executes.
 
 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
         period).
 
         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-order 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
 example:
 
         X ->rcu-gp Y ->rcu-link Z ->rcu-rscsi T ->rcu-link U ->rcu-gp V
 
 would imply that X ->rcu-order V, because this sequence contains two
 rcu-gp links and one rcu-rscsi link.  (It also implies that
 X ->rcu-order T and Z ->rcu-order 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-order V, because the sequence contains only
 one rcu-gp link but two rcu-rscsi links.
 
 The rcu-order relation is important because the Grace Period Guarantee
 means that rcu-order links act kind of like strong fences.  In
 particular, E ->rcu-order 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-order 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);
 
         3. 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-order.
 
 The rcu-fence relation is a simple extension of rcu-order.  While
 rcu-order only links certain fence events (calls to synchronize_rcu(),
 rcu_read_lock(), or rcu_read_unlock()), rcu-fence links any events
 that are separated by an rcu-order link.  This is analogous to the way
 the strong-fence relation links events that are separated by an
 smp_mb() fence event (as mentioned above, rcu-order links act kind of
 like strong fences).  Written symbolically, X ->rcu-fence Y means
 there are fence events E and F such that:
 
         X ->po E ->rcu-order F ->po Y.
 
 From the discussion above, we see this implies not only that X
 executes before Y, but also (if X is a store) that X propagates to
 every CPU before Y executes.  Thus rcu-fence is sort of a
 "super-strong" fence: Unlike the original strong fences (smp_mb() and
 synchronize_rcu()), rcu-fence is able to link events on different
 CPUs.  (Perhaps this fact should lead us to say that rcu-fence isn't
 really a fence at all!)
 
 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-order 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;
 
         P0()
         {
                 L: rcu_read_lock();
                 X: WRITE_ONCE(x, 1);
                 Y: WRITE_ONCE(y, 1);
                 U: rcu_read_unlock();
         }
 
         P1()
         {
                 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;
 
         P0()
         {
                 int r0;
 
                 L0: rcu_read_lock();
                     r0 = READ_ONCE(x);
                     WRITE_ONCE(y, 1);
                 U0: rcu_read_unlock();
         }
 
         P1()
         {
                 int r1;
 
                     r1 = READ_ONCE(y);
                 S1: synchronize_rcu();
                     WRITE_ONCE(z, 1);
         }
 
         P2()
         {
                 int r2;
 
                 L2: rcu_read_lock();
                     r2 = READ_ONCE(z);
                     WRITE_ONCE(x, 1);
                 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
 occur:
 
 P0                      P1                      P2
 --------------------    --------------------    --------------------
 rcu_read_lock()
 WRITE_ONCE(y, 1)
                         r1 = READ_ONCE(y)
                         synchronize_rcu() starts
                         .                       rcu_read_lock()
                         .                       WRITE_ONCE(x, 1)
 r0 = READ_ONCE(x)       .
 rcu_read_unlock()       .
                         synchronize_rcu() ends
                         WRITE_ONCE(z, 1)
                                                 r2 = READ_ONCE(z)
                                                 rcu_read_unlock()
 
 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-order 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.
 
 
 LOCKING
 -------
 
 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)
                 cpu_relax();
 
 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 or is po-after 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 and accessed the same lock variable, but the LKMM says
 it also holds when they are on the same CPU, even if they access
 different lock variables.  For example:
 
         int x, y;
         spinlock_t s, t;
 
         P0()
         {
                 int r1, r2;
 
                 spin_lock(&s);
                 r1 = READ_ONCE(x);
                 spin_unlock(&s);
                 spin_lock(&t);
                 r2 = READ_ONCE(y);
                 spin_unlock(&t);
         }
 
         P1()
         {
                 WRITE_ONCE(y, 1);
                 smp_wmb();
                 WRITE_ONCE(x, 1);
         }
 
 Here the second spin_lock() is po-after the first spin_unlock(), and
 therefore the load of x must execute before the load of y, even though
 the two locking operations use different locks.  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:
 
         P0()
         {
                 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 or is po-after 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 s;
 
         P0()
         {
                 spin_lock(&s);
                 WRITE_ONCE(x, 1);
                 spin_unlock(&s);
         }
 
         P1()
         {
                 int r1;
 
                 spin_lock(&s);
                 r1 = READ_ONCE(x);
                 WRITE_ONCE(y, 1);
                 spin_unlock(&s);
         }
 
         P2()
         {
                 int r2, r3;
 
                 r2 = READ_ONCE(y);
                 smp_rmb();
                 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.  But
 if P1 had used a lock variable different from s, the writes could have
 propagated in either order.  (On the other hand, if the code in P0 and
 P1 had all executed on a single CPU, as in the example before this
 one, then the writes would have propagated in order even if the two
 critical sections used different lock variables.)
 
 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.
 
 
 PLAIN ACCESSES AND DATA RACES
 -----------------------------
 
 In the LKMM, memory accesses such as READ_ONCE(x), atomic_inc(&y),
 smp_load_acquire(&z), and so on are collectively referred to as
 "marked" accesses, because they are all annotated with special
 operations of one kind or another.  Ordinary C-language memory
 accesses such as x or y = 0 are simply called "plain" accesses.
 
 Early versions of the LKMM had nothing to say about plain accesses.
 The C standard allows compilers to assume that the variables affected
 by plain accesses are not concurrently read or written by any other
 threads or CPUs.  This leaves compilers free to implement all manner
 of transformations or optimizations of code containing plain accesses,
 making such code very difficult for a memory model to handle.
 
 Here is just one example of a possible pitfall:
 
         int a = 6;
         int *x = &a;
 
         P0()
         {
                 int *r1;
                 int r2 = 0;
 
                 r1 = x;
                 if (r1 != NULL)
                         r2 = READ_ONCE(*r1);
         }
 
         P1()
         {
                 WRITE_ONCE(x, NULL);
         }
 
 On the face of it, one would expect that when this code runs, the only
 possible final values for r2 are 6 and 0, depending on whether or not
 P1's store to x propagates to P0 before P0's load from x executes.
 But since P0's load from x is a plain access, the compiler may decide
 to carry out the load twice (for the comparison against NULL, then again
 for the READ_ONCE()) and eliminate the temporary variable r1.  The
 object code generated for P0 could therefore end up looking rather
 like this:
 
         P0()
         {
                 int r2 = 0;
 
                 if (x != NULL)
                         r2 = READ_ONCE(*x);
         }
 
 And now it is obvious that this code runs the risk of dereferencing a
 NULL pointer, because P1's store to x might propagate to P0 after the
 test against NULL has been made but before the READ_ONCE() executes.
 If the original code had said "r1 = READ_ONCE(x)" instead of "r1 = x",
 the compiler would not have performed this optimization and there
 would be no possibility of a NULL-pointer dereference.
 
 Given the possibility of transformations like this one, the LKMM
 doesn't try to predict all possible outcomes of code containing plain
 accesses.  It is instead content to determine whether the code
 violates the compiler's assumptions, which would render the ultimate
 outcome undefined.
 
 In technical terms, the compiler is allowed to assume that when the
 program executes, there will not be any data races.  A "data race"
 occurs when there are two memory accesses such that:
 
 1.      they access the same location,
 
 2.      at least one of them is a store,
 
 3.      at least one of them is plain,
 
 4.      they occur on different CPUs (or in different threads on the
         same CPU), and
 
 5.      they execute concurrently.
 
 In the literature, two accesses are said to "conflict" if they satisfy
 1 and 2 above.  We'll go a little farther and say that two accesses
 are "race candidates" if they satisfy 1 - 4.  Thus, whether or not two
 race candidates actually do race in a given execution depends on
 whether they are concurrent.
 
 The LKMM tries to determine whether a program contains race candidates
 which may execute concurrently; if it does then the LKMM says there is
 a potential data race and makes no predictions about the program's
 outcome.
 
 Determining whether two accesses are race candidates is easy; you can
 see that all the concepts involved in the definition above are already
 part of the memory model.  The hard part is telling whether they may
 execute concurrently.  The LKMM takes a conservative attitude,
 assuming that accesses may be concurrent unless it can prove they
 are not.
 
 If two memory accesses aren't concurrent then one must execute before
 the other.  Therefore the LKMM decides two accesses aren't concurrent
 if they can be connected by a sequence of hb, pb, and rb links
 (together referred to as xb, for "executes before").  However, there
 are two complicating factors.
 
 If X is a load and X executes before a store Y, then indeed there is
 no danger of X and Y being concurrent.  After all, Y can't have any
 effect on the value obtained by X until the memory subsystem has
 propagated Y from its own CPU to X's CPU, which won't happen until
 some time after Y executes and thus after X executes.  But if X is a
 store, then even if X executes before Y it is still possible that X
 will propagate to Y's CPU just as Y is executing.  In such a case X
 could very well interfere somehow with Y, and we would have to
 consider X and Y to be concurrent.
 
 Therefore when X is a store, for X and Y to be non-concurrent the LKMM
 requires not only that X must execute before Y but also that X must
 propagate to Y's CPU before Y executes.  (Or vice versa, of course, if
 Y executes before X -- then Y must propagate to X's CPU before X
 executes if Y is a store.)  This is expressed by the visibility
 relation (vis), where X ->vis Y is defined to hold if there is an
 intermediate event Z such that:
 
         X is connected to Z by a possibly empty sequence of
         cumul-fence links followed by an optional rfe link (if none of
         these links are present, X and Z are the same event),
 
 and either:
 
         Z is connected to Y by a strong-fence link followed by a
         possibly empty sequence of xb links,
 
 or:
 
         Z is on the same CPU as Y and is connected to Y by a possibly
         empty sequence of xb links (again, if the sequence is empty it
         means Z and Y are the same event).
 
 The motivations behind this definition are straightforward:
 
         cumul-fence memory barriers force stores that are po-before
         the barrier to propagate to other CPUs before stores that are
         po-after the barrier.
 
         An rfe link from an event W to an event R says that R reads
         from W, which certainly means that W must have propagated to
         R's CPU before R executed.
 
         strong-fence memory barriers force stores that are po-before
         the barrier, or that propagate to the barrier's CPU before the
         barrier executes, to propagate to all CPUs before any events
         po-after the barrier can execute.
 
 To see how this works out in practice, consider our old friend, the MP
 pattern (with fences and statement labels, but without the conditional
 test):
 
         int buf = 0, flag = 0;
 
         P0()
         {
                 X: WRITE_ONCE(buf, 1);
                    smp_wmb();
                 W: WRITE_ONCE(flag, 1);
         }
 
         P1()
         {
                 int r1;
                 int r2 = 0;
 
                 Z: r1 = READ_ONCE(flag);
                    smp_rmb();
                 Y: r2 = READ_ONCE(buf);
         }
 
 The smp_wmb() memory barrier gives a cumul-fence link from X to W, and
 assuming r1 = 1 at the end, there is an rfe link from W to Z.  This
 means that the store to buf must propagate from P0 to P1 before Z
 executes.  Next, Z and Y are on the same CPU and the smp_rmb() fence
 provides an xb link from Z to Y (i.e., it forces Z to execute before
 Y).  Therefore we have X ->vis Y: X must propagate to Y's CPU before Y
 executes.
 
 The second complicating factor mentioned above arises from the fact
 that when we are considering data races, some of the memory accesses
 are plain.  Now, although we have not said so explicitly, up to this
 point most of the relations defined by the LKMM (ppo, hb, prop,
 cumul-fence, pb, and so on -- including vis) apply only to marked
 accesses.
 
 There are good reasons for this restriction.  The compiler is not
 allowed to apply fancy transformations to marked accesses, and
 consequently each such access in the source code corresponds more or
 less directly to a single machine instruction in the object code.  But
 plain accesses are a different story; the compiler may combine them,
 split them up, duplicate them, eliminate them, invent new ones, and
 who knows what else.  Seeing a plain access in the source code tells
 you almost nothing about what machine instructions will end up in the
 object code.
 
 Fortunately, the compiler isn't completely free; it is subject to some
 limitations.  For one, it is not allowed to introduce a data race into
 the object code if the source code does not already contain a data
 race (if it could, memory models would be useless and no multithreaded
 code would be safe!).  For another, it cannot move a plain access past
 a compiler barrier.
 
 A compiler barrier is a kind of fence, but as the name implies, it
 only affects the compiler; it does not necessarily have any effect on
 how instructions are executed by the CPU.  In Linux kernel source
 code, the barrier() function is a compiler barrier.  It doesn't give
 rise directly to any machine instructions in the object code; rather,
 it affects how the compiler generates the rest of the object code.
 Given source code like this:
 
         ... some memory accesses ...
         barrier();
         ... some other memory accesses ...
 
 the barrier() function ensures that the machine instructions
 corresponding to the first group of accesses will all end po-before
 any machine instructions corresponding to the second group of accesses
 -- even if some of the accesses are plain.  (Of course, the CPU may
 then execute some of those accesses out of program order, but we
 already know how to deal with such issues.)  Without the barrier()
 there would be no such guarantee; the two groups of accesses could be
 intermingled or even reversed in the object code.
 
 The LKMM doesn't say much about the barrier() function, but it does
 require that all fences are also compiler barriers.  In addition, it
 requires that the ordering properties of memory barriers such as
 smp_rmb() or smp_store_release() apply to plain accesses as well as to
 marked accesses.
 
 This is the key to analyzing data races.  Consider the MP pattern
 again, now using plain accesses for buf:
 
         int buf = 0, flag = 0;
 
         P0()
         {
                 U: buf = 1;
                    smp_wmb();
                 X: WRITE_ONCE(flag, 1);
         }
 
         P1()
         {
                 int r1;
                 int r2 = 0;
 
                 Y: r1 = READ_ONCE(flag);
                    if (r1) {
                            smp_rmb();
                         V: r2 = buf;
                    }
         }
 
 This program does not contain a data race.  Although the U and V
 accesses are race candidates, the LKMM can prove they are not
 concurrent as follows:
 
         The smp_wmb() fence in P0 is both a compiler barrier and a
         cumul-fence.  It guarantees that no matter what hash of
         machine instructions the compiler generates for the plain
         access U, all those instructions will be po-before the fence.
         Consequently U's store to buf, no matter how it is carried out
         at the machine level, must propagate to P1 before X's store to
         flag does.
 
         X and Y are both marked accesses.  Hence an rfe link from X to
         Y is a valid indicator that X propagated to P1 before Y
         executed, i.e., X ->vis Y.  (And if there is no rfe link then
         r1 will be 0, so V will not be executed and ipso facto won't
         race with U.)
 
         The smp_rmb() fence in P1 is a compiler barrier as well as a
         fence.  It guarantees that all the machine-level instructions
         corresponding to the access V will be po-after the fence, and
         therefore any loads among those instructions will execute
         after the fence does and hence after Y does.
 
 Thus U's store to buf is forced to propagate to P1 before V's load
 executes (assuming V does execute), ruling out the possibility of a
 data race between them.
 
 This analysis illustrates how the LKMM deals with plain accesses in
 general.  Suppose R is a plain load and we want to show that R
 executes before some marked access E.  We can do this by finding a
 marked access X such that R and X are ordered by a suitable fence and
 X ->xb* E.  If E was also a plain access, we would also look for a
 marked access Y such that X ->xb* Y, and Y and E are ordered by a
 fence.  We describe this arrangement by saying that R is
 "post-bounded" by X and E is "pre-bounded" by Y.
 
 In fact, we go one step further: Since R is a read, we say that R is
 "r-post-bounded" by X.  Similarly, E would be "r-pre-bounded" or
 "w-pre-bounded" by Y, depending on whether E was a store or a load.
 This distinction is needed because some fences affect only loads
 (i.e., smp_rmb()) and some affect only stores (smp_wmb()); otherwise
 the two types of bounds are the same.  And as a degenerate case, we
 say that a marked access pre-bounds and post-bounds itself (e.g., if R
 above were a marked load then X could simply be taken to be R itself.)
 
 The need to distinguish between r- and w-bounding raises yet another
 issue.  When the source code contains a plain store, the compiler is
 allowed to put plain loads of the same location into the object code.
 For example, given the source code:
 
         x = 1;
 
 the compiler is theoretically allowed to generate object code that
 looks like:
 
         if (x != 1)
                 x = 1;
 
 thereby adding a load (and possibly replacing the store entirely).
 For this reason, whenever the LKMM requires a plain store to be
 w-pre-bounded or w-post-bounded by a marked access, it also requires
 the store to be r-pre-bounded or r-post-bounded, so as to handle cases
 where the compiler adds a load.
 
 (This may be overly cautious.  We don't know of any examples where a
 compiler has augmented a store with a load in this fashion, and the
 Linux kernel developers would probably fight pretty hard to change a
 compiler if it ever did this.  Still, better safe than sorry.)
 
 Incidentally, the other tranformation -- augmenting a plain load by
 adding in a store to the same location -- is not allowed.  This is
 because the compiler cannot know whether any other CPUs might perform
 a concurrent load from that location.  Two concurrent loads don't
 constitute a race (they can't interfere with each other), but a store
 does race with a concurrent load.  Thus adding a store might create a
 data race where one was not already present in the source code,
 something the compiler is forbidden to do.  Augmenting a store with a
 load, on the other hand, is acceptable because doing so won't create a
 data race unless one already existed.
 
 The LKMM includes a second way to pre-bound plain accesses, in
 addition to fences: an address dependency from a marked load.  That
 is, in the sequence:
 
         p = READ_ONCE(ptr);
         r = *p;
 
 the LKMM says that the marked load of ptr pre-bounds the plain load of
 *p; the marked load must execute before any of the machine
 instructions corresponding to the plain load.  This is a reasonable
 stipulation, since after all, the CPU can't perform the load of *p
 until it knows what value p will hold.  Furthermore, without some
 assumption like this one, some usages typical of RCU would count as
 data races.  For example:
 
         int a = 1, b;
         int *ptr = &a;
 
         P0()
         {
                 b = 2;
                 rcu_assign_pointer(ptr, &b);
         }
 
         P1()
         {
                 int *p;
                 int r;
 
                 rcu_read_lock();
                 p = rcu_dereference(ptr);
                 r = *p;
                 rcu_read_unlock();
         }
 
 (In this example the rcu_read_lock() and rcu_read_unlock() calls don't
 really do anything, because there aren't any grace periods.  They are
 included merely for the sake of good form; typically P0 would call
 synchronize_rcu() somewhere after the rcu_assign_pointer().)
 
 rcu_assign_pointer() performs a store-release, so the plain store to b
 is definitely w-post-bounded before the store to ptr, and the two
 stores will propagate to P1 in that order.  However, rcu_dereference()
 is only equivalent to READ_ONCE().  While it is a marked access, it is
 not a fence or compiler barrier.  Hence the only guarantee we have
 that the load of ptr in P1 is r-pre-bounded before the load of *p
 (thus avoiding a race) is the assumption about address dependencies.
 
 This is a situation where the compiler can undermine the memory model,
 and a certain amount of care is required when programming constructs
 like this one.  In particular, comparisons between the pointer and
 other known addresses can cause trouble.  If you have something like:
 
         p = rcu_dereference(ptr);
         if (p == &x)
                 r = *p;
 
 then the compiler just might generate object code resembling:
 
         p = rcu_dereference(ptr);
         if (p == &x)
                 r = x;
 
 or even:
 
         rtemp = x;
         p = rcu_dereference(ptr);
         if (p == &x)
                 r = rtemp;
 
 which would invalidate the memory model's assumption, since the CPU
 could now perform the load of x before the load of ptr (there might be
 a control dependency but no address dependency at the machine level).
 
 Finally, it turns out there is a situation in which a plain write does
 not need to be w-post-bounded: when it is separated from the other
 race-candidate access by a fence.  At first glance this may seem
 impossible.  After all, to be race candidates the two accesses must
 be on different CPUs, and fences don't link events on different CPUs.
 Well, normal fences don't -- but rcu-fence can!  Here's an example:
 
         int x, y;
 
         P0()
         {
                 WRITE_ONCE(x, 1);
                 synchronize_rcu();
                 y = 3;
         }
 
         P1()
         {
                 rcu_read_lock();
                 if (READ_ONCE(x) == 0)
                         y = 2;
                 rcu_read_unlock();
         }
 
 Do the plain stores to y race?  Clearly not if P1 reads a non-zero
 value for x, so let's assume the READ_ONCE(x) does obtain 0.  This
 means that the read-side critical section in P1 must finish executing
 before the grace period in P0 does, because RCU's Grace-Period
 Guarantee says that otherwise P0's store to x would have propagated to
 P1 before the critical section started and so would have been visible
 to the READ_ONCE().  (Another way of putting it is that the fre link
 from the READ_ONCE() to the WRITE_ONCE() gives rise to an rcu-link
 between those two events.)
 
 This means there is an rcu-fence link from P1's "y = 2" store to P0's
 "y = 3" store, and consequently the first must propagate from P1 to P0
 before the second can execute.  Therefore the two stores cannot be
 concurrent and there is no race, even though P1's plain store to y
 isn't w-post-bounded by any marked accesses.
 
 Putting all this material together yields the following picture.  For
 race-candidate stores W and W', where W ->co W', the LKMM says the
 stores don't race if W can be linked to W' by a
 
         w-post-bounded ; vis ; w-pre-bounded
 
 sequence.  If W is plain then they also have to be linked by an
 
         r-post-bounded ; xb* ; w-pre-bounded
 
 sequence, and if W' is plain then they also have to be linked by a
 
         w-post-bounded ; vis ; r-pre-bounded
 
 sequence.  For race-candidate load R and store W, the LKMM says the
 two accesses don't race if R can be linked to W by an
 
         r-post-bounded ; xb* ; w-pre-bounded
 
 sequence or if W can be linked to R by a
 
         w-post-bounded ; vis ; r-pre-bounded
 
 sequence.  For the cases involving a vis link, the LKMM also accepts
 sequences in which W is linked to W' or R by a
 
         strong-fence ; xb* ; {w and/or r}-pre-bounded
 
 sequence with no post-bounding, and in every case the LKMM also allows
 the link simply to be a fence with no bounding at all.  If no sequence
 of the appropriate sort exists, the LKMM says that the accesses race.
 
 There is one more part of the LKMM related to plain accesses (although
 not to data races) we should discuss.  Recall that many relations such
 as hb are limited to marked accesses only.  As a result, the
 happens-before, propagates-before, and rcu axioms (which state that
 various relation must not contain a cycle) doesn't apply to plain
 accesses.  Nevertheless, we do want to rule out such cycles, because
 they don't make sense even for plain accesses.
 
 To this end, the LKMM imposes three extra restrictions, together
 called the "plain-coherence" axiom because of their resemblance to the
 rules used by the operational model to ensure cache coherence (that
 is, the rules governing the memory subsystem's choice of a store to
 satisfy a load request and its determination of where a store will
 fall in the coherence order):
 
         If R and W are race candidates and it is possible to link R to
         W by one of the xb* sequences listed above, then W ->rfe R is
         not allowed (i.e., a load cannot read from a store that it
         executes before, even if one or both is plain).
 
         If W and R are race candidates and it is possible to link W to
         R by one of the vis sequences listed above, then R ->fre W is
         not allowed (i.e., if a store is visible to a load then the
         load must read from that store or one coherence-after it).
 
         If W and W' are race candidates and it is possible to link W
         to W' by one of the vis sequences listed above, then W' ->co W
         is not allowed (i.e., if one store is visible to a second then
         the second must come after the first in the coherence order).
 
 This is the extent to which the LKMM deals with plain accesses.
 Perhaps it could say more (for example, plain accesses might
 contribute to the ppo relation), but at the moment it seems that this
 minimal, conservative approach is good enough.
 
 
 ODDS AND ENDS
 -------------
 
 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
 CPU.
 
 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,
 given:
 
         atomic_inc(&x);
         smp_rmb();
         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
 them.
 
 Although we said that plain accesses are not linked by the ppo
 relation, they do contribute to it indirectly.  Namely, when there is
 an address dependency from a marked load R to a plain store W,
 followed by smp_wmb() and then a marked store W', the LKMM creates a
 ppo link from R to W'.  The reasoning behind this is perhaps a little
 shaky, but essentially it says there is no way to generate object code
 for this source code in which W' could execute before R.  Just as with
 pre-bounding by address dependencies, it is possible for the compiler
 to undermine this relation if sufficient care is not taken.
 
 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
         events.
 
 Interestingly, RCU and locking each introduce the possibility of
 deadlock.  When faced with code sequences such as:
 
         spin_lock(&s);
         spin_lock(&s);
         spin_unlock(&s);
         spin_unlock(&s);
 
 or:
 
         rcu_read_lock();
         synchronize_rcu();
         rcu_read_unlock();
 
 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;
 
         P0()
         {
                 int r0;
 
                 WRITE_ONCE(x, 1);
                 r0 = READ_ONCE(y);
         }
 
         P1()
         {
                 rcu_read_lock();
                 if (READ_ONCE(x) > 0) {
                         WRITE_ONCE(y, 36);
                         synchronize_rcu();
                 }
                 rcu_read_unlock();
         }
 
 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.

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