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X86-TSO: A Rigorous and Usable Programmer's Model For X86 Multiprocessors

Tilera multicore Tile GX processor

Credit: Tilera

Exploiting the multiprocessors that have recently become ubiquitous requires high-performance and reliable concurrent systems code, for concurrent data structures, operating system kernels, synchronization libraries, compilers, and so on. However, concurrent programming, which is always challenging, is made much more so by two problems. First, real multiprocessors typically do not provide the sequentially consistent memory that is assumed by most work on semantics and verification. Instead, they have relaxed memory models, varying in subtle ways between processor families, in which different hardware threads may have only loosely consistent views of a shared memory. Second, the public vendor architectures, supposedly specifying what programmers can rely on, are often in ambiguous informal prose (a particularly poor medium for loose specifications), leading to widespread confusion.

In this paper we focus on x86 processors. We review several recent Intel and AMD specifications, showing that all contain serious ambiguities, some are arguably too weak to program above, and some are simply unsound with respect to actual hardware. We present a new x86-TSO programmer's model that, to the best of our knowledge, suffers from none of these problems. It is mathematically precise (rigorously defined in HOL4) but can be presented as an intuitive abstract machine which should be widely accessible to working programmers. We illustrate how this can be used to reason about the correctness of a Linux spinlock implementation and describe a general theory of data-race freedom for x86-TSO. This should put x86 multiprocessor system building on a more solid foundation; it should also provide a basis for future work on verification of such systems.

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1. Introduction

Multiprocessor machines, with many processors acting on a shared memory, have been developed since the 1960s; they are now ubiquitous. Meanwhile, the difficulty of programming concurrent systems has motivated extensive research on programming language design, semantics, and verification, from semaphores and monitors to program logics, software model checking, and so forth. This work has almost always assumed that concurrent threads share a single sequentially consistent memory,21 with their reads and writes interleaved in some order. In fact, however, real multiprocessors use sophisticated techniques to achieve high performance: store buffers, hierarchies of local cache, speculative execution, etc. These optimizations are not observable by sequential code, but in multithreaded programs different threads may see subtly different views of memory; such machines exhibit relaxed, or weak, memory models.6, 7, 17, 19

For a simple example, consider the following assembly language program (SB) for modern Intel or AMD x86 multiprocessors: given two distinct memory locations x and y (initially holding 0), if two processors respectively write 1 to x and y and then read from y and x (into register EAX on processor 0 and EBX on processor 1), it is possible for both to read 0 in the same execution. It is easy to check that this result cannot arise from any interleaving of the reads and writes of the two processors; modern x86 multiprocessors do not have a sequentially consistent semantics.


Microarchitecturally, one can view this particular example as a visible consequence of store buffering: if each processor effectively has a FIFO buffer of pending memory writes (to avoid the need to block while a write completes), then the reads from y and x could occur before the writes have propagated from the buffers to main memory.

Other families of multiprocessors, dating back at least to the IBM 370, and including ARM, Itanium, POWER, and SPARC, also exhibit relaxed-memory behavior. Moreover, there are major and subtle differences between different processor families (arising from their different internal design choices): in the details of exactly what non-sequentially-consistent executions they permit, and of what memory barrier and synchronization instructions they provide to let the programmer regain control.

For any of these processors, relaxed-memory behavior exacerbates the difficulties of writing concurrent software, as systems programmers cannot reason, at the level of abstraction of memory reads and writes, in terms of an intuitive concept of global time.

Still worse, while some vendors' architectural specifications clearly define what they guarantee, others do not, despite the extensive previous research on relaxed memory models. We focus in this paper on x86 processors. In Section 2, we introduce the key examples and discuss several vendor specifications, showing that they all leave key questions ambiguous, some give unusably weak guarantees, and some are simply wrong, prohibiting behavior that actual processors do exhibit.

For there to be any hope of building reliable multiprocessor software, systems programmers need to understand what relaxed-memory behavior they can rely on, but at present that understanding exists only in folklore, not in clear public specifications. To remedy this, we aim to produce mathematically precise (but still appropriately loose) programmer's models for real-world multiprocessors, to inform the intuition of systems programmers, to provide a sound foundation for rigorous reasoning about multiprocessor programs, and to give a clear correctness criterion for hardware. In Section 3, we describe a simple x86 memory model, x86-TSO.27 In contrast to those vendor specifications, it is unambiguous, defined in rigorous mathematics, but it is also accessible, presented in an operational abstract-machine style. To the best of our knowledge it is consistent with the behavior of actual processors. We consider the relevant vendor litmus tests in Section 3.2 and describe some empirical test results in Section 3.3.

Relaxed memory behavior is particularly critical for low-level systems code: synchronization libraries, concurrent data structure libraries, language runtime systems, compilers for concurrent languages, and so on. To reason (even informally) about such code, such as the implementation of an OS mutual exclusion lock, one would necessarily depend on the details of a specific model. Higher-level application code, on the other hand, should normally be oblivious to the underlying processor memory model. The usual expectation is that such code is in some sense race free, with all access to shared memory (except for accesses within the library code) protected by locks or clearly identified as synchronization accesses. Most memory models are designed with the intention that such race-free code behaves as if it were executing on a sequentially consistent machine. In Section 4, we describe an implementation of spin locks for x86, from one version of the Linux kernel, and discuss informally why it is correct with respect to x86-TSO. In Section 5, we define a precise notion of data race for x86 and discuss results showing that programs that use spin locks but are otherwise race-free (except for the races within the lock implementation) do indeed behave as if executing on a sequentially consistent machine.26

To support formal reasoning about programs, a memory model must be integrated with a semantics for machine instructions (a problem which has usually been neglected in the relaxed-memory literature). In previous work31, §3 we describe a semantics for core x86 instructions, with several innovations. We take care not to over-sequentialize the memory accesses within each instruction, parameterizing the instruction semantics over parallel and sequential combinators. A single definition, with all the intricacies of flag-setting, addressing modes, etc., can then be used to generate both an event-based semantics that can be integrated with memory models, and a state-based semantics for sequential programs; the latter enables us to test the semantics against implementations. We also build an instruction decoding function, directly from the vendor documentation, to support reasoning about concrete machine code.

The intended scope of x86-TSO is typical user code and most kernel code: we cover programs using coherent writeback memory, without exceptions, misaligned or mixed-size accesses, "nontemporal" operations (e.g., MOVNTI), self-modifying code, or page-table changes. Within this domain, and together with our earlier instruction semantics, x86-TSO thus defines a complete semantics of programs.

Relaxed memory models play an important role also in the design of high-level concurrent languages such as Java or C++0x, where programs are subject not just to the memory model of the underlying processor but also to reorderings introduced by compiler optimizations. The Java Memory Model24 attempts to ensure that data-race free (DRF) programs are sequentially consistent; all programs satisfy memory safety/security properties; and common compiler optimizations are sound. Unfortunately, as shown by Ševčík and Aspinall,33 the last goal is not met. In the future, we hope that it will be possible to prove correctness of implementations of language-level memory models above the models provided by real-world processors; ensuring that both are precisely and clearly specified is a first step towards that goal.

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2. Architecture Specifications

To describe what programmers can rely on, processor vendors document architectures. These are loose specifications, claimed to cover a range of past and future processor implementations, which should specify processor behavior tightly enough to enable effective programming, but without unduly constraining future processor designs. For some architectures, the memory-model aspects of these specifications are expressed in reasonably precise mathematics, as in the normative Appendix K of the SPARC v.8 specification.2 For x86, however, the vendor architecture specifications are informal prose documents. Informal prose is a poor medium for loose specification of subtle properties, and, as we shall see, such documents are almost inevitably ambiguous and sometimes wrong. Moreover, one cannot test programs above such a vague specification (one can only run programs on particular actual processors), and one cannot use them as criteria for testing processor implementations. In this section, we review the informal-prose Intel and AMD x86 specifications: the Intel 64 and IA-32 Architectures Software Developer's Manual (SDM)5 and the AMD64 Architecture Programmer's Manual (APM).3 There have been several versions of these, some differing radically; we contrast them with each other, and with what we have discovered of the behavior of actual processors. In the process we introduce the key discriminating examples.

* 2.1. Pre-IWP (before aug. 2007)

Early revisions of the Intel SDM (e.g. rev. 22, Nov. 2006) gave an informal-prose model called "processor ordering," unsupported by any examples. It is hard to see precisely what this prose means, especially without additional knowledge or assumptions about the microarchitecture of particular implementations. The uncertainty about x86 behavior that at least some systems programmers had about earlier IA-32 processors can be gauged from an extensive discussion about the correctness of a proposed optimization to a Linux spinlock implementation.1 The discussion is largely in microarchitectural terms, not just in terms of the specified architecture, and seems to have been resolved only with input from Intel staff. We return to this optimization in Section 4, where we can explain why it is sound with respect to x86-TSO.

* 2.2. IWP/AMD3.14/x86-CC

In August 2007, an Intel White Paper4 (IWP) gave a somewhat more precise model, with 8 informal-prose principles P1–P8 supported by 10 examples (known as litmus tests). This was incorporated, essentially unchanged, into later revisions of the Intel SDM (including rev. 26–28), and AMD gave similar, though not identical, prose and tests in rev. 3.14 of their manual3, vol. 2, §7.2 (AMD3.14). These are essentially causal-consistency models,9 and they allow different processors to see writes to independent locations in different orders, as in the IRIW litmus test11 below.a AMD3.14 allows this explicitly, while IWP allows it implicitly, as IRIW is not ruled out by the stated principles. Microarchitecturally, IRIW can arise


from store buffers that are shared between some but not all processors.

However, both require that, in some sense, causality is respected, as in the IWP principle "P5. In a multiprocessor system, memory ordering obeys causality (memory ordering respects transitive visibility)."

We used these informal specifications as the basis for a formal model, x86-CC,31 for which a key issue was giving a reasonable interpretation to this "causality," which is not defined in IWP or AMD3.14. Apart from that, the informal specifications were reasonably unambiguous—but they turned out to have two serious flaws.

First, they are arguably rather weak for programmers. In particular, they admit the IRIW behavior above but, under reasonable assumptions on the strongest x86 memory barrier, MFENCE, adding MFENCEs would not suffice to recover sequential consistency (instead, one would have to make liberal use of x86 LOCK'd instructions).31, §2.12 Here, the specifications seem to be much looser than the behavior of implemented processors: to the best of our knowledge, and following some testing, IRIW is not observable in practice, even without MFENCEs. It appears that some JVM implementations depend on this fact, and would not be correct if one assumed only the IWP/AMD3.14/x86-CC architecture.15

Second, more seriously, x86-CC and IWP are unsound with respect to current processors. The following example, n6, due to Paul Loewenstein [personal communication, Nov. 2008] shows a behavior that is observable (e.g., on an Intel Core 2


duo) but that is disallowed by x86-CC and by any interpretation we can make of IWP principles P1, 2, 4 and 6.27, A.5

To see why this could be allowed by multiprocessors with FIFO store buffers, suppose that first the Proc 1 write of [y]=2 is buffered, then Proc 0 buffers its write of [x]=1, reads [x]=1 from its own store buffer, and reads [y]=0 from main memory, then Proc 1 buffers its [x]=2 write and flushes its buffered [y]=2 and [x]=2 writes to memory, then finally Proc 0 flushes its [x]=1 write to memory.

The AMD3.14 manual is not expressed in terms of a clearly identified set of principles, and the main text (vol. 2, §7.2) leaves the ordering of stores to a single location unconstrained, though elsewhere the manual describes a microarchitecture with store buffers and cache protocols that strongly implies that memory is coherent. In the absence of an analogue of the IWP P6, the reasoning prohibiting n6 does not carry over.

* 2.3. Intel SDM rev. 29–34 (nov. 2008–Mar. 2010)

The most recent substantial change to the Intel memorymodel specification, at the time of writing, was in revision 29 of the Intel SDM (revisions 29–34 are essentially identical except for the LFENCE text). This is in a similar informal-prose style to previous versions, again supported by litmus tests, but is significantly different to IWP/x86-CC/AMD3.14. First, the IRIW final state above is forbidden, 5, Example 8–7, vol. 3A and the previous coherence condition: "P6. In a multiprocessor system, stores to the same location have a total order" has been replaced by: "Any two stores are seen in a consistent order by processors other than those performing the stores" (we label this P9).

Second, the memory barrier instructions are now included. It is stated that reads and writes cannot pass MFENCE instructions, together with more refined properties for SFENCE and LFENCE

Third, same-processor writes are now explicitly ordered: "Writes by a single processor are observed in the same order by all processors" (P10) (we regarded this as implicit in the IWP "P2. Stores are not reordered with other stores").

This revision appears to deal with the unsoundness, admitting the n6 behavior above, but, unfortunately, it is still problematic. The first issue is, again, how to interpret "causality" as used in P5. The second issue is one of weakness: the new P9 says nothing about observations of two stores by those two processors themselves (or by one of those processors and one other). The following examples (which we call n5 and n4b) illustrate potentially surprising behavior that arguably violates coherence. Their final states are not allowed in x86-CC, are not allowed in a pure store-buffer implementation or in x86-TSO, and we have not observed them on actual processors. However, the principles stated in revisions 29–34 of the Intel SDM appear, presumably unintentionally, to allow them. The AMD3.14 vol. 2, §7.2 text taken alone would allow them, but the implied coherence from elsewhere in the AMD manual would forbid them. These points illustrate



once again the difficulty of writing unambiguous and correct loose specifications in informal prose.

* 2.4. AMD3.15 (Nov. 2009)

In November 2009, AMD produced a new revision, 3.15, of their manuals. The main difference in the memory-model specification is that IRIW is now explicitly forbidden.

Summarizing the key litmus-test differences, we have the


following, where radic.gif and × entries are explicit in the specification text and starred entries indicate possible deductions, some of which may not have been intended.

There are also many non-differences: tests for which the behaviors coincide in all three cases. We return to these, and go through the other tests from the Intel and AMD documentation, in Section 3.2.

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3. Our x86-TSO Programmer's Model

Given these problems with the informal specifications, we cannot produce a useful rigorous model by formalizing the "principles" they contain, as we attempted with x86-CC.31 Instead, we have to build a reasonable model that is consistent with the given litmus tests, with observed processor behavior, and with what we know of the needs of programmers, the vendors' intentions, and the folklore in the area.

We emphasize that our aim is a programmer's model, of the allowable behaviors of x86 processors as observed by assembly programs, not of the internal structure of processor implementations, or of what could be observed on hardware interfaces. We present the model in an abstract-machine style to make it accessible, but are concerned only with its external behavior; its buffers and locks are highly abstracted from the microarchitecture of processor implementations.

The fact that store buffering is observable, as in the SB and n6 examples, but IRIW is not (and IRIW is explicitly forbidden in the SDM revs. 29–34 and AMD3.15), together with additional tests that prohibit many other reorderings, strongly suggests that, apart from store buffering, all processors share the same view of memory. Moreover, different processors or hardware threads do not observably share store buffers. This is in sharp contrast to x86-CC, where each processor has a separate view order of its memory accesses and other processors' writes. To the best of our knowledge, for the usual write-back memory, no other aspects of the microarchitecture (the out-of-order execution, cache hierarchies and protocols, interconnect topology, and so on) are observable to the programmer, except in so far as they affect performance.

This is broadly similar to the SPARC Total Store Ordering (TSO) memory model,2, 32 which is essentially an axiomatic description of the behavior of store-buffer multiprocessors. Accordingly, we have designed a TSO-like model for x86, called x86-TSO.27 It is defined mathematically in two styles: an abstract machine with explicit store buffers and an axiomatic model that defines valid executions in terms of memory orders; they are formalized in HOL420 and are proved equivalent. The abstract machine conveys the programmer-level operational intuition behind x86-TSO; we describe it informally in the next subsection. The axiomatic model supports constraint-based reasoning about example programs (e.g. by our memevents tool in Section 3.3); it is similar to that of SPARCv8,2, App. K but we also deal with x86 CISC instructions with multiple memory accesses and with x86 barriers and atomic (or LOCK'd) instructions. The x86 supports a range of atomic instructions: one can add a LOCK prefix to many read–modify–write instructions (ADD, INC, etc.), and the XCHG instruction is implicitly LOCK'd. There are three main memory barriers: MFENCE, SFENCE and LFENCE.

* 3.1. The abstract machine

Our programmer's model of a multiprocessor x86 system is illustrated in Figure 1. At the top of the figure are a number of hardware threads, each corresponding to a single in-order stream of instruction execution. (In this programmer's model there is no need to consider physical processors explicitly; it is the hardware threads that correspond to the Proc N columns in the tests we give.) They interact with a storage subsystem, drawn as the dotted box.

The state of the storage subsystem comprises a shared memory that maps addresses to values, a global lock to indicate when a particular hardware thread has exclusive access to memory, and one store buffer per hardware thread.

The behavior of the storage subsystem is described in more detail below, but the main points are:

  • The store buffers are FIFO and a reading thread must read its most recent buffered write, if there is one, to that address; otherwise reads are satisfied from shared memory.
  • An MFENCE instruction flushes the store buffer of that thread.
  • To execute a LOCK'd instruction, a thread must first obtain the global lock. At the end of the instruction, it flushes its store buffer and relinquishes the lock. While the lock is held by one thread, no other thread can read.
  • A buffered write from a thread can propagate to the shared memory at any time except when some other thread holds the lock.

More precisely, the possible interactions between the threads and the storage subsystem are described by the following events:

  • Wp [a]=υ, for a write of value υ to address a by thread p
  • Rp [a]=υ, for a read of υ from a by thread p
  • Fp, for an MFENCE memory barrier by thread p
  • Lp, at the start of a LOCK'd instruction by thread p
  • Up, at the end of a LOCK'd instruction by thread p
  • τp, for an internal action of the storage subsystem, propagating a write from p's store buffer to the shared memory

For example, suppose a particular hardware thread p has come to the instruction INC [56] (which adds 1 to the value at address 56), and p's store buffer contains a single write to 56, of value 0. In one execution we might see read and write events, Rp[56]=0 and Wp[56]=1, followed by two τp events as the two writes propagate to shared memory. Another execution might start with the write of 0 propagating to shared memory, where it could be overwritten by another thread. Executions of LOCK;INC [56] would be similar but bracketed by Lp and Up events.

The behavior of the storage subsystem is specified by the following rules, where we define a hardware thread to be blocked if the storage subsystem lock is taken by another hardware thread, i.e., while another hardware thread is executing a LOCK'd instruction.

  1. Rp[a]=υ: p can read υ from memory at address a if p is not blocked, there are no writes to a in p's store buffer, and the memory does contain υ at a.
  2. Rp[a]=υ: p can read υ from its store buffer for address a if p is not blocked and has υ as the newest write to a in its buffer.
  3. Wp[a]=υ: p can write υ to its store buffer for address a at any time.
  4. τp: if p is not blocked, it can silently dequeue the oldest write from its store buffer and place the value in memory at the given address, without coordinating with any hardware thread.
  5. Fp: if p's store buffer is empty, it can execute an MFENCE (note that if a hardware thread encounters an MFENCE instruction when its store buffer is not empty, it can take one or more τp steps to empty the buffer and proceed, and similarly in 7 below).
  6. Lp: if the lock is not held, it can begin a LOCK'd instruction.
  7. Up: if p holds the lock, and its store buffer is empty, it can end a LOCK'd instruction.

Technically, the formal versions of these rules27 define a labeled transition system (with the events as labels) for the storage subsystem, and we define the behavior of the whole system as a parallel composition of that and transition systems for each thread, synchronizing on the non-τ labels as in CCS.25

Additionally, we tentatively impose a progress condition, that each memory write is eventually propagated from the relevant store buffer to the shared memory. This is not stated in the documentation and is hard to test. We are assured that it holds at least for AMD processors.

For write-back cacheable memory, and the fragment of the instruction set that we consider, we treat LFENCE and SFENCE semantically as no-ops. This follows the Intel and AMD documentation, both of which imply that these fences do not order store/load pairs which are the only reorderings allowed in x86-TSO. Note, though, that else-where it is stated that the Intel SFENCE flushes the store buffer.5, vol.3A, §11.10

* 3.2. Litmus tests

For our introductory SB example from Section 1, x86-TSO permits the given behavior for the same reasons as set forth there. For each of the examples in Section 2 (IRIW, n6, and n5/n4b), x86-TSO permits the given final state if and only if it is observable in our testing of actual processors, i.e., for IRIW it is forbidden (in contrast to IWP and AMD3.14), for n6 it is allowed (in contrast to IWP), and for n5/n4b it is forbidden (in contrast to the Intel SDM rev. 29–34). For all the other relevant tests from the current Intel and AMD manuals the stated behavior agrees with x86-TSO. We now go through Examples 8–1 to 8–10 from rev. 34 of the Intel SDM, and the three other tests from AMD3.15, and explain the x86-TSO behavior in each case.

For completeness we repeat the Intel SDM short descriptions of these tests, e.g. "stores are not reordered with other stores," but note that "not reordered with" is not defined there and is open to misinterpretation.27, §3.2



This test implies that the writes by Proc 0 are seen in order by Proc 1's reads, which also execute in order. x86-TSO forbids the final state because Proc 0's store buffer is FIFO, and Proc 0 communicates with Proc 1 only through shared memory.



x86-TSO forbids the final state because reads are never delayed.

EXAMPLE 8–3. LOADS MAY BE REORDERED WITH OLDER STORES. This test is just the SB example from Section 1, which x86-TSO permits. The third AMD test (amd3) is similar but with additional writes inserted in the middle of each thread, of 2 to x and y respectively.



x86-TSO requires the specified result because reads must check the local store buffer.

EXAMPLE 8–5. INTRA-PROCESSOR FORWARDING IS ALLOWED. This test is similar to Example 8–3.



x86-TSO forbids the given final state because otherwise the Proc 2 constraints imply that y was written to shared memory before x. Hence the write to x must be in Proc 0's store buffer (or the instruction has not executed), when the write to y is initiated. Note that this test contains the only mention of "transitive visibility" in the Intel SDM, leaving its meaning unclear.

EXAMPLE 8–7. STORES ARE SEEN IN A CONSISTENT ORDER BY OTHER PROCESSORS. This test rules out the IRIW behavior as described in Section 2.2. x86-TSO forbids the given final state because the Proc 2 constraints imply that x was written to shared memory before y whereas the Proc 3 constraints imply that y was written to shared memory before x.

EXAMPLE 8–8. LOCKED INSTRUCTIONS HAVE A TOTAL ORDER. This is the same as the IRIW Example 8–7 but with LOCK'd instructions for the writes; x86-TSO forbids the final state for the same reason as above.



This test indicates that locking both writes in Example 8–3 would forbid the nonsequentially consistent result. x86-TSO forbids the final state because LOCK'd instructions flush the local store buffer. If only one write were LOCK'd (say the write to x), the Example 8–3 final state would be allowed as follows: on Proc 1, buffer the write to y and execute the read x, then on Proc 0 write to x in shared memory then read from y.


EXAMPLE 8–10. STORES ARE NOT REORDERED WITH LOCKS. This is implied by Example 8–1, as we treat the memory writes of LOCK'd instructions as stores.



For x86-TSO, this test has the same force as Example 8.8, but using MFENCE instructions to flush the buffers instead of LOCK'd instructions. The tenth AMD test is similar. None of the Intel litmus tests include fence instructions.

In x86-TSO adding MFENCE between every instruction would clearly suffice to regain sequential consistency (though obviously in practice one would insert fewer barriers), in contrast to IWP/x86-CC/AMD3.14.

* 3.3. Empirical testing

To build confidence that we have a sound model of the behavior of actual x86 processors we have tested the correspondence between them in various ways.

First, for the memory model, we have a litmus tool that takes a litmus test (essentially as given in this paper) and builds a C program with embedded assembly to run the test repeatedly to try to produce all possible results, taking care to synchronize the different threads and with some randomization of memory usage. We have run these on the Intel and AMD processors that we have access to. The results can be compared with the output of a memevents tool that takes such tests and computes the set of all possible executions allowed by the x86-TSO model. We use a verified witness checker, extracted from the HOL4 definition of the model, to verify that any executions found are indeed allowed.

The results correspond exactly for all the tests given here and others we have tried, including amd3, n1,31 n7,27 the single-XCHG variant of Example 8–9, and an unfenced variant of RWC.11 In general, though, there may be tests where x86-TSO allows some final state that cannot be observed in practice, perhaps because litmus does not drive the processor into the correct internal state (of store buffers, cache lines, etc.) to exhibit it, or perhaps because the particular implementations we tested cannot exhibit it. For example, we have only seen amd3 on a four-processor (×2 hyperthread) machine and only very rarely, 4 out of 3.2e9 times. Testing, especially this black-box testing of a complex and time-dependent system, is obviously subject to the usual limitations; it cannot conclusively prove that some outcome is not possible.

Second, for the behavior of individual instructions, we have an x86sem tool that generates random instances of instructions, runs them on an actual machine, and generates a HOL4 conjecture relating the memory and register state before and after. These conjectures are then automatically verified, by a HOL4 script, for the 4600 instances that we tried.

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4. A Linux x86 Spinlock Implementation

In Section 2.1, we mentioned the uncertainty that arose in a discussion on a particular optimization for Linux spin-locks.1 In this section, we present a spinlock from the Linux kernel (version that incorporates the proposed optimization, as an example of a small but nontrivial concurrent programming idiom. We show how one can reason about this code using the x86-TSO programmer's model, explaining in terms of the model why it works and why the optimization is sound—thus making clear what (we presume) the developer's informal reasoning depended on. For accessibility we do this in prose, but the argument could easily be formalized as a proof.

The implementation comprises code to acquire and release a spinlock. It is assumed that these are properly bracketed around critical sections and that spinlocks are not mutated by any other code.


A spinlock is represented by a signed integer which is 1 if the lock is free and 0 or less if the lock is held. To acquire a lock, a thread atomically decrements the integer (which will not wrap around assuming there are fewer than 231 hardware threads). If the lock was free, it is now held and the thread can proceed to the critical section. If the lock was held, the thread loops, waiting for it to become free. Because there might be multiple threads waiting for the lock, once it is freed, each waiting thread must again attempt to enter through the LOCK'd decrement. To release the lock, a thread simply sets its value to 1.

The optimization in question made the releasing MOV instruction not LOCK'd (removing a LOCK prefix and hence letting the releasing thread proceed without flushing its buffer).

For example, consider a spinlock at address x and let y be another shared memory address. Suppose that several threads want to access y, and that they use spinlocks to ensure mutual exclusion. Initially, no one has the lock and [x] = 1. The first thread t to try to acquire the lock atomically decrements x by 1 (using a LOCK prefix); it then jumps into the critical section. Because a store buffer flush is part of LOCK'd instructions, [x] will be 0 in shared memory after the decrement.

Now if another thread attempts to acquire the lock, it will not jump into the critical section after performing the atomic decrement, since x was not 1. It will thus enter the spin loop. In this loop, the waiting thread continually reads the value of x until it gets a positive result.

Returning to the original thread t, it can read and write y inside of its critical section while the others are spinning. These writes are initially placed in t's store buffer, and some may be propagated to shared memory. However, it does not matter how many (if any) are written to main memory, because (by assumption) no other thread is attempting to read (or write) y. When t is ready to exit the critical section, it releases the lock by writing the value 1 to x; this write is put in t's store buffer. It can now continue after the critical section (in the text below, we assume it does not try to reacquire the lock).

If the releasing MOV had the LOCK prefix then all of the buffered writes to y would be sent to main memory, as would the write of 1 to x. Another thread could then acquire the spinlock.

However, since it does not, the other threads continue to spin until the write setting x to 1 is removed from t's write buffer and sent to shared memory at some point in the future. At that point, the spinning threads will read 1 and restart the acquisition with atomic decrements, and another thread could enter its critical section. However, because t's write buffer is emptied in FIFO order, any writes to y from within t's critical section must have been propagated to shared memory (in order) before the write to x. Thus, the next thread to enter a critical section will not be able to see y in an inconsistent state.

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5. Data-Race Freedom

To make a relaxed-memory architecture usable for large-scale programming, it is highly desirable (perhaps essential) to identify programming idioms which ensure that one can reason in terms of a traditional interleaving model of concurrency, showing that any relaxed memory execution is equivalent to one that is possible above a sequentially consistent memory model. One common idiom with this property is data-race freedom. Informally, a program has a data race if multiple threads can access the same location (where at least one is writing to the location) without a synchronization operation separating the accesses. Programs where every shared access is in a critical section are one common example of DRF programs.

A variety of relaxed models, both for processors and for programming languages, have been proved to support sequentially consistent semantics for DRF programs.8, 9, 10, 12, 16, 23 Saraswat et al.30 call supporting sequentially consistent semantics for DRF programs the "fundamental property" of a relaxed memory model, and indeed memory models have sometimes been defined in these terms.6 However, for a processor architecture, we prefer to define a memory model that is applicable to arbitrary programs, to support reasoning about low-level code, and have results about well-behaved programs as theorems above it.

The details of what constitutes a data race, or a synchronization operation, vary from model to model. For x86-TSO, we define two events on different threads to be competing if they access the same address, one is a write, and the other is a read (for aligned x86 accesses, it is not necessary to consider write/write pairs as competing). We say that a program is data race free if it is impossible for a competing read/write pair to execute back-to-back. Critically, we require this property only of sequentially consistent executions (equivalently, the x86-TSO executions where store buffers are always flushed immediately after each write).

We have proved that x86-TSO supports interleaving semantics for DRF programs. However, this theorem alone is not often useful, because most programs do contain data races at this level of abstraction. For example, the read in the spin loop of Section 4's spinlock races with the write in the release. We have, therefore, identified an extended notion of data race freedom that the spinlock code does satisfy, and we have used it to prove that, for well-synchronized programs using the spinlock, every x86-TSO execution has an equivalent sequentially consistent execution.26

Thus, the relaxed nature of x86-TSO is provably not a concern for low-level systems code that uses spinlocks to synchronize. Extending this result to other synchronization primitives, and to code compiled from high-level languages, is a major topic for future work.

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6. Related Work

There is an extensive literature on relaxed memory models, but most of it does not address x86. We touch here on some of the most closely related work.

There are several surveys of weak memory models, including those by Adve and Gharachorloo,6 Luchango,23 and Higham et al.19 The latter, in particular, formalizes a range of models, including a TSO model, in both operational and axiomatic styles, and proves equivalence results. Their axiomatic TSO model is rather closer to the operational style than ours is, and is idealized rather than x86-specific. Park and Dill28 verify programs by model checking them directly above TSO. Burckhardt and Musuvathi13, App. A also give operational and axiomatic definitions of a TSO model and prove equivalence, but only for finite executions. Their models treat memory reads and writes and barrier events, but lack instruction semantics and LOCK'd instructions with multiple events that happen atomically. Hangal et al.18 describe the Sun TSOtool, checking the observed behavior of pseudo-randomly generated programs against a TSO model. Roy et al.29 describe an efficient algorithm for checking whether an execution lies within an approximation to a TSO model, used in Intel's Random Instruction Test (RIT) generator. Loewenstein et al.22 describe a "golden memory model" for SPARC TSO, somewhat closer to a particular implementation microarchitecture than the abstract machine we give in Section 3, that they use for testing implementations. They argue that the additional intensional detail increases the effectiveness of simulation-based verification. Boudol and Petri12 give an operational model with hierarchical write buffers (thereby permitting IRIW behaviors), and prove sequential consistency for DRF programs. Burckhardt et al.14 define an x86 memory model based on IWP.4 The mathematical form of their definitions is rather different to our axiomatic and abstract-machine models, using rewrite rules to reorder or eliminate memory accesses in sets of traces. Their model validates the 10 IWP tests and also some instances of IRIW (depending on how parallel compositions are associated), so it will not coincide with x86-TSO or x86-CC. Saraswat et al.30 also define memory models in terms of local reordering, and prove a DRF theorem, but focus on high-level languages.

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7. Conclusion

We have described x86-TSO, a memory model for x86 processors that does not suffer from the ambiguities, weaknesses, or unsoundnesses of earlier models. Its abstract-machine definition should be intuitive for programmers, and its equivalent axiomatic definition supports the memevents exhaustive search and permits an easy comparison with related models; the similarity with SPARCv8 suggests x86-TSO is strong enough to program above. This work high-lights the clarity of mathematically rigorous definitions, in contrast to informal prose, for subtle loose specifications.

We do not speak for any x86 vendor, and it is, of course, entirely possible that x86-TSO is not a good description of some existing or future x86 implementation (we would be very interested to hear of any such example). Nonetheless, we hope that this will clarify the semantics of x86 architectures as they exist, for systems programmers, hardware developers, and those working on the verification of concurrent software.

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We thank Luc Maranget for his work on memevents and litmus, Tom Ridge, Thomas Braibant and Jade Alglave for their other work on the project, and Hans Boehm, David Christie, Dave Dice, Doug Lea, Paul Loewenstein, and Gil Neiger for helpful remarks. We acknowledge funding from EPSRC grants EP/F036345 and EP/H005633 and ANR grant ANR-06-SETI-010-02.

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Peter Sewell (, University of Cambridge.

Susmit Sarkar (, University of Cambridge.

Scott Owens (, University of Cambridge.

Francesco Zappa Nardelli (, INRIA.

Magnus O. Myreen (, University of Cambridge.

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This paper is based on work that first appeared in the Proceedings of the 36th SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), 2009, and in the Proceedings of the 22nd International Conference on Theorem Proving in Higher-Order Logics (TPHOLs), 2009.

a. We use Intel assembly syntax throughout except that we use an arrow ← to indicate the direction of data flow, so MOV [x]←1 is a write of 1 to address x and MOV EAX←[x] is a read from address x into register EAX. Initial states are all 0 unless otherwise specified.


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F1Figure 1. x89-TSO block diagram.

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