diff options
author | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-27 10:05:51 +0000 |
---|---|---|
committer | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-27 10:05:51 +0000 |
commit | 5d1646d90e1f2cceb9f0828f4b28318cd0ec7744 (patch) | |
tree | a94efe259b9009378be6d90eb30d2b019d95c194 /Documentation/locking/lockdep-design.rst | |
parent | Initial commit. (diff) | |
download | linux-upstream/5.10.209.tar.xz linux-upstream/5.10.209.zip |
Adding upstream version 5.10.209.upstream/5.10.209upstream
Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>
Diffstat (limited to '')
-rw-r--r-- | Documentation/locking/lockdep-design.rst | 663 |
1 files changed, 663 insertions, 0 deletions
diff --git a/Documentation/locking/lockdep-design.rst b/Documentation/locking/lockdep-design.rst new file mode 100644 index 000000000..9f3cfca9f --- /dev/null +++ b/Documentation/locking/lockdep-design.rst @@ -0,0 +1,663 @@ +Runtime locking correctness validator +===================================== + +started by Ingo Molnar <mingo@redhat.com> + +additions by Arjan van de Ven <arjan@linux.intel.com> + +Lock-class +---------- + +The basic object the validator operates upon is a 'class' of locks. + +A class of locks is a group of locks that are logically the same with +respect to locking rules, even if the locks may have multiple (possibly +tens of thousands of) instantiations. For example a lock in the inode +struct is one class, while each inode has its own instantiation of that +lock class. + +The validator tracks the 'usage state' of lock-classes, and it tracks +the dependencies between different lock-classes. Lock usage indicates +how a lock is used with regard to its IRQ contexts, while lock +dependency can be understood as lock order, where L1 -> L2 suggests that +a task is attempting to acquire L2 while holding L1. From lockdep's +perspective, the two locks (L1 and L2) are not necessarily related; that +dependency just means the order ever happened. The validator maintains a +continuing effort to prove lock usages and dependencies are correct or +the validator will shoot a splat if incorrect. + +A lock-class's behavior is constructed by its instances collectively: +when the first instance of a lock-class is used after bootup the class +gets registered, then all (subsequent) instances will be mapped to the +class and hence their usages and dependecies will contribute to those of +the class. A lock-class does not go away when a lock instance does, but +it can be removed if the memory space of the lock class (static or +dynamic) is reclaimed, this happens for example when a module is +unloaded or a workqueue is destroyed. + +State +----- + +The validator tracks lock-class usage history and divides the usage into +(4 usages * n STATEs + 1) categories: + +where the 4 usages can be: + +- 'ever held in STATE context' +- 'ever held as readlock in STATE context' +- 'ever held with STATE enabled' +- 'ever held as readlock with STATE enabled' + +where the n STATEs are coded in kernel/locking/lockdep_states.h and as of +now they include: + +- hardirq +- softirq + +where the last 1 category is: + +- 'ever used' [ == !unused ] + +When locking rules are violated, these usage bits are presented in the +locking error messages, inside curlies, with a total of 2 * n STATEs bits. +A contrived example:: + + modprobe/2287 is trying to acquire lock: + (&sio_locks[i].lock){-.-.}, at: [<c02867fd>] mutex_lock+0x21/0x24 + + but task is already holding lock: + (&sio_locks[i].lock){-.-.}, at: [<c02867fd>] mutex_lock+0x21/0x24 + + +For a given lock, the bit positions from left to right indicate the usage +of the lock and readlock (if exists), for each of the n STATEs listed +above respectively, and the character displayed at each bit position +indicates: + + === =================================================== + '.' acquired while irqs disabled and not in irq context + '-' acquired in irq context + '+' acquired with irqs enabled + '?' acquired in irq context with irqs enabled. + === =================================================== + +The bits are illustrated with an example:: + + (&sio_locks[i].lock){-.-.}, at: [<c02867fd>] mutex_lock+0x21/0x24 + |||| + ||| \-> softirq disabled and not in softirq context + || \--> acquired in softirq context + | \---> hardirq disabled and not in hardirq context + \----> acquired in hardirq context + + +For a given STATE, whether the lock is ever acquired in that STATE +context and whether that STATE is enabled yields four possible cases as +shown in the table below. The bit character is able to indicate which +exact case is for the lock as of the reporting time. + + +--------------+-------------+--------------+ + | | irq enabled | irq disabled | + +--------------+-------------+--------------+ + | ever in irq | '?' | '-' | + +--------------+-------------+--------------+ + | never in irq | '+' | '.' | + +--------------+-------------+--------------+ + +The character '-' suggests irq is disabled because if otherwise the +charactor '?' would have been shown instead. Similar deduction can be +applied for '+' too. + +Unused locks (e.g., mutexes) cannot be part of the cause of an error. + + +Single-lock state rules: +------------------------ + +A lock is irq-safe means it was ever used in an irq context, while a lock +is irq-unsafe means it was ever acquired with irq enabled. + +A softirq-unsafe lock-class is automatically hardirq-unsafe as well. The +following states must be exclusive: only one of them is allowed to be set +for any lock-class based on its usage:: + + <hardirq-safe> or <hardirq-unsafe> + <softirq-safe> or <softirq-unsafe> + +This is because if a lock can be used in irq context (irq-safe) then it +cannot be ever acquired with irq enabled (irq-unsafe). Otherwise, a +deadlock may happen. For example, in the scenario that after this lock +was acquired but before released, if the context is interrupted this +lock will be attempted to acquire twice, which creates a deadlock, +referred to as lock recursion deadlock. + +The validator detects and reports lock usage that violates these +single-lock state rules. + +Multi-lock dependency rules: +---------------------------- + +The same lock-class must not be acquired twice, because this could lead +to lock recursion deadlocks. + +Furthermore, two locks can not be taken in inverse order:: + + <L1> -> <L2> + <L2> -> <L1> + +because this could lead to a deadlock - referred to as lock inversion +deadlock - as attempts to acquire the two locks form a circle which +could lead to the two contexts waiting for each other permanently. The +validator will find such dependency circle in arbitrary complexity, +i.e., there can be any other locking sequence between the acquire-lock +operations; the validator will still find whether these locks can be +acquired in a circular fashion. + +Furthermore, the following usage based lock dependencies are not allowed +between any two lock-classes:: + + <hardirq-safe> -> <hardirq-unsafe> + <softirq-safe> -> <softirq-unsafe> + +The first rule comes from the fact that a hardirq-safe lock could be +taken by a hardirq context, interrupting a hardirq-unsafe lock - and +thus could result in a lock inversion deadlock. Likewise, a softirq-safe +lock could be taken by an softirq context, interrupting a softirq-unsafe +lock. + +The above rules are enforced for any locking sequence that occurs in the +kernel: when acquiring a new lock, the validator checks whether there is +any rule violation between the new lock and any of the held locks. + +When a lock-class changes its state, the following aspects of the above +dependency rules are enforced: + +- if a new hardirq-safe lock is discovered, we check whether it + took any hardirq-unsafe lock in the past. + +- if a new softirq-safe lock is discovered, we check whether it took + any softirq-unsafe lock in the past. + +- if a new hardirq-unsafe lock is discovered, we check whether any + hardirq-safe lock took it in the past. + +- if a new softirq-unsafe lock is discovered, we check whether any + softirq-safe lock took it in the past. + +(Again, we do these checks too on the basis that an interrupt context +could interrupt _any_ of the irq-unsafe or hardirq-unsafe locks, which +could lead to a lock inversion deadlock - even if that lock scenario did +not trigger in practice yet.) + +Exception: Nested data dependencies leading to nested locking +------------------------------------------------------------- + +There are a few cases where the Linux kernel acquires more than one +instance of the same lock-class. Such cases typically happen when there +is some sort of hierarchy within objects of the same type. In these +cases there is an inherent "natural" ordering between the two objects +(defined by the properties of the hierarchy), and the kernel grabs the +locks in this fixed order on each of the objects. + +An example of such an object hierarchy that results in "nested locking" +is that of a "whole disk" block-dev object and a "partition" block-dev +object; the partition is "part of" the whole device and as long as one +always takes the whole disk lock as a higher lock than the partition +lock, the lock ordering is fully correct. The validator does not +automatically detect this natural ordering, as the locking rule behind +the ordering is not static. + +In order to teach the validator about this correct usage model, new +versions of the various locking primitives were added that allow you to +specify a "nesting level". An example call, for the block device mutex, +looks like this:: + + enum bdev_bd_mutex_lock_class + { + BD_MUTEX_NORMAL, + BD_MUTEX_WHOLE, + BD_MUTEX_PARTITION + }; + + mutex_lock_nested(&bdev->bd_contains->bd_mutex, BD_MUTEX_PARTITION); + +In this case the locking is done on a bdev object that is known to be a +partition. + +The validator treats a lock that is taken in such a nested fashion as a +separate (sub)class for the purposes of validation. + +Note: When changing code to use the _nested() primitives, be careful and +check really thoroughly that the hierarchy is correctly mapped; otherwise +you can get false positives or false negatives. + +Annotations +----------- + +Two constructs can be used to annotate and check where and if certain locks +must be held: lockdep_assert_held*(&lock) and lockdep_*pin_lock(&lock). + +As the name suggests, lockdep_assert_held* family of macros assert that a +particular lock is held at a certain time (and generate a WARN() otherwise). +This annotation is largely used all over the kernel, e.g. kernel/sched/ +core.c:: + + void update_rq_clock(struct rq *rq) + { + s64 delta; + + lockdep_assert_held(&rq->lock); + [...] + } + +where holding rq->lock is required to safely update a rq's clock. + +The other family of macros is lockdep_*pin_lock(), which is admittedly only +used for rq->lock ATM. Despite their limited adoption these annotations +generate a WARN() if the lock of interest is "accidentally" unlocked. This turns +out to be especially helpful to debug code with callbacks, where an upper +layer assumes a lock remains taken, but a lower layer thinks it can maybe drop +and reacquire the lock ("unwittingly" introducing races). lockdep_pin_lock() +returns a 'struct pin_cookie' that is then used by lockdep_unpin_lock() to check +that nobody tampered with the lock, e.g. kernel/sched/sched.h:: + + static inline void rq_pin_lock(struct rq *rq, struct rq_flags *rf) + { + rf->cookie = lockdep_pin_lock(&rq->lock); + [...] + } + + static inline void rq_unpin_lock(struct rq *rq, struct rq_flags *rf) + { + [...] + lockdep_unpin_lock(&rq->lock, rf->cookie); + } + +While comments about locking requirements might provide useful information, +the runtime checks performed by annotations are invaluable when debugging +locking problems and they carry the same level of details when inspecting +code. Always prefer annotations when in doubt! + +Proof of 100% correctness: +-------------------------- + +The validator achieves perfect, mathematical 'closure' (proof of locking +correctness) in the sense that for every simple, standalone single-task +locking sequence that occurred at least once during the lifetime of the +kernel, the validator proves it with a 100% certainty that no +combination and timing of these locking sequences can cause any class of +lock related deadlock. [1]_ + +I.e. complex multi-CPU and multi-task locking scenarios do not have to +occur in practice to prove a deadlock: only the simple 'component' +locking chains have to occur at least once (anytime, in any +task/context) for the validator to be able to prove correctness. (For +example, complex deadlocks that would normally need more than 3 CPUs and +a very unlikely constellation of tasks, irq-contexts and timings to +occur, can be detected on a plain, lightly loaded single-CPU system as +well!) + +This radically decreases the complexity of locking related QA of the +kernel: what has to be done during QA is to trigger as many "simple" +single-task locking dependencies in the kernel as possible, at least +once, to prove locking correctness - instead of having to trigger every +possible combination of locking interaction between CPUs, combined with +every possible hardirq and softirq nesting scenario (which is impossible +to do in practice). + +.. [1] + + assuming that the validator itself is 100% correct, and no other + part of the system corrupts the state of the validator in any way. + We also assume that all NMI/SMM paths [which could interrupt + even hardirq-disabled codepaths] are correct and do not interfere + with the validator. We also assume that the 64-bit 'chain hash' + value is unique for every lock-chain in the system. Also, lock + recursion must not be higher than 20. + +Performance: +------------ + +The above rules require **massive** amounts of runtime checking. If we did +that for every lock taken and for every irqs-enable event, it would +render the system practically unusably slow. The complexity of checking +is O(N^2), so even with just a few hundred lock-classes we'd have to do +tens of thousands of checks for every event. + +This problem is solved by checking any given 'locking scenario' (unique +sequence of locks taken after each other) only once. A simple stack of +held locks is maintained, and a lightweight 64-bit hash value is +calculated, which hash is unique for every lock chain. The hash value, +when the chain is validated for the first time, is then put into a hash +table, which hash-table can be checked in a lockfree manner. If the +locking chain occurs again later on, the hash table tells us that we +don't have to validate the chain again. + +Troubleshooting: +---------------- + +The validator tracks a maximum of MAX_LOCKDEP_KEYS number of lock classes. +Exceeding this number will trigger the following lockdep warning:: + + (DEBUG_LOCKS_WARN_ON(id >= MAX_LOCKDEP_KEYS)) + +By default, MAX_LOCKDEP_KEYS is currently set to 8191, and typical +desktop systems have less than 1,000 lock classes, so this warning +normally results from lock-class leakage or failure to properly +initialize locks. These two problems are illustrated below: + +1. Repeated module loading and unloading while running the validator + will result in lock-class leakage. The issue here is that each + load of the module will create a new set of lock classes for + that module's locks, but module unloading does not remove old + classes (see below discussion of reuse of lock classes for why). + Therefore, if that module is loaded and unloaded repeatedly, + the number of lock classes will eventually reach the maximum. + +2. Using structures such as arrays that have large numbers of + locks that are not explicitly initialized. For example, + a hash table with 8192 buckets where each bucket has its own + spinlock_t will consume 8192 lock classes -unless- each spinlock + is explicitly initialized at runtime, for example, using the + run-time spin_lock_init() as opposed to compile-time initializers + such as __SPIN_LOCK_UNLOCKED(). Failure to properly initialize + the per-bucket spinlocks would guarantee lock-class overflow. + In contrast, a loop that called spin_lock_init() on each lock + would place all 8192 locks into a single lock class. + + The moral of this story is that you should always explicitly + initialize your locks. + +One might argue that the validator should be modified to allow +lock classes to be reused. However, if you are tempted to make this +argument, first review the code and think through the changes that would +be required, keeping in mind that the lock classes to be removed are +likely to be linked into the lock-dependency graph. This turns out to +be harder to do than to say. + +Of course, if you do run out of lock classes, the next thing to do is +to find the offending lock classes. First, the following command gives +you the number of lock classes currently in use along with the maximum:: + + grep "lock-classes" /proc/lockdep_stats + +This command produces the following output on a modest system:: + + lock-classes: 748 [max: 8191] + +If the number allocated (748 above) increases continually over time, +then there is likely a leak. The following command can be used to +identify the leaking lock classes:: + + grep "BD" /proc/lockdep + +Run the command and save the output, then compare against the output from +a later run of this command to identify the leakers. This same output +can also help you find situations where runtime lock initialization has +been omitted. + +Recursive read locks: +--------------------- +The whole of the rest document tries to prove a certain type of cycle is equivalent +to deadlock possibility. + +There are three types of lockers: writers (i.e. exclusive lockers, like +spin_lock() or write_lock()), non-recursive readers (i.e. shared lockers, like +down_read()) and recursive readers (recursive shared lockers, like rcu_read_lock()). +And we use the following notations of those lockers in the rest of the document: + + W or E: stands for writers (exclusive lockers). + r: stands for non-recursive readers. + R: stands for recursive readers. + S: stands for all readers (non-recursive + recursive), as both are shared lockers. + N: stands for writers and non-recursive readers, as both are not recursive. + +Obviously, N is "r or W" and S is "r or R". + +Recursive readers, as their name indicates, are the lockers allowed to acquire +even inside the critical section of another reader of the same lock instance, +in other words, allowing nested read-side critical sections of one lock instance. + +While non-recursive readers will cause a self deadlock if trying to acquire inside +the critical section of another reader of the same lock instance. + +The difference between recursive readers and non-recursive readers is because: +recursive readers get blocked only by a write lock *holder*, while non-recursive +readers could get blocked by a write lock *waiter*. Considering the follow +example:: + + TASK A: TASK B: + + read_lock(X); + write_lock(X); + read_lock_2(X); + +Task A gets the reader (no matter whether recursive or non-recursive) on X via +read_lock() first. And when task B tries to acquire writer on X, it will block +and become a waiter for writer on X. Now if read_lock_2() is recursive readers, +task A will make progress, because writer waiters don't block recursive readers, +and there is no deadlock. However, if read_lock_2() is non-recursive readers, +it will get blocked by writer waiter B, and cause a self deadlock. + +Block conditions on readers/writers of the same lock instance: +-------------------------------------------------------------- +There are simply four block conditions: + +1. Writers block other writers. +2. Readers block writers. +3. Writers block both recursive readers and non-recursive readers. +4. And readers (recursive or not) don't block other recursive readers but + may block non-recursive readers (because of the potential co-existing + writer waiters) + +Block condition matrix, Y means the row blocks the column, and N means otherwise. + + +---+---+---+---+ + | | E | r | R | + +---+---+---+---+ + | E | Y | Y | Y | + +---+---+---+---+ + | r | Y | Y | N | + +---+---+---+---+ + | R | Y | Y | N | + +---+---+---+---+ + + (W: writers, r: non-recursive readers, R: recursive readers) + + +acquired recursively. Unlike non-recursive read locks, recursive read locks +only get blocked by current write lock *holders* other than write lock +*waiters*, for example:: + + TASK A: TASK B: + + read_lock(X); + + write_lock(X); + + read_lock(X); + +is not a deadlock for recursive read locks, as while the task B is waiting for +the lock X, the second read_lock() doesn't need to wait because it's a recursive +read lock. However if the read_lock() is non-recursive read lock, then the above +case is a deadlock, because even if the write_lock() in TASK B cannot get the +lock, but it can block the second read_lock() in TASK A. + +Note that a lock can be a write lock (exclusive lock), a non-recursive read +lock (non-recursive shared lock) or a recursive read lock (recursive shared +lock), depending on the lock operations used to acquire it (more specifically, +the value of the 'read' parameter for lock_acquire()). In other words, a single +lock instance has three types of acquisition depending on the acquisition +functions: exclusive, non-recursive read, and recursive read. + +To be concise, we call that write locks and non-recursive read locks as +"non-recursive" locks and recursive read locks as "recursive" locks. + +Recursive locks don't block each other, while non-recursive locks do (this is +even true for two non-recursive read locks). A non-recursive lock can block the +corresponding recursive lock, and vice versa. + +A deadlock case with recursive locks involved is as follow:: + + TASK A: TASK B: + + read_lock(X); + read_lock(Y); + write_lock(Y); + write_lock(X); + +Task A is waiting for task B to read_unlock() Y and task B is waiting for task +A to read_unlock() X. + +Dependency types and strong dependency paths: +--------------------------------------------- +Lock dependencies record the orders of the acquisitions of a pair of locks, and +because there are 3 types for lockers, there are, in theory, 9 types of lock +dependencies, but we can show that 4 types of lock dependencies are enough for +deadlock detection. + +For each lock dependency:: + + L1 -> L2 + +, which means lockdep has seen L1 held before L2 held in the same context at runtime. +And in deadlock detection, we care whether we could get blocked on L2 with L1 held, +IOW, whether there is a locker L3 that L1 blocks L3 and L2 gets blocked by L3. So +we only care about 1) what L1 blocks and 2) what blocks L2. As a result, we can combine +recursive readers and non-recursive readers for L1 (as they block the same types) and +we can combine writers and non-recursive readers for L2 (as they get blocked by the +same types). + +With the above combination for simplification, there are 4 types of dependency edges +in the lockdep graph: + +1) -(ER)->: + exclusive writer to recursive reader dependency, "X -(ER)-> Y" means + X -> Y and X is a writer and Y is a recursive reader. + +2) -(EN)->: + exclusive writer to non-recursive locker dependency, "X -(EN)-> Y" means + X -> Y and X is a writer and Y is either a writer or non-recursive reader. + +3) -(SR)->: + shared reader to recursive reader dependency, "X -(SR)-> Y" means + X -> Y and X is a reader (recursive or not) and Y is a recursive reader. + +4) -(SN)->: + shared reader to non-recursive locker dependency, "X -(SN)-> Y" means + X -> Y and X is a reader (recursive or not) and Y is either a writer or + non-recursive reader. + +Note that given two locks, they may have multiple dependencies between them, +for example:: + + TASK A: + + read_lock(X); + write_lock(Y); + ... + + TASK B: + + write_lock(X); + write_lock(Y); + +, we have both X -(SN)-> Y and X -(EN)-> Y in the dependency graph. + +We use -(xN)-> to represent edges that are either -(EN)-> or -(SN)->, the +similar for -(Ex)->, -(xR)-> and -(Sx)-> + +A "path" is a series of conjunct dependency edges in the graph. And we define a +"strong" path, which indicates the strong dependency throughout each dependency +in the path, as the path that doesn't have two conjunct edges (dependencies) as +-(xR)-> and -(Sx)->. In other words, a "strong" path is a path from a lock +walking to another through the lock dependencies, and if X -> Y -> Z is in the +path (where X, Y, Z are locks), and the walk from X to Y is through a -(SR)-> or +-(ER)-> dependency, the walk from Y to Z must not be through a -(SN)-> or +-(SR)-> dependency. + +We will see why the path is called "strong" in next section. + +Recursive Read Deadlock Detection: +---------------------------------- + +We now prove two things: + +Lemma 1: + +If there is a closed strong path (i.e. a strong circle), then there is a +combination of locking sequences that causes deadlock. I.e. a strong circle is +sufficient for deadlock detection. + +Lemma 2: + +If there is no closed strong path (i.e. strong circle), then there is no +combination of locking sequences that could cause deadlock. I.e. strong +circles are necessary for deadlock detection. + +With these two Lemmas, we can easily say a closed strong path is both sufficient +and necessary for deadlocks, therefore a closed strong path is equivalent to +deadlock possibility. As a closed strong path stands for a dependency chain that +could cause deadlocks, so we call it "strong", considering there are dependency +circles that won't cause deadlocks. + +Proof for sufficiency (Lemma 1): + +Let's say we have a strong circle:: + + L1 -> L2 ... -> Ln -> L1 + +, which means we have dependencies:: + + L1 -> L2 + L2 -> L3 + ... + Ln-1 -> Ln + Ln -> L1 + +We now can construct a combination of locking sequences that cause deadlock: + +Firstly let's make one CPU/task get the L1 in L1 -> L2, and then another get +the L2 in L2 -> L3, and so on. After this, all of the Lx in Lx -> Lx+1 are +held by different CPU/tasks. + +And then because we have L1 -> L2, so the holder of L1 is going to acquire L2 +in L1 -> L2, however since L2 is already held by another CPU/task, plus L1 -> +L2 and L2 -> L3 are not -(xR)-> and -(Sx)-> (the definition of strong), which +means either L2 in L1 -> L2 is a non-recursive locker (blocked by anyone) or +the L2 in L2 -> L3, is writer (blocking anyone), therefore the holder of L1 +cannot get L2, it has to wait L2's holder to release. + +Moreover, we can have a similar conclusion for L2's holder: it has to wait L3's +holder to release, and so on. We now can prove that Lx's holder has to wait for +Lx+1's holder to release, and note that Ln+1 is L1, so we have a circular +waiting scenario and nobody can get progress, therefore a deadlock. + +Proof for necessary (Lemma 2): + +Lemma 2 is equivalent to: If there is a deadlock scenario, then there must be a +strong circle in the dependency graph. + +According to Wikipedia[1], if there is a deadlock, then there must be a circular +waiting scenario, means there are N CPU/tasks, where CPU/task P1 is waiting for +a lock held by P2, and P2 is waiting for a lock held by P3, ... and Pn is waiting +for a lock held by P1. Let's name the lock Px is waiting as Lx, so since P1 is waiting +for L1 and holding Ln, so we will have Ln -> L1 in the dependency graph. Similarly, +we have L1 -> L2, L2 -> L3, ..., Ln-1 -> Ln in the dependency graph, which means we +have a circle:: + + Ln -> L1 -> L2 -> ... -> Ln + +, and now let's prove the circle is strong: + +For a lock Lx, Px contributes the dependency Lx-1 -> Lx and Px+1 contributes +the dependency Lx -> Lx+1, and since Px is waiting for Px+1 to release Lx, +so it's impossible that Lx on Px+1 is a reader and Lx on Px is a recursive +reader, because readers (no matter recursive or not) don't block recursive +readers, therefore Lx-1 -> Lx and Lx -> Lx+1 cannot be a -(xR)-> -(Sx)-> pair, +and this is true for any lock in the circle, therefore, the circle is strong. + +References: +----------- +[1]: https://en.wikipedia.org/wiki/Deadlock +[2]: Shibu, K. (2009). Intro To Embedded Systems (1st ed.). Tata McGraw-Hill |