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1 Introduction 2 MCS-Lock Algorithm A simple correctness pro of of the MCS contentionfree lo ck Theo dore Johnson Krishna Harathi Computer and Information Sciences Department University of Florida Abstract MellorCrummey and Scott present a spinlo ck that avoids network contention byhaving pro cessors spin on lo cal memory lo cations Their algorithm is equivalent to a lo ckfree queue with a sp ecial access pattern The authors provide a complex and unintuitive pro of of the correctness of their algorithm In this pap er weprovide a simple pro of that the MCS lo ckisa correct critical section solution whichprovides insightinto why the algorithm is correct Keywords mutual exclusion serializability parallel pro cessing spin lo ck Intro duction MellorCrummey and Scott present a simple spinlo ckformutual exclusion which they call the MCS lock Their algorithm has several desirable prop erties it is is fair b ecause it is equivalentto aFIFO queue ADT with sp ecial access pattern and it is is contention free b ecause each pro cessor spins on a lo cal variable In contrast testandset spin waiting is not fair and will severely limit p erformance The authors of the MCS lo cks provide a complex pro of of the correctness of their algorithm In they show correctness through a series of algorithm transformations Since this pro of provides little insight they also provide intuitive correctness arguments In this pap er we present a simple correctness pro of for the MCS lo ck We show that the underlying queue is decisiveinstruction serializable and that no op eration access a garbage address We conclude that the MCS lo ck is a correct solution to the critical section problem MCSLo ck Algorithm The co de for the MCS lo ck which is shown b elow relies on two atomic readmo difywrite op era tions The swap instruction takes two parameters Lapointer to a memory lo cation and Iavalue The execution of swapLI puts I into L and returns the old value of LThe compare and swap instruction takes a third parameter Xavalue The execution of compare and swapLIX puts and swap returns the old I into L only if L currently contains the value X In either case compare value in L Each pro cessor that sets a lo ck rst allo cates a lo callyaccessible record If pro cess p inserts record r into the queue then wesay that r is ps record and p is r s pro cess This record contains a b o olean ag for spin waiting and a link for forming a queue The tail of the queue is p ointed to by L and the head of the queue is implicitly p ointed to by the lo ck holder Tosetalock a pro cessor executes the acquire lock pro cedure and adds its record p ointed to by I to the end of the queue The enqueuing is p erformed by executing the fetch and store instruction which atomically stores I in L and returns Ls old value If the pro cessor has a predecessor in the queue the pro cessor completes the link otherwise the pro cessor is the lo ck holder To release a lo ck the pro cessor must reset the busywaiting bit of its successor If the pro cessor and swap has no successor it sets L to nil atomically with the compare type qnode record next qnode ptr to successor in queue locked Boolean busywaiting necessary type lock qnode ptr to tail of queue I points to a queue link record allocated in an enclosing scope in shared memory locally accessible to the invoking processor lock L lock I qnode procedure acquire var pred qnode Inext nil Initially no successor pred swapL I Queue for lock if pred nil Lock was not free ilocked true Prepare to spin prednext I Link behind predecessor repeat while Ilocked Busy wait for lock procedure release lock L lock I qnode if Inext nil No known successor if compare and swapL I nil return No successor lock free repeat while Inext nil Wait for successor Inextlocked false Pass lock MCSLo ck Algorithm from Correctness We show that the MCS lo ck is correct by showing that it maintains a queue and the head of the queue is the pro cess that holds the lo ck The MCS lo ckisdecisiveinstruction serializable Each op eration has a single decisive instruction and corresp onding to a concurrent execution C of the queue op erations there is an equivalent serial execution S such that if op eration O executes its d decisive instruction b efore op eration O do es in C then O O in S The decisiveinstruction d serializabili ty of the MCS lo ck greatly simplies its correctness pro of b ecause the equivalent queue is in a single state at any instant In contrast a concurrent data structure that is linearizable but not serializable might b e in several states simultaneously The Queue ADT A queue is an Abstract Data Typ e that consists of nite set Q and two op erations enq ueue and q q deq ueue The elements of Q are totally ordered We write the state a queue as Q q n where q q q Q Q Q n The enqueue op eration is sp ecied by enq ueueq q q q q q q q n n The dequeue op eration on a nonempty queue is sp ecied by deq ueueq q q q q n n where the return value is q A dequeue op eration on an empty queue is undened We dene two functions on a queue Q headQ and tail Q The head function returns the least element of the queue and the tail function returns the greatest element of the queue Corresp onding to an actual MCS lo ck M there is an abstract queue Q Initially b oth M and Q are empty When pro cess p with record r p erforms the decisive instruction for an acquire lock op eration Q changes state to enq ueueQ r When a pro cess p erforms the decisive instruction for lock op eration Q changes state to deq ueueQ We will show that the actual MCS lo ck a release corresp onds to the abstract queue Execution Sequences The MCS lo ck executes correctly b ecause it is presented with a sp ecial sequence of concurrent op erations We assume that each pro cessor uses the acquire lock and release lock op erations to synchronize access to a resource acquire lockLr critical section lockL release If the MCS lo ck is correct then it is a multipleenqueuesingledequeue queue Anynumber of acquire lock op erations might execute concurrently but we are guaranteed that At most one release lock op eration executes at any given time Let D and D be two dierent release lo ck executions executing in the time intervals I and I resp ectively Then I and I do not overlap No release lock op eration executes b etween the time that a release lock sets L to Nil and the time that the rst subsequent acquire lock op eration terminates Let D b e the execution of a release lock op eration that sets L to NIL and let D complete its execution at time t Let E b e the execution of the acquire lock op eration that p erforms the rst decisive d lock instruction after time t If E completes its execution at time t then no release d e op eration executes in the time interval t t d e Queue Invariants The MCS lo ckmaintains three invariant conditions Tail Invariant If the abstract queue is nonempty then L p oints to the record at the tail of the abstract queue If the queue is empty L is Nil We dene the head process to b e the pro cess whose record is at the head of the abstract queue A pro cess that is waiting for Ilocked to b ecome false is blocked Head Invariant If the head pro cess is blo ckeditwillbeunblo cked by the time that the preceding release lock op eration terminates Blo cking Invariant A nonhead pro cess will not exit the acquire lock pro cedure Decisive Op erations lock pro cedure is the fetch and store instruction The The decisive instruction in the acquire decisive instruction in the release lock pro cedure is the reading of Inext if the fetch returns and swap instruction otherwise a nonnil value and is the compare Theorem The MCS lock is decisiveinstruction serializable Proof Toshow that the MCS lo ck is decisiveinstruction serializable we rst show that the three invariants are always maintained To show that the invariants are maintained we assume that only lock op eration We then show that this assumption holds by the head pro cess executes a release induction Lemma The tail invariant is always maintained Proof When a pro cess p erforms the decisive instruction for an acquire lock op eration it sets the tail p ointer L to its record Therefore acquire lock op erations maintain the tail invariant lock op eration mo dies the tail p ointer by the compare and swap instruction only A release The compare and swap will succeed only if the queue contains exactly one element Therefore the lock op eration also maintains the queue invariant release Lemma If only the head process executes a release lock operation the blocking invariant is always maintained Supp ose that the record of a pro cess is in the queue but isnt the head record By the tail invariant the pro cess must have read a nonnil tail p ointer when it p erformed the decisive instruction for the acquire lock op eration The pro cess will therefore wait until the locked eld of its record which is initialized to trueissettofalse This bit will only b e reset when the pro cess of the predecessor record executes a release lock op eration But after the release lock decisive instruction the pro cess b ecomes the head pro cess Lemma The head invariant is always maintained Supp ose that when a head pro cess P p erforms its decisive instruction on a release lock op eration deq ueueQ is nonempty In this case the decisive instruction will rep ort that tail p ointer is not the pro cess record Then P will wait until the next eld of its record R is nonnil By the tail invariant this eld will eventually p ointtoRs successor in Q r b ecause of the execution
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