Concurrent Objects Companion slides for The Art of Multiprocessor Programming by Maurice Herlihy & Nir Shavit

Concurrent Computation memory object object Art of Multiprocessor Programming

Art of Multiprocessor Programming Objectivism What is a concurrent object? How do we describe one? How do we implement one? How do we tell if we’re right? In this lecture we will focus on the first and last items. Art of Multiprocessor Programming

FIFO Queue: Enqueue Method q.enq( ) Consider a FIFO queue. Art of Multiprocessor Programming

FIFO Queue: Dequeue Method q.deq()/ Art of Multiprocessor Programming

Art of Multiprocessor Programming A Lock-Based Queue class LockBasedQueue<T> { int head, tail; T[] items; Lock lock; public LockBasedQueue(int capacity) { head = 0; tail = 0; lock = new ReentrantLock(); items = (T[]) new Object[capacity]; } Here is code for implementing a concurrent FIFO queue. Explain it IN DETAIL to the students. It’s the first algorithm they see which is not a mutual exclusion algorithm. The queue is implemented in an array. Initially the Head and Tail fields are equal and the queue is empty. All operations are synchronized Via the objects lock. If the Head and Tail differ by the queue size, then the queue is full. The Enq() method reads the Head() field, and if the queue is full, it waits until the queue is no longer full. It then stores the object in the array, and increments the Tail() field. Art of Multiprocessor Programming

Art of Multiprocessor Programming A Lock-Based Queue 1 capacity-1 2 head tail y z class LockBasedQueue<T> { int head, tail; T[] items; Lock lock; public LockBasedQueue(int capacity) { head = 0; tail = 0; lock = new ReentrantLock(); items = (T[]) new Object[capacity]; } Here is code for implementing a concurrent FIFO queue. Explain it IN DETAIL to the students. It’s the first algorithm they see which is not a mutual exclusion algorithm. The queue is implemented in an array. Initially the Head and Tail fields are equal and the queue is empty. All operations are synchronized Via the objects lock. If the Head and Tail differ by the queue size, then the queue is full. The Enq() method reads the Head() field, and if the queue is full, it waits until the queue is no longer full. It then stores the object in the array, and increments the Tail() field. Queue fields protected by single shared lock Art of Multiprocessor Programming

Art of Multiprocessor Programming A Lock-Based Queue 1 capacity-1 2 head tail y z class LockBasedQueue<T> { int head, tail; T[] items; Lock lock; public LockBasedQueue(int capacity) { head = 0; tail = 0; lock = new ReentrantLock(); items = (T[]) new Object[capacity]; } Here is code for implementing a concurrent FIFO queue. Explain it IN DETAIL to the students. It’s the first algorithm they see which is not a mutual exclusion algorithm. The queue is implemented in an array. Initially the Head and Tail fields are equal and the queue is empty. All operations are synchronized Via the objects lock. If the Head and Tail differ by the queue size, then the queue is full. The Enq() method reads the Head() field, and if the queue is full, it waits until the queue is no longer full. It then stores the object in the array, and increments the Tail() field. Initially head = tail Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Method calls mutually exclusive Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } If queue empty throw exception Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Queue not empty: remove item and update head Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Return result Art of Multiprocessor Programming

Implementation: Deq Release lock no matter what! y z y z 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Release lock no matter what! Art of Multiprocessor Programming

Art of Multiprocessor Programming Implementation: Deq public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Should be correct because modifications are mutually exclusive… Art of Multiprocessor Programming

Now consider the following implementation The same thing without mutual exclusion For simplicity, only two threads One thread enq only The other deq only This is sometimes called a producer-consumer channel or buffer. Art of Multiprocessor Programming

Wait-free 2-Thread Queue public class WaitFreeQueue { int head = 0, tail = 0; items = (T[]) new Object[capacity]; public void enq(Item x) { if (tail-head == capacity) throw new FullException(); items[tail % capacity] = x; tail++; } public Item deq() { if (tail == head) throw new EmptyException(); Item item = items[head % capacity]; head++; return item; }} 1 capacity-1 2 head tail y z Art of Multiprocessor Programming

Wait-free 2-Thread Queue public class WaitFreeQueue { int head = 0, tail = 0; items = (T[]) new Object[capacity]; public void enq(Item x) { if (tail-head == capacity) throw new FullException(); items[tail % capacity] = x; tail++; } public Item deq() { if (tail == head) throw new EmptyException(); Item item = items[head % capacity]; head++; return item; }} 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z Art of Multiprocessor Programming

Wait-free 2-Thread Queue public class WaitFreeQueue { int head = 0, tail = 0; items = (T[]) new Object[capacity]; public void enq(Item x) { if (tail-head == capacity) throw new FullException(); items[tail % capacity] = x; tail++; } public Item deq() { if (tail == head) throw new EmptyException(); Item item = items[head % capacity]; head++; return item; }} 1 capacity-1 2 head tail y z How do we define “correct” when modifications are not mutually exclusive? Queue is updated without a lock! Art of Multiprocessor Programming

Defining concurrent queue implementations Need a way to specify a concurrent queue object Need a way to prove that an algorithm implements the object’s specification Lets talk about object specifications … In general, to make our intuitive understanding that the algorithm is correct we need a way to specify a concurrent queue object. Need a way to prove that the algorithm implements the object’s specification. We saw two types of object implementations, one that used locks and intuitively felt like a queue, and one that did not use locks, it was infact, wait-free. What properties are they actually providing, are they both implementations of a concurrent queue? In what ways are they similar and in what ways do they differ? Lets try and answer these questions. Art of Multiprocessor Programming

Correctness and Progress In a concurrent setting, we need to specify both the safety and the liveness properties of an object Need a way to define when an implementation is correct the conditions under which it guarantees progress There are two elements in which a concurrent specification imposes terms on the implementation: safety and liveness, or, in our case, correctness and progress. Lets begin with correctness Art of Multiprocessor Programming

Art of Multiprocessor Programming Sequential Objects Each object has a state Usually given by a set of fields Queue example: sequence of items Each object has a set of methods Only way to manipulate state Queue example: enq and deq methods Keep going back to the queue example throughout this lecture. Later, if you need to talk about multiple objects, give as an example a program that uses a stack and a queue. Art of Multiprocessor Programming

Sequential Specifications If (precondition) the object is in such-and-such a state before you call the method, Then (postcondition) the method will return a particular value or throw a particular exception. and (postcondition, con’t) the object will be in some other state when the method returns, Art of Multiprocessor Programming

Pre and PostConditions for Dequeue Precondition: Queue is non-empty Postcondition: Returns first item in queue Removes first item in queue Art of Multiprocessor Programming

Pre and PostConditions for Dequeue Precondition: Queue is empty Postcondition: Throws Empty exception Queue state unchanged Art of Multiprocessor Programming

Why Sequential Specifications Totally Rock Interactions among methods captured by side-effects on object state State meaningful between method calls Documentation size linear in number of methods Each method described in isolation Can add new methods Without changing descriptions of old methods Art of Multiprocessor Programming

What About Concurrent Specifications ? Methods? Documentation? Adding new methods? Art of Multiprocessor Programming

Art of Multiprocessor Programming Methods Take Time In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time. time time Art of Multiprocessor Programming

Art of Multiprocessor Programming Methods Take Time invocation 12:00 q.enq(...) In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time. time time Art of Multiprocessor Programming

Art of Multiprocessor Programming Methods Take Time invocation 12:00 q.enq(...) Method call In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time. time time Art of Multiprocessor Programming

Art of Multiprocessor Programming Methods Take Time invocation 12:00 q.enq(...) Method call In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time. time time Art of Multiprocessor Programming

Art of Multiprocessor Programming Methods Take Time invocation 12:00 void response 12:01 q.enq(...) Method call In the sequential world, we treat method calls as if they were instantaneous, but in fact, they take time. time time Art of Multiprocessor Programming

Sequential vs Concurrent Methods take time? Who knew? Concurrent Method call is not an event Method call is an interval. Art of Multiprocessor Programming

Concurrent Methods Take Overlapping Time Art of Multiprocessor Programming

Concurrent Methods Take Overlapping Time Method call time time Art of Multiprocessor Programming

Concurrent Methods Take Overlapping Time Method call Method call time time Art of Multiprocessor Programming

Concurrent Methods Take Overlapping Time Method call Method call Method call time time Art of Multiprocessor Programming

Sequential vs Concurrent Object needs meaningful state only between method calls Concurrent Because method calls overlap, object might never be between method calls Art of Multiprocessor Programming

Sequential vs Concurrent Each method described in isolation Concurrent Must characterize all possible interactions with concurrent calls What if two enqs overlap? Two deqs? enq and deq? … Art of Multiprocessor Programming

Sequential vs Concurrent Can add new methods without affecting older methods Concurrent: Everything can potentially interact with everything else Art of Multiprocessor Programming

Sequential vs Concurrent Can add new methods without affecting older methods Concurrent: Everything can potentially interact with everything else Panic! Art of Multiprocessor Programming

Art of Multiprocessor Programming The Big Question What does it mean for a concurrent object to be correct? What is a concurrent FIFO queue? FIFO means strict temporal order Concurrent means ambiguous temporal order Art of Multiprocessor Programming

Art of Multiprocessor Programming Intuitively… public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Art of Multiprocessor Programming

Art of Multiprocessor Programming Intuitively… public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } All modifications of queue are done mutually exclusive Art of Multiprocessor Programming

Art of Multiprocessor Programming Intuitively Lets capture the idea of describing the concurrent via the sequential q.deq lock() unlock() enq deq deq q.enq enq Behavior is “Sequential” On an intuitive level, since we use mutual exclusion locks, then each of the actual updates happens in a non-overlapping interval so the behavior as a whole looks sequential. This fits our intuition, and the question is if we can generalize this intuition. Stress the fact that we understand the concurrent execution because we have a way of mapping it to the sequential one, in which we know how a FIFO queue behaves. time Art of Multiprocessor Programming

Art of Multiprocessor Programming Linearizability Each method should “take effect” Instantaneously Between invocation and response events Object is correct if this “sequential” behavior is correct Any such concurrent object is Linearizable™ Linearizability is the generalization of this intuition to general objects, with our without mutual exclusion. Art of Multiprocessor Programming

Is it really about the object? Each method should “take effect” Instantaneously Between invocation and response events Sounds like a property of an execution… A linearizable object: one all of whose possible executions are linearizable Art of Multiprocessor Programming

Art of Multiprocessor Programming Example Take you time with this example…don’t let them guess this one, show them, and let them guess the linearizability of the subsequent ones. time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.enq(y) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.enq(y) q.deq(x) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(y) q.enq(y) q.deq(x) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example linearizable q.enq(x) q.enq(x) q.enq(y) q.deq(x) q.deq(y) q.deq(y) q.enq(y) q.deq(x) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.enq(x) Valid? q.deq(y) q.deq(y) q.enq(y) q.enq(y) q.deq(x) q.deq(x) time time (6) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(y) time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(y) q.enq(y) time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.enq(y) q.enq(x) q.deq(y) q.enq(y) time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example not linearizable q.enq(x) q.enq(x) q.deq(y) q.enq(y) q.enq(y) time (5) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example time time (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) time time (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(x) time time (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(x) q.enq(x) q.deq(x) time time (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example linearizable q.enq(x) q.enq(x) q.deq(x) q.deq(x) time time (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) time time (8) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.enq(y) time time (8) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(y) q.enq(y) time time (8) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example q.enq(x) q.deq(y) q.enq(y) q.deq(x) time time (8) Art of Multiprocessor Programming

Art of Multiprocessor Programming Example Comme ci Comme ça multiple orders OK q.enq(x) q.deq(y) linearizable q.enq(y) q.deq(x) time Art of Multiprocessor Programming

Read/Write Register Example time time (4) Art of Multiprocessor Programming

Read/Write Register Example write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example not linearizable write(0) read(1) write(2) write(1) write(1) read(0) write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example not linearizable write(0) read(1) write(2) write(2) write(1) write(1) read(1) write(1) already happened time time (4) Art of Multiprocessor Programming

Read/Write Register Example time time (4) Art of Multiprocessor Programming

Read/Write Register Example time time (4) Art of Multiprocessor Programming

Read/Write Register Example linearizable write(0) write(1) write(2) write(2) write(1) read(1) time time (4) Art of Multiprocessor Programming

Read/Write Register Example time time (2) Art of Multiprocessor Programming

Read/Write Register Example time time (2) Art of Multiprocessor Programming

Read/Write Register Example time time (2) Art of Multiprocessor Programming

Read/Write Register Example linearizable Not write(0) read(1) write(2) write(2) write(1) write(1) read(2) time time (2) Art of Multiprocessor Programming

Talking About Executions Why? Can’t we specify the linearization point of each operation without describing an execution? Not Always In some cases, linearization point depends on the execution Art of Multiprocessor Programming

Formal Model of Executions Define precisely what we mean Ambiguity is bad when intuition is weak Allow reasoning Formal But mostly informal In the long run, actually more important Ask me why! We now define a formal model to allow us to precisely define linearizability and other consistency conditions. Art of Multiprocessor Programming

Split Method Calls into Two Events Invocation method name & args q.enq(x) Response result or exception q.enq(x) returns void q.deq() returns x q.deq() throws empty Art of Multiprocessor Programming

Art of Multiprocessor Programming Invocation Notation A q.enq(x) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Invocation Notation thread A q.enq(x) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Invocation Notation thread method A q.enq(x) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Invocation Notation thread object method A q.enq(x) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Invocation Notation thread object method arguments A q.enq(x) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation A q: void (2) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation thread A q: void (2) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation thread A q: void result (2) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation thread object A q: void result (2) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation thread object A q: void Method is implicit result (2) Art of Multiprocessor Programming

Art of Multiprocessor Programming Response Notation thread object exception A q: empty() Method is implicit (2) Art of Multiprocessor Programming

History - Describing an Execution A q.enq(3) A q:void A q.enq(5) B p.enq(4) B p:void B q.deq() B q:3 H = Sequence of invocations and responses Art of Multiprocessor Programming

Art of Multiprocessor Programming Definition Invocation & response match if Object names agree Thread names agree A q.enq(3) A q:void Method call (1) Art of Multiprocessor Programming

Art of Multiprocessor Programming Object Projections A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 H = Art of Multiprocessor Programming

Art of Multiprocessor Programming Object Projections A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 H|q = Art of Multiprocessor Programming

Art of Multiprocessor Programming Thread Projections A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 H = Art of Multiprocessor Programming

Art of Multiprocessor Programming Thread Projections A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 H|B = Art of Multiprocessor Programming

An invocation is pending if it has no matching response Complete Subhistory A q.enq(3) A q:void A q.enq(5) B p.enq(4) B p:void B q.deq() B q:3 H = An invocation is pending if it has no matching response Art of Multiprocessor Programming

May or may not have taken effect Complete Subhistory A q.enq(3) A q:void A q.enq(5) B p.enq(4) B p:void B q.deq() B q:3 H = May or may not have taken effect Art of Multiprocessor Programming

discard pending invocations Complete Subhistory A q.enq(3) A q:void A q.enq(5) B p.enq(4) B p:void B q.deq() B q:3 H = discard pending invocations Art of Multiprocessor Programming

Art of Multiprocessor Programming Complete Subhistory A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 Complete(H) = Art of Multiprocessor Programming

Art of Multiprocessor Programming Sequential Histories A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Sequential Histories match A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Sequential Histories match A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) match (4) Art of Multiprocessor Programming

Art of Multiprocessor Programming Sequential Histories match A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) match match (4) Art of Multiprocessor Programming

Final pending invocation OK Sequential Histories match A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) match match Final pending invocation OK (4) Art of Multiprocessor Programming

Method calls of different threads do not interleave Sequential Histories match A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 A q:enq(5) Method calls of different threads do not interleave match match Final pending invocation OK (4) Art of Multiprocessor Programming

Well-Formed Histories A q.enq(3) B p.enq(4) B p:void B q.deq() A q:void B q:3 H= Art of Multiprocessor Programming

Well-Formed Histories Per-thread projections sequential B p.enq(4) B p:void B q.deq() B q:3 H|B= A q.enq(3) B p.enq(4) B p:void B q.deq() A q:void B q:3 H= Art of Multiprocessor Programming

Well-Formed Histories Per-thread projections sequential B p.enq(4) B p:void B q.deq() B q:3 H|B= A q.enq(3) B p.enq(4) B p:void B q.deq() A q:void B q:3 H= A q.enq(3) A q:void H|A= Art of Multiprocessor Programming

Threads see the same thing in both Equivalent Histories H|A = G|A H|B = G|B Threads see the same thing in both A q.enq(3) B p.enq(4) B p:void B q.deq() A q:void B q:3 A q.enq(3) A q:void B p.enq(4) B p:void B q.deq() B q:3 H= G= Art of Multiprocessor Programming

Sequential Specifications A sequential specification is some way of telling whether a Single-thread, single-object history Is legal For example: Pre and post-conditions But plenty of other techniques exist … Art of Multiprocessor Programming

Art of Multiprocessor Programming Legal Histories A sequential (multi-object) history H is legal if For every object x H|x is in the sequential spec for x Art of Multiprocessor Programming

Art of Multiprocessor Programming Precedence A q.enq(3) B p.enq(4) B p.void A q:void B q.deq() B q:3 A method call precedes another if response event precedes invocation event Method call Method call (1) Art of Multiprocessor Programming

Some method calls overlap one another Non-Precedence A q.enq(3) B p.enq(4) B p.void B q.deq() A q:void B q:3 Some method calls overlap one another . Method call Method call (1) Art of Multiprocessor Programming

Art of Multiprocessor Programming Notation Given History H method executions m0 and m1 in H We say m0 H m1, if m0 precedes m1 Relation m0 H m1 is a Partial order Total order if H is sequential m0 m1 Maybe the main point to notice is that in general, given a history, precedence only defines a partial order – because we have methods that overlaps. So in a sequential history, this is a total order, because we have no overlapping methods. The fact that this is only partial, is exactly the ambiguity about the order that we do not like, and that linearizability comes to fix. So what we really want, is to take the partial order, and somehow complete it to a total order – and that’s how we’ll define linearizability in the next slide. Art of Multiprocessor Programming

Art of Multiprocessor Programming Linearizability History H is linearizable if it can be extended to G by Appending zero or more responses to pending invocations Discarding other pending invocations So that G is equivalent to Legal sequential history S where G  S First part: takes care of pending invocations. If the operation took effect, we want to append a response, otherwise, let’s drop it. Second part, takes care of completing the missing part in the partial order – it tells us to find an equivalent legal sequential history (in which precedence relation defines total order), such that this total order respects the partial order in the original history. Of course, the correctness part hides in the fact that the sequential history is legal. Explain that equivalence captures, apart from the ordering, the fact that the input/output relation of the methods must be the same as in the sequential execution. The ordering part is also covered by the G \subset S requirement but the fac that the values must be the same is captured only by the equivalence requirement. When we later go to seq consistency the only ordering property will come from the equivalence property. Art of Multiprocessor Programming

Art of Multiprocessor Programming What is G  S G = {ac,bc} S = {ab,ac,bc} A limitation on the Choice of S! a G b c The G  S is a limitation on S NOT ON G! Its saying that you can just pick any S, you must pick an  S so that it contains G, that is, its saying that when we pick the linearization points, they have got to be within the intervals, and we can only pick an arbitrary order when the intervals are unordered. G time time S (8) Art of Multiprocessor Programming

Art of Multiprocessor Programming Remarks Some pending invocations Took effect, so keep them Discard the rest Condition G  S Means that S respects “real-time order” of G Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 B q:enq(6) A. q.enq(3) B.q.enq(4) B.q.deq(4) B. q.enq(6) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 B q:enq(6) Complete this pending invocation A. q.enq(3) B.q.enq(4) B.q.deq(3) B. q.enq(6) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 B q:enq(6) A q:void Complete this pending invocation A.q.enq(3) B.q.enq(4) B.q.deq(4) B. q.enq(6) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 B q:enq(6) A q:void discard this one A.q.enq(3) B.q.enq(4) B.q.deq(4) B. q.enq(6) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 A q:void discard this one A.q.enq(3) B.q.enq(4) B.q.deq(4) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 A q:void A.q.enq(3) B.q.enq(4) B.q.deq(4) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 A q:void B q.enq(4) B q:void A q.enq(3) A q:void B q.deq() B q:4 A.q.enq(3) B.q.enq(4) B.q.deq(4) time Art of Multiprocessor Programming

Art of Multiprocessor Programming Example Equivalent sequential history A q.enq(3) B q.enq(4) B q:void B q.deq() B q:4 A q:void B q.enq(4) B q:void A q.enq(3) A q:void B q.deq() B q:4 A.q.enq(3) B.q.enq(4) B.q.deq(4) time Art of Multiprocessor Programming

Composability Theorem History H is linearizable if and only if For every object x H|x is linearizable We care about objects only! (Materialism?) Art of Multiprocessor Programming

Why Does Composability Matter? Modularity Can prove linearizability of objects in isolation Can compose independently-implemented objects Art of Multiprocessor Programming

Reasoning About Linearizability: Locking 1 capacity-1 2 head tail y z public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Art of Multiprocessor Programming

Reasoning About Linearizability: Locking public T deq() throws EmptyException { lock.lock(); try { if (tail == head) throw new EmptyException(); T x = items[head % items.length]; head++; return x; } finally { lock.unlock(); } Linearization points are when locks are released Art of Multiprocessor Programming

More Reasoning: Wait-free public class WaitFreeQueue { int head = 0, tail = 0; items = (T[]) new Object[capacity]; public void enq(Item x) { if (tail-head == capacity) throw new FullException(); items[tail % capacity] = x; tail++; } public Item deq() { if (tail == head) throw new EmptyException(); Item item = items[head % capacity]; head++; return item; }} 1 capacity-1 2 head tail y z 1 capacity-1 2 head tail y z Art of Multiprocessor Programming

More Reasoning: Wait-free public class WaitFreeQueue { int head = 0, tail = 0; items = (T[]) new Object[capacity]; public void enq(Item x) { if (tail-head == capacity) throw new FullException(); items[tail % capacity] = x; tail++; } public Item deq() { if (tail == head) throw new EmptyException(); Item item = items[head % capacity]; head++; return item; }} 1 capacity-1 2 head tail y z Linearization order is order head and tail fields modified Remember that there is only one enqueuer and only one dequeuer Art of Multiprocessor Programming

Art of Multiprocessor Programming Strategy Identify one atomic step where method “happens” Critical section Machine instruction Doesn’t always work Might need to define several different steps for a given method We will see this phenomena that you might need different steps for different executions in lists chapter. Art of Multiprocessor Programming

Linearizability: Summary Powerful specification tool for shared objects, good for high level objects (composability) Allows us to capture the notion of objects being “atomic” Operation takes effect instantaneously between invocation and response Uses sequential specification Don’t leave home without it We will see this phenomena that you might need different steps for different executions in lists chapter. Art of Multiprocessor Programming

Art of Multiprocessor Programming Progress We saw an implementation whose methods were lock-based (deadlock-free) We saw an implementation whose methods did not use locks (lock-free) How do they relate? So far we have talked about correctness. Now its time to discuss progress. Art of Multiprocessor Programming

Art of Multiprocessor Programming Progress Conditions Deadlock-free: some thread trying to acquire the lock eventually succeeds. Starvation-free: every thread trying to acquire the lock eventually succeeds. Lock-free: some thread calling a method eventually returns. Wait-free: every thread calling a method eventually returns. The above are definitions of progress conditions we have used and will use in the coming lectures. We give here informal definitions of progress conditions…formal ones need to talk about fair histories which is beyond the scope of this lecture…for the above conditions: A method implementation is deadlock-free if it guarantees min- imal progress in every fair history, and maximal progress in some fair history. A method implementation is starvation-free if it guarantees maximal progress in every fair history. A method implementation is lock-free if it guarantees minimal progress in every history and maximal progress in some history. A method implementation is wait-free if it guarantees maximal progress in every history. Art of Multiprocessor Programming

Art of Multiprocessor Programming Progress Conditions Non-Blocking Blocking Everyone makes progress Wait-free Starvation-free Someone makes progress Lock-free Deadlock-free Art of Multiprocessor Programming

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