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Outline 0024 Spring 2010 21 :: 2 Source [1] Talking About Executions Why? Can t we specify the linearization point of each operation without describing an execution? Not Always In some

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Outline

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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

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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!

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Split Method Calls into Two Events

Invocation

method name & args

  • q. enq( x)

Response

result or exception

  • q. enq( x) returns voi d
  • q. deq( ) returns x
  • q. deq( ) throws em

pt y

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Invocation Notation

A q.enq(x)

(4)

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Invocation Notation

A q.enq(x) thread

(4)

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Invocation Notation

A q.enq(x) thread method

(4)

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Invocation Notation

A q.enq(x) thread

  • bject

(4)

method

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Invocation Notation

A q.enq(x) thread

  • bject

method arguments

(4)

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Response Notation

A q: void

(2)

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Response Notation

A q: void thread

(2)

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Response Notation

A q: void thread result

(2)

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Response Notation

A q: void thread

  • bject

result

(2)

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Response Notation

A q: void thread

  • bject

result

(2)

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Response Notation

A q: empty() thread

  • bject

(2)

exception

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History - Describing an Execution

A q. enq( 3) A q: voi d A q. enq( 5) B p. enq( 4) B p: voi d B q. deq( ) B q: 3 Sequence of invocations and responses H =

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Definition

Invocation & response match if

A q. enq( 3) A q: voi d Thread names agree Object names agree

Method call

(1)

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Object Projections

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3

H =

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Object Projections

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3

H|q =

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Thread Projections

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3

H =

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Thread Projections

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3

H|B =

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Complete Subhistory

A q. enq( 3) A q: voi d A q. enq( 5) B p. enq( 4) B p: voi d B q. deq( ) B q: 3 An invocation is pending if it has no matching respnse H =

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Complete Subhistory

A q. enq( 3) A q: voi d A q. enq( 5) B p. enq( 4) B p: voi d B q. deq( ) B q: 3 May or may not have taken effect H =

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Complete Subhistory

A q. enq( 3) A q: voi d A q. enq( 5) B p. enq( 4) B p: voi d B q. deq( ) B q: 3 discard pending invocations H =

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Complete Subhistory

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 Complete(H) =

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5)

(4)

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5) match

(4)

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5) match match

(4)

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5) match match match

(4)

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5) match match match Final pending invocation OK

(4)

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Sequential Histories

A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q: enq( 5) match match match Final pending invocation OK

(4)

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Well-Formed Histories

H=

A q. enq( 3) B p. enq( 4) B p: voi d B q. deq( ) A q: voi d B q: 3

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Well-Formed Histories

H=

A q. enq( 3) B p. enq( 4) B p: voi d B q. deq( ) A q: voi d B q: 3

H| B=

B p. enq( 4) B p: voi d B q. deq( ) B q: 3 Per-thread projections sequential

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Well-Formed Histories

H=

A q. enq( 3) B p. enq( 4) B p: voi d B q. deq( ) A q: voi d B q: 3

H| B=

B p. enq( 4) B p: voi d B q. deq( ) B q: 3 A q. enq( 3) A q: voi d

H| A=

Per-thread projections sequential

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Equivalent Histories

H=

A q. enq( 3) B p. enq( 4) B p: voi d B q. deq( ) A q: voi d B q: 3 Threads see the same thing in both A q. enq( 3) A q: voi d B p. enq( 4) B p: voi d B q. deq( ) B q: 3

G = H| A = G | A H| B = G | B

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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 …

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Legal Histories

A sequential (multi-object) history H is legal if

For every object x H|x is in the sequential spec for x

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Precedence

A q. enq( 3) B p. enq( 4) B p. voi d A q: voi d B q. deq( ) B q: 3 A method call precedes another if response event precedes invocation event

Method call Method call

(1)

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Non-Precedence

A q. enq( 3) B p. enq( 4) B p. voi d B q. deq( ) A q: voi d B q: 3 Some method calls

  • verlap one another

Method call Method call

(1)

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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

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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

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What is G S

time a b time

(8)

S c G

G = { ac, bc} S = { ab, ac, bc}

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Remarks

Some pending invocations

Took effect, so keep them Discard the rest

Condition G S

Means that S respects “real-time order” of G

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A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 B q: enq( 6)

Example

time B.q.enq(4)

  • A. q.enq(3)

B.q.deq(4)

  • B. q.enq(6)
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Example

Complete this pending invocation

time B.q.enq(4) B.q.deq(4)

  • B. q.enq(6)

A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 B q: enq( 6)

  • A. q.enq(3)
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Example

Complete this pending invocation

time B.q.enq(4) B.q.deq(4)

  • B. q.enq(6)

A.q.enq(3)

A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 B q: enq( 6) A q: voi d

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Example

time B.q.enq(4) B.q.deq(4)

  • B. q.enq(6)

A.q.enq(3)

A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 B q: enq( 6) A q: voi d discard this one

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Example

time B.q.enq(4) B.q.deq(4) A.q.enq(3)

A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 A q: voi d discard this one

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A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 A q: voi d

Example

time B.q.enq(4) B.q.deq(4) A.q.enq(3)

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A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 A q: voi d

Example

time

B q. enq( 4) B q: voi d A q. enq( 3) A q: voi d B q. deq( ) B q: 4

B.q.enq(4) B.q.deq(4) A.q.enq(3)

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B.q.enq(4) B.q.deq(4) A.q.enq(3)

A q. enq( 3) B q. enq( 4) B q: voi d B q. deq( ) B q: 4 A q: voi d

Example

time

B q. enq( 4) B q: voi d A q. enq( 3) A q: voi d B q. deq( ) B q: 4 Equivalent sequential history

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Composability Theorem

History H is linearizable if and only if

For every object x H|x is linearizable

We care about objects only!

(Materialism?)

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Why Does Composability Matter?

Modularity Can prove linearizability of objects in isolation Can compose independently-implemented objects

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Reasoning About Linearizability: Locking

publ i c O bj ect deq( ) t hr ows Em pt yExcept i on { l ock. l ock( ) ; t r y { i f ( t ai l == head) t hr ow new Em pt yExcept i on( ) ; O bj ect x = i t em s[ head % i t em

  • s. l engt h] ;

head++; r et ur n x; } f i nal l y { l ock. unl ock( ) ; } }

1 capaci t y- 1 2

head tail

y z

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Reasoning About Linearizability: Locking

publ i c O bj ect deq( ) t hr ows Em pt yExcept i on { l ock. l ock( ) ; t r y { i f ( t ai l == head) t hr ow new Em pt yExcept i on( ) ; O bj ect x = i t em s[ head % i t em

  • s. l engt h] ;

head++; r et ur n x; } f i nal l y { l ock. unl ock( ) ; } }

Linearization points are when locks are released

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More Reasoning: Lock-free

publ i c cl ass LockFr eeQ ueue { vol at i l e i nt head = 0, t ai l = 0; i t em s = new O bj ect [ capaci t y] ; publ i c voi d enq( O bj ect x) { whi l e ( t ai l - head == capaci t y) ; / / busy- wai t i t em s[ t ai l % capaci t y] = x; t ai l ++; } publ i c I t em deq( ) { whi l e ( t ai l == head) ; / / busy- wai t O bj ect i t em = i t em s[ head % capaci t y] ; head++; r et ur n i t em ; } }

1 capaci t y- 1 2

head tail

y z

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publ i c cl ass LockFr eeQ ueue { vol at i l e i nt head = 0, t ai l = 0; i t em s = new O bj ect [ capaci t y] ; publ i c voi d enq( O bj ect x) { whi l e ( t ai l - head == capaci t y) ; / / busy- wai t i t em s[ t ai l % capaci t y] = x; t ai l ++; } publ i c I t em deq( ) { whi l e ( t ai l == head) ; / / busy- wai t O bj ect i t em = i t em s[ head % capaci t y] ; head++; r et ur n i t em ; } }

Linearization order is

  • rder head and tail

fields modified

More Reasoning

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More Reasoning

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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

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Linearizability: Summary

Powerful specification tool for shared objects Allows us to capture the notion of objects being “atomic” Don’t leave home without it

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Source