background Technical Development Case-Study Conclusion

Separation-Based Reasoning for DeterministicChannel-Passing Concurrent Programs

Aimee Borda

December 18, 2013

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background Technical Development Case-Study Conclusion

Table of Content

backgroundCompositional ReasoningSeparation Logic

Technical DevelopmentResource ReuseResearch Time-Line

Case-StudyOverviewResource Reuse Patterns

ConclusionFuture Work & ContributionsAppendix

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Compositional (localized) Proof Systems

A B

C D A

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Compositional (localized) Proof Systems

A B

C D A

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Compositional (localized) Proof Systems

A B

C D A

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Localized Reasoning

sum(l) = sum(l1) sum(l2)+

l = l1 · l2

r1 r2

r1 ∩ r2 = ∅

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background Technical Development Case-Study Conclusion

Localized Reasoning

sum(l) = sum(l1) sum(l2)+

l = l1 · l2

r1 r2

r1 ∩ r2 = ∅

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Localized Reasoning

sum(l) = sum(l1) sum(l2)+

l = l1 · l2

r1 r2

r1 ∩ r2 = ∅

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background Technical Development Case-Study Conclusion

Separation Logic [Rey02]

A

r1

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Separation Logic [Rey02]

A

r1

B

r2⊥

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Resource Transfer [O’H07]

P1

P2 {r1, r2, r3}

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Separation-Based Reasoning for Message PassingPrograms [FRS11]

c ↑c ↓

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Communication Channels as Synchronization Mechanism

c!

P1

c?

P2

{r1, r2, r3}

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Channel Reuse - Dynamic Resource TransferP1

P2

P3

{r1, r2, r3}

{r4, r5}

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Multiple-Sender and Single Receiver Pattern

SUM

65

DIFF

65

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Multiple-Sender and Single Receiver Pattern

SUM

65

= 11 DIFF

65

= 1 or -1

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Semantic Satisfaction

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

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Technical Development Timeline

FRS11 Channel Reuse Multiple-Sender GoalSingle- Receiver

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Proof of Soundness

(P is deterministic)

(Data Analysis) (Behavioral Analysis)

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Sorting Networks [Knu98]

7

9

5

3

9

7

5

7

3

9

9

7

3

5

9

7

5

3

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background Technical Development Case-Study Conclusion

Sorting Networks [Knu98]

7

9

5

3

9

7

5

7

3

9

9

7

3

5

9

7

5

3

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background Technical Development Case-Study Conclusion

Our Implementation of SNs

c2

c1

c4

c3c1?(x1).c2?(x2).

if x1≤x2 then(c3!〈x1〉‖c4!〈x2〉

)else

(c3!〈x2〉‖c4!〈x1〉

)

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Regular Pattern in SNs

A B

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Naıve Solution for SNs

c8

c7

c6

c5

c4

c3

c2

c1

d7

d5

d3

d1

d8

d6

d4

d2

e8

e7

e6

e5

e4

e3

e2

e1

f5

f1

f6

f2

f7

f3

f8

f4

g7

g5

g3

g1

g8

g6

g4

g2

h8

h7

h6

h5

h4

h3

h2

h1

i8

i7

i6

i5

i4

i3

i2

i1

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Horizontal Reuse in SNs

c1 A d1 c1 B e1 d1 c1 C

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Vertical Reuse in SNs

5

6

5

6c2

c1

c4

c3

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Vertical Reuse in SNs

5

6

6

5c2

c1

c4

c3

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Vertical Reuse in SNs

c1

c1

c4

c3

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background Technical Development Case-Study Conclusion

Contributions

• Separation-Based Logic for Stable Process for the pre- andpostconditions

• Separation-based Proof System for Message-Passing,Deterministic and Terminating Programs

• Proof of Soundness of Proof System

• Message-passing Implementation of Sorting Network resortingto resource reuse

• Proof of Correctness for the Implementation

• Preliminary Design of Second Proof System where channelscan be shared

• An innovative Proof Technique for proving Soundness

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background Technical Development Case-Study Conclusion

Future Work

• More Resource Reuse Pattern

• Enhanced Languages• Name-Passing Channels• Scoping Construct

• Logical Framework Improvement

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background Technical Development Case-Study Conclusion

BibliographyAdrian Francalanza, Julian Rathke, and Vladimiro Sassone.

Permission-based separation logic for message-passing concurrency.

Logical Methods in Computer Science, 7(3), 2011.

Donald E. Knuth.

The art of computer programming, volume 3: (2nd ed.) sorting andsearching.

Addison Wesley Longman Publishing Co., Inc., Redwood City, CA,USA, 1998.

Peter W. O’Hearn.

Resources, concurrency, and local reasoning.

Theor. Comput. Sci., 375(1-3):271–307, 2007.

John C. Reynolds.

Separation logic: A logic for shared mutable data structures.

In LICS, pages 55–74, 2002.22 / 23

background Technical Development Case-Study Conclusion

Conclusion Remarks

Questions ?

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The lPar Rule

P1

P2

Q1

Q2

R1

R2

T

P1 ‖P2

Q1

Q2

R1

R2

Γi ; Γo\Γ; b ` {ϕ1} P1 {ϕ3 ? ψ1} : ρ1 dom(Γ) ⊆ fn(ϕ3)

Γi\Γ ; Γo ; b ` {ϕ2 ? ϕ3} P2 {ψ2} : ρ2 ϕ2 ⊥ ϕ3 ψ1 ⊥ ψ2

Γi ; Γo ; b ` {ϕ1 ? ϕ2} P1 ‖P2 {ψ1 ? ψ2} : ρ1 ] ρ2

The lPar Rule

P1

P2

ϕ1

ϕ2

ψ1

ψ2

ϕ3

ρ1

ρ2

P1 ‖P2

Q1

Q2

R1

R2

Γi ; Γo\Γ; b ` {ϕ1} P1 {ϕ3 ? ψ1} : ρ1 dom(Γ) ⊆ fn(ϕ3)

Γi\Γ ; Γo ; b ` {ϕ2 ? ϕ3} P2 {ψ2} : ρ2 ϕ2 ⊥ ϕ3 ψ1 ⊥ ψ2

Γi ; Γo ; b ` {ϕ1 ? ϕ2} P1 ‖P2 {ψ1 ? ψ2} : ρ1 ] ρ2

Vertical Reuse - SN

B2

B1

Multiple Sender and Single Receiver Checklist

c!〈5〉 ‖ c!〈6〉 ‖ c?(x).(c?(y).d!〈x + y〉)

• Permissions Analysis• Frozen Permissions• Permission Bags

• Data Analysis• Number of I/O operations• Operation performed on the Data• Frozen Data

Multiple Sender and Single Receiver Checklist

c!〈7〉 ‖ c!〈5〉 ‖ c!〈6〉 ‖ c?(x).(c?(y).d!〈x + y〉)

• Permissions Analysis• Frozen Permissions• Permission Bags

• Data Analysis• Number of I/O operations• Operation performed on the Data• Frozen Data

Multiple Sender and Single Receiver Checklist

c!〈5〉 ‖ c!〈6〉 ‖ c?(x).(c?(y).d!〈x − y〉)

• Permissions Analysis• Frozen Permissions• Permission Bags

• Data Analysis• Number of I/O operations• Operation performed on the Data• Frozen Data

Multiple Sender and Single Receiver Checklist

c!〈5〉 ‖ c!〈6〉 ‖ c?(x).(d!〈x〉‖c?(y).d!〈x + y + y〉)

• Permissions Analysis• Frozen Permissions• Permission Bags

• Data Analysis• Number of I/O operations• Operation performed on the Data• Frozen Data

The lNil Rule

lNilfn(ϕ) ⊆ dom(Γi ∩ Γo)

Γi ; Γo ; b ` {ϕ} nil {ϕ} : ρ

{c〈5〉} nil‖c?(x).c!〈e〉 {c〈5〉}

Nested Permission Environment Update

Γi ; Γo\Γ; b ` {ϕ1} P1 {ϕ3 ? ψ1} : ρ1 dom(Γ) ⊆ fn(ϕ3)

Γi\Γ ; Γo ; b ` {ϕ2 ? ϕ3} P2 {ψ2} : ρ2 ϕ2 ⊥ ϕ3 ψ1 ⊥ ψ2

Γi ; Γo ; b ` {ϕ1 ? ϕ2} P1 ‖P2 {ψ1 ? ψ2} : ρ1 ] ρ2

{c〈5〉} c?(x).(c!‖d!)‖c?(x).d?(y).c!〈5〉 {c〈5〉}

Changes from [FRS11]

• Logical Formula Satisfaction

• Proof of Soundness - from 2 tier to 1 tier

• Removed the Confined Processes Semantics – permissiondescribe the sequent’s footprint rather then the process’s

Sequent Definition

Γi ; Γo ; b

` {ϕ} P {ψ}

: ρ

P,Q , nil | c?(x).P | c!〈e〉 | P ‖ Q | if b thenP else Q | f (~x)

ϕ,ψ , emp | blk(c) | c〈e〉 | ϕ ? ψ

Sequent Definition

Γi ; Γo ; b

` {ϕ} P {ψ} : ρ

E.g., {c ↑, d ↓}

Sequent Definition

Γi ; Γo ;

b ` {ϕ} P {ψ} : ρ

E.g., x = y + 1 ` {c〈x〉} P {c〈y〉}

Sequent Definition

Γi ; Γo ; b ` {ϕ} P {ψ} : ρ

E.g.,c : {c ↑, d ↓}

E.g.,Γi = c : {c ↑, d ↓}Γo = c : {c ↑, e ↑}

Logical Formula Satisfaction

Γ , P , µ � emp iff P ≡ nil

Γ , P , µ � c〈e〉 iff P ≡ c!〈e′〉 and e ⇓ v , e′ ⇓ v and Γ(c) ⊆ µ

Γ , P , µ � ϕ1 ? ϕ2 iff P ≡ P1 ‖P2 and Γ , P1 , µ1 � ϕ1 and

Γ , P2 , µ2 � ϕ2 and µ = µ1 ] µ2

Γ , P , µ � blk(c) iff P ≡ c?(x).P ′ and c ∈ dom(Γ) and c ↓ µ

Semantic Definition

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

∀σ,Q, µ. Γin,Qσ, µ � ϕσ and ρ ⊥ µ and bσ ⇓ tt

implies (P ‖Q)σ ⇓ Rσ and Γout ,Rσ, µ ] ρ � ψσ

Semantic Definition

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

∀σ,Q, µ. Γin,Qσ, µ � ϕσ and ρ ⊥ µ and bσ ⇓ tt

implies (P ‖Q)σ ⇓ Rσ and Γout ,Rσ, µ ] ρ � ψσ

Semantic Definition

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

∀σ,Q, µ. Γin,Qσ, µ � ϕσ and ρ ⊥ µ and bσ ⇓ tt

implies (P ‖Q)σ ⇓ Rσ and Γout ,Rσ, µ ] ρ � ψσ

Semantic Definition

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

∀σ,Q, µ. Γin,Qσ, µ � ϕσ and ρ ⊥ µ and bσ ⇓ tt

implies (P ‖Q)σ ⇓ Rσ and Γout ,Rσ, µ ] ρ � ψσ

Semantic Definition

Γin; Γout ; b ` {ϕ} P {ψ} : ρ

implies

Γin; Γout ; b � {ϕ} P {ψ} : ρ

∀σ,Q, µ. Γin,Qσ, µ � ϕσ and ρ ⊥ µ and bσ ⇓ tt

implies (P ‖Q)σ ⇓ Rσ and Γout ,Rσ, µ ] ρ � ψσ

Race Conditions in SNs

Race Conditions in SNs

7

6

5

6

7

5

7

6

5

7

5

6

Deadlocks

P1

P2

emp

emp

emp

emp

c〈x〉

d〈x〉

c?(x).d!〈x〉 ‖ d?(y).c!〈y〉