I NVARIANT S AFETY FOR D ISTRIBUTED A PPLICATIONS Sreeja Nair - - PowerPoint PPT Presentation

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I NVARIANT S AFETY FOR D ISTRIBUTED A PPLICATIONS Sreeja Nair - - PowerPoint PPT Presentation

I NVARIANT S AFETY FOR D ISTRIBUTED A PPLICATIONS Sreeja Nair Gustavo Petri Marc Shapiro S TATEFUL D ISTRIBUTED S YSTEMS W E WANT : } Scalability Replicated State Availability Programmability Strong Consistency S


slide-1
SLIDE 1

INVARIANT SAFETY FOR
 DISTRIBUTED APPLICATIONS

Sreeja Nair Gustavo Petri Marc Shapiro

slide-2
SLIDE 2

STATEFUL DISTRIBUTED SYSTEMS

WE WANT:

  • Scalability
  • Availability
  • Programmability


≈ Strong Consistency

}

Replicated State

slide-3
SLIDE 3

STATEFUL DISTRIBUTED SYSTEMS

WE WANT:

  • Scalability
  • Availability
  • Programmability


≈ Strong Consistency

}

Replicated State

CAP* Theorem

* Consistency, Availability, Partition Tolerance

[Gilbert&Lynch’02]

slide-4
SLIDE 4

STATEFUL DISTRIBUTED SYSTEMS

WE WANT:

  • Scalability
  • Availability
  • Programmability


≈ Strong Consistency

}

Replicated State

CAP* Theorem

WE GET:

  • Availability

Availability

* Consistency, Availability, Partition Tolerance

[Gilbert&Lynch’02]

slide-5
SLIDE 5

STATEFUL DISTRIBUTED SYSTEMS

WE WANT:

  • Scalability
  • Availability
  • Programmability


≈ Strong Consistency OR:

  • Programmability

}

Replicated State

CAP* Theorem

WE GET:

  • Availability

Availability Consistency

* Consistency, Availability, Partition Tolerance

[Gilbert&Lynch’02]

slide-6
SLIDE 6

DISTRIBUTED STATE (CRDTS)

slide-7
SLIDE 7

DISTRIBUTED STATE (CRDTS)

CONFLICT-FREE REPLICATED DATA TYPES

  • Availability
  • Network Partition Tolerance
  • (Strong) Eventual Consistency
  • Distributed Data Type Abstractions
  • Deterministic Conflict Resolution


⟹ Eventual Convergence

slide-8
SLIDE 8

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

slide-9
SLIDE 9

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

slide-10
SLIDE 10

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

slide-11
SLIDE 11

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M:

slide-12
SLIDE 12

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M:

slide-13
SLIDE 13

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M:

slide-14
SLIDE 14

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M:

slide-15
SLIDE 15

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M: G:

slide-16
SLIDE 16

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M: G: M:

slide-17
SLIDE 17

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M: G: G: M: M:

slide-18
SLIDE 18

REPLICATED ONLINE AUCTION

Sreeja Marc Gustavo

M: M: M: G: G: M: M:

slide-19
SLIDE 19

SAFETY FOR DISTRIBUTED APPLICATIONS

slide-20
SLIDE 20

SAFETY FOR DISTRIBUTED APPLICATIONS

  • High Availability
  • Strong Consistency

slide-21
SLIDE 21

SAFETY FOR DISTRIBUTED APPLICATIONS

  • High Availability
  • Strong Consistency

  • High Availability
  • Eventual Consistency
slide-22
SLIDE 22

SAFETY FOR DISTRIBUTED APPLICATIONS

  • High Availability
  • Strong Consistency

  • High Availability
  • Eventual Consistency

  • High Availability
  • Eventual Consistency
  • Data Safety
slide-23
SLIDE 23

SAFETY FOR DISTRIBUTED APPLICATIONS

  • High Availability
  • Strong Consistency

  • High Availability
  • Eventual Consistency

  • High Availability
  • Eventual Consistency
  • Data Safety

PROOF RULE FOR STATEFUL DISTRIBUTED APPLICATION SAFETY

  • Modular
  • Automated verification
slide-24
SLIDE 24

STATE-BASED CRDTS

  • State-based CRDTs
  • Propagation of states (instead of operations)
slide-25
SLIDE 25

STATE-BASED CRDTS

  • State-based CRDTs
  • Propagation of states (instead of operations)
  • rigin replica

client replica replica

u? u v?

slide-26
SLIDE 26

STATE-BASED CRDTS

  • State-based CRDTs
  • Propagation of states (instead of operations)
  • States are merged on receive
  • Convergence: concurrent conflicting operations result

deterministically on a unique state

  • No delivery assumptions
slide-27
SLIDE 27

STATE-BASED CRDTS

… … … …

slide-28
SLIDE 28

STATE-BASED CRDTS

  • State is a (join semi-)Lattice

… … … …

slide-29
SLIDE 29

STATE-BASED CRDTS

  • State is a (join semi-)Lattice
  • Effectors send the state at the origin
  • Lazy update propagation

… … … …

slide-30
SLIDE 30

STATE-BASED CRDTS

  • State is a (join semi-)Lattice
  • Effectors send the state at the origin
  • Lazy update propagation
  • Each operation is an inflation in the lattice

… … … …

slide-31
SLIDE 31

STATE-BASED CRDTS

  • State is a (join semi-)Lattice
  • Effectors send the state at the origin
  • Lazy update propagation
  • Each operation is an inflation in the lattice
  • merge function joins the state of two replicas
  • Join of the lattice

… … … …

slide-32
SLIDE 32

INVARIANTS FOR SB-CRDTS

  • CRDT (lattice) constraints

… … … …

slide-33
SLIDE 33

INVARIANTS FOR SB-CRDTS

  • CRDT (lattice) constraints
  • Operations are inflations


… … … …

σ v σ0

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∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒

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slide-34
SLIDE 34

INVARIANTS FOR SB-CRDTS

  • CRDT (lattice) constraints
  • Operations are inflations

  • merge is join (LUB)

… … … …

∀σ, σ0, merge(σ, σ0) = σ00 ⇒ σ00 = LUBv(σ, σ0)

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σ v σ0

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∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒

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sha1_base64="uvB8XQ+ebGjiQJe0uZg0V3WYgoQ=">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</latexit><latexit 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slide-35
SLIDE 35

AUCTION STATE EVOLUTION

slide-36
SLIDE 36

AUCTION STATE EVOLUTION

AUCTION STATUS: OPEN Closed

slide-37
SLIDE 37

AUCTION STATE EVOLUTION

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

slide-38
SLIDE 38

AUCTION STATE EVOLUTION

100

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS:

slide-39
SLIDE 39

AUCTION STATE EVOLUTION

100

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS: AUCTION OPERATIONS:

  • Open auction
  • Place bid
  • Close auction
  • Select winner
slide-40
SLIDE 40

AUCTION STATE EVOLUTION

100

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS: AUCTION OPERATIONS:

  • Open auction
  • Place bid
  • pen auction
  • Close auction
  • Select winner
slide-41
SLIDE 41

AUCTION STATE EVOLUTION

100 100 110

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS: place bid AUCTION OPERATIONS:

  • Open auction
  • Place bid
  • pen auction
  • Close auction
  • Select winner
slide-42
SLIDE 42

AUCTION STATE EVOLUTION

100 100 110 100 110

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS: place bid close auction AUCTION OPERATIONS:

  • Open auction
  • Place bid
  • pen auction
  • Close auction
  • Select winner
slide-43
SLIDE 43

AUCTION STATE EVOLUTION

100 100 110 110 100 100 110

AUCTION STATUS: OPEN Closed AUCTION RESULT: NO WINNER WINNER MARKER

100 110

AUCTION BIDDERS: place bid close auction select winner AUCTION OPERATIONS:

  • Open auction
  • Place bid
  • pen auction
  • Close auction
  • Select winner
slide-44
SLIDE 44

AUCTION STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

slide-45
SLIDE 45

AUCTION STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

slide-46
SLIDE 46

AUCTION STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

slide-47
SLIDE 47

AUCTION STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

slide-48
SLIDE 48

AUCTION STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100
slide-49
SLIDE 49

AUCTION STATE EVOLUTION

110

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100
slide-50
SLIDE 50

AUCTION STATE EVOLUTION

110

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100
slide-51
SLIDE 51

AUCTION STATE EVOLUTION

110

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100 110 100
slide-52
SLIDE 52

AUCTION STATE EVOLUTION

110

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100 110 100 100
slide-53
SLIDE 53

AUCTION STATE EVOLUTION

110 100

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100 110 100 100
slide-54
SLIDE 54

AUCTION STATE EVOLUTION

110 100

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100 110 100 100 110 100
slide-55
SLIDE 55

AUCTION STATE EVOLUTION

110 100

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100 100 110 100 100 110 100
slide-56
SLIDE 56

CONCURRENCY CONTROL

110 100 100
slide-57
SLIDE 57

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

slide-58
SLIDE 58

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
slide-59
SLIDE 59

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
slide-60
SLIDE 60

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
slide-61
SLIDE 61

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental
slide-62
SLIDE 62

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental

How do we enforce invariants?

slide-63
SLIDE 63

CONCURRENCY CONTROL

110 100 100

INVARIANTS UNDER CONCURRENCY

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental

How do we verify these invariants? How do we enforce invariants?

slide-64
SLIDE 64

INVARIANTS FOR SB-CRDTS

  • Invariant constraints

… … … …

slide-65
SLIDE 65

INVARIANTS FOR SB-CRDTS

  • Invariant constraints
  • Operations preserve the invariant


… … … …

∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒ σ0 ✏ Inv

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slide-66
SLIDE 66

INVARIANTS FOR SB-CRDTS

  • Invariant constraints
  • Operations preserve the invariant

  • merge preserves the invariant

… … … …

∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒ σ0 ✏ Inv

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∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

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σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">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</latexit>
slide-67
SLIDE 67

INVARIANTS FOR SB-CRDTS

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

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∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

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σ ✏ Inv ∧ σ00 ✏ Inv ∧

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slide-68
SLIDE 68

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

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( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">AGI3icjVRNb9NAEHVLAyV8tXDksiVCLVJbxaUSXCqiwgEkDkEibaU4qtbrcbLKetfaHTekW/+c/pyQuXAgf/COgkVcRJgpdjmfmzc5ONkwFN1iv/1havrVSuX1n9W713v0HDx+trT8+MirTDFpMCaVPQmpAcAkt5CjgJNVAk1DAcdh/W8SPz0AbruRnHKbQSWhX8pgzis51uvYmMLybUBKQ4OwdNT35vLMPQcQdYGMo5ubC+JVcrpWq+/WR4vMGv7EqHmT1TxdX/kaRIplCUhkghrT9uspdizVyJmAvBpkBlLK+rQLbWdKmoDp2NFOc/LceSISK+1+EsnI+yfD0sSYRI6ZEKxZ8qxwjkv1s4wft2xXKYZgmRjoTgTBUp2kYiroGhGDqDMs1drYT1qKYMXOnVIrcabEbPbUXazCheqijaW9xFx2zbT3y3i7Uz7k/fMdVqKnBnekilzbqoEwaABtlRAwjKbApeERHNjGRW4vGvm2Cwgqu5njHTSH2FOycLnh4czgUMBECuFUiEYfg4BYkwTLoYFyPTUwDhNMAcxFQYKH1NJcYT6qhg29UAcmO/vhEKV6CTLOqSMCiwVEY2UGluA5Am01D0yQajk4it8+f5NSN2XzsBxcog5MC3fY7badg85C3qmt+TNk94dhSkb/4jc1LOArg/+TwDYdMJ+fw0CykC5E6Aat7842JzWfBFrfJfTuDLiUtdCNUAOZaDIwry9N7/Oj63DfKvmb5Pa3otZgSlWJIwbk5mFAyOFNq/2ztfyB1V9VCuYFIx0lUOp8/SP9SqowFZ+ViJ8Sb6Mwoge5CuYkT0uiNaEeYvKC6+8v3azxtHerv9yd+/Tfq1xOLkJV72n3jNvy/O9V17De+81vZbHvEvyrv2vlcuK1eVb5XrMXR5acJ54k2tys9f18M2Vg=</latexit>
slide-69
SLIDE 69

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

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( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">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</latexit> ✘
slide-70
SLIDE 70

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

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( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

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∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

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σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">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</latexit>

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧ reachableσi(σ) ∧ reachableσi(σ00) ∧

<latexit sha1_base64="qlTzsUOuTJpWtUFU5aQdFuG20qw=">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</latexit> ✘
slide-71
SLIDE 71

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

<latexit sha1_base64="r7e1DZjO1EjcRJqwcoJkNgq3pZg=">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</latexit>

( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">AGI3icjVRNb9NAEHVLAyV8tXDksiVCLVJbxaUSXCqiwgEkDkEibaU4qtbrcbLKetfaHTekW/+c/pyQuXAgf/COgkVcRJgpdjmfmzc5ONkwFN1iv/1havrVSuX1n9W713v0HDx+trT8+MirTDFpMCaVPQmpAcAkt5CjgJNVAk1DAcdh/W8SPz0AbruRnHKbQSWhX8pgzis51uvYmMLybUBKQ4OwdNT35vLMPQcQdYGMo5ubC+JVcrpWq+/WR4vMGv7EqHmT1TxdX/kaRIplCUhkghrT9uspdizVyJmAvBpkBlLK+rQLbWdKmoDp2NFOc/LceSISK+1+EsnI+yfD0sSYRI6ZEKxZ8qxwjkv1s4wft2xXKYZgmRjoTgTBUp2kYiroGhGDqDMs1drYT1qKYMXOnVIrcabEbPbUXazCheqijaW9xFx2zbT3y3i7Uz7k/fMdVqKnBnekilzbqoEwaABtlRAwjKbApeERHNjGRW4vGvm2Cwgqu5njHTSH2FOycLnh4czgUMBECuFUiEYfg4BYkwTLoYFyPTUwDhNMAcxFQYKH1NJcYT6qhg29UAcmO/vhEKV6CTLOqSMCiwVEY2UGluA5Am01D0yQajk4it8+f5NSN2XzsBxcog5MC3fY7badg85C3qmt+TNk94dhSkb/4jc1LOArg/+TwDYdMJ+fw0CykC5E6Aat7842JzWfBFrfJfTuDLiUtdCNUAOZaDIwry9N7/Oj63DfKvmb5Pa3otZgSlWJIwbk5mFAyOFNq/2ztfyB1V9VCuYFIx0lUOp8/SP9SqowFZ+ViJ8Sb6Mwoge5CuYkT0uiNaEeYvKC6+8v3azxtHerv9yd+/Tfq1xOLkJV72n3jNvy/O9V17De+81vZbHvEvyrv2vlcuK1eVb5XrMXR5acJ54k2tys9f18M2Vg=</latexit>

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧ reachableσi(σ) ∧ reachableσi(σ00) ∧

<latexit sha1_base64="qlTzsUOuTJpWtUFU5aQdFuG20qw=">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</latexit> ✘
slide-72
SLIDE 72

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

<latexit sha1_base64="r7e1DZjO1EjcRJqwcoJkNgq3pZg=">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</latexit>

( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">AGi3icjVThbts2EJbzmuzZUvbn/3DzhiSAmlgpQVaFDUQpC3QAvhDUtSwDICijrJRChSIE9xXUbP05fpC/RtelKUYJadbQMnr+7j8cS4UNLhcPitd+v2nR/6P969t/HTz5u/Lp1/8GxM6UVcCSMvZjzB0oqeEIJSr4WFjgeazgJD57U/tPzsE6afTfuChgmvNMy1QKjgSdbn1hLEqN5UpFLHIy/lu29fG9u70QZjOx39hMWnb/lbsYiUhRGJ5MoB5tBOqVUc0gyqGktoY9urIp/C+ZzZBba+ZXIva9LRLfX6NRjuDZvFVo2wNQZBu8an9+98jRIjyhw0CsWdm4TDAqeW5RCQbURlQ4KLs54BhMyNc/BTX3T0Yr9TkjCqDH08ga9J8Mz3PnFnlMkTnHmev6anCdb1Ji+nLqpS5KBC0uhdJSMTSsvh6WSOolqgUZXFhJtTIx45YLpEtcUqlzF/Vp7NJZvMOc24VNltF6lKTO3DL6fK4SxjKs89PRYdeOHyqTUJt24iUQwfo6W7BCV6A1E4mMPIHF5W/OKh2yaG4zkrijcYLnBldQzSkUjhcKBjRwBnUBsHJzxAhpjyXalEHuZmZO9IEN0q5clBjwuT1HbUpmCfWQD9+PnwcayoQJKs69Iwr2O5TnxkispHoF1poe6Tj5qbSD3hVbUcSsO1PvYDObrBeR09CacTUvDV9EbeqR+EK+T2M/kv/tjCDXzj8P8k8GMKrNbncJDfSFcqpkE7o7ut2CBkbX/7tpqIy07XYzFCN1CVcTXZX1/nH0eH1c4g3GWD/SerAnBsSPhaE5WFBw2CpOr9q7XcsjpSbxRbq6KySmWM+fF/9Sqk6VFN1iW+K1d2WUmndxPanZEX0TU9VUev/C7mu3ahzv74XDvfDP54ODw/YlvBs8Cn4LdoIweBEcBO+DcXAUiN5m71nvdW/U3+w/67/qv74MvdVrOQ+DpdV/9x1klPr</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">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</latexit>

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧ reachableσi(σ) ∧ reachableσi(σ00) ∧

<latexit sha1_base64="qlTzsUOuTJpWtUFU5aQdFuG20qw=">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</latexit> ✘

slide-73
SLIDE 73

INVARIANTS FOR SB-CRDTS

110 100 110 200

merge

<latexit sha1_base64="r7e1DZjO1EjcRJqwcoJkNgq3pZg=">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</latexit>

( ) ,

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">AGi3icjVThbts2EJbzmuzZUvbn/3DzhiSAmlgpQVaFDUQpC3QAvhDUtSwDICijrJRChSIE9xXUbP05fpC/RtelKUYJadbQMnr+7j8cS4UNLhcPitd+v2nR/6P969t/HTz5u/Lp1/8GxM6UVcCSMvZjzB0oqeEIJSr4WFjgeazgJD57U/tPzsE6afTfuChgmvNMy1QKjgSdbn1hLEqN5UpFLHIy/lu29fG9u70QZjOx39hMWnb/lbsYiUhRGJ5MoB5tBOqVUc0gyqGktoY9urIp/C+ZzZBba+ZXIva9LRLfX6NRjuDZvFVo2wNQZBu8an9+98jRIjyhw0CsWdm4TDAqeW5RCQbURlQ4KLs54BhMyNc/BTX3T0Yr9TkjCqDH08ga9J8Mz3PnFnlMkTnHmev6anCdb1Ji+nLqpS5KBC0uhdJSMTSsvh6WSOolqgUZXFhJtTIx45YLpEtcUqlzF/Vp7NJZvMOc24VNltF6lKTO3DL6fK4SxjKs89PRYdeOHyqTUJt24iUQwfo6W7BCV6A1E4mMPIHF5W/OKh2yaG4zkrijcYLnBldQzSkUjhcKBjRwBnUBsHJzxAhpjyXalEHuZmZO9IEN0q5clBjwuT1HbUpmCfWQD9+PnwcayoQJKs69Iwr2O5TnxkispHoF1poe6Tj5qbSD3hVbUcSsO1PvYDObrBeR09CacTUvDV9EbeqR+EK+T2M/kv/tjCDXzj8P8k8GMKrNbncJDfSFcqpkE7o7ut2CBkbX/7tpqIy07XYzFCN1CVcTXZX1/nH0eH1c4g3GWD/SerAnBsSPhaE5WFBw2CpOr9q7XcsjpSbxRbq6KySmWM+fF/9Sqk6VFN1iW+K1d2WUmndxPanZEX0TU9VUev/C7mu3ahzv74XDvfDP54ODw/YlvBs8Cn4LdoIweBEcBO+DcXAUiN5m71nvdW/U3+w/67/qv74MvdVrOQ+DpdV/9x1klPr</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">AGi3icjVThbts2EJbzmuzZUvbn/3DzhiSAmlgpQVaFDUQpC3QAvhDUtSwDICijrJRChSIE9xXUbP05fpC/RtelKUYJadbQMnr+7j8cS4UNLhcPitd+v2nR/6P969t/HTz5u/Lp1/8GxM6UVcCSMvZjzB0oqeEIJSr4WFjgeazgJD57U/tPzsE6afTfuChgmvNMy1QKjgSdbn1hLEqN5UpFLHIy/lu29fG9u70QZjOx39hMWnb/lbsYiUhRGJ5MoB5tBOqVUc0gyqGktoY9urIp/C+ZzZBba+ZXIva9LRLfX6NRjuDZvFVo2wNQZBu8an9+98jRIjyhw0CsWdm4TDAqeW5RCQbURlQ4KLs54BhMyNc/BTX3T0Yr9TkjCqDH08ga9J8Mz3PnFnlMkTnHmev6anCdb1Ji+nLqpS5KBC0uhdJSMTSsvh6WSOolqgUZXFhJtTIx45YLpEtcUqlzF/Vp7NJZvMOc24VNltF6lKTO3DL6fK4SxjKs89PRYdeOHyqTUJt24iUQwfo6W7BCV6A1E4mMPIHF5W/OKh2yaG4zkrijcYLnBldQzSkUjhcKBjRwBnUBsHJzxAhpjyXalEHuZmZO9IEN0q5clBjwuT1HbUpmCfWQD9+PnwcayoQJKs69Iwr2O5TnxkispHoF1poe6Tj5qbSD3hVbUcSsO1PvYDObrBeR09CacTUvDV9EbeqR+EK+T2M/kv/tjCDXzj8P8k8GMKrNbncJDfSFcqpkE7o7ut2CBkbX/7tpqIy07XYzFCN1CVcTXZX1/nH0eH1c4g3GWD/SerAnBsSPhaE5WFBw2CpOr9q7XcsjpSbxRbq6KySmWM+fF/9Sqk6VFN1iW+K1d2WUmndxPanZEX0TU9VUev/C7mu3ahzv74XDvfDP54ODw/YlvBs8Cn4LdoIweBEcBO+DcXAUiN5m71nvdW/U3+w/67/qv74MvdVrOQ+DpdV/9x1klPr</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧

<latexit sha1_base64="HrTaiNF+2Oe9gXtFzsat/PSANAk=">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</latexit>

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">AGi3icjVTdbhM7EN7wkwOFcgpcmNOhFqkUmULEgRUfEjnSNxERAFpGxUeb2zG6te2XPNgSz8PL8AK8BY/A7CatTjYp51iKPlmvnG41nHhZIO+/0fnQsXL13u/nHl6sa165s3/ty6euDM6UVcCiMvZTzB0oqeEQJSr4VFjgeazgY3z8svZ/PAHrpNHvcVbAOeZlqkUHAk62vrGWJQay5WKWORklvPdxb59ZmzvRhuM7bTc2/dZdPKuwmLSFEYnYyiHGwG6ZhSTSHJoKYtsDXswalN4e9kNkFurZmegixapKd6pOjrV5/r98stmqEC6P3/GfQrOHRzUvfo8SIMgeNQnHnRmG/wLHnFqVQUG1EpYOCi2OewYhMzXNwY90tGL3CEkYNYZ+GlmD/pvhe7cLI8pMuc4cW1fDa7zjUpMn4y91EWJoMVcKC0VQ8Pq62GJpF6impHBhZVUKxMTbrlAusQlTp3UZ/GLp3FO8y5ndlkGa1HSerMLaOf58dwlAef3kgWvTC4QNtEmrbRqQcOkBPdwtO8AKkdjKBgT/4WvmvB9UuORTXWUm8wXCGE6NriIZUCoczBQMaOIPaIDj5BSLElOdSzeogNzFTR5rgBilXDmpMmLy+wW1KdhnFkDfdS/GysqkCTrujRM61iuEx+ZovIRaFdaqPvko+YmUk94VS2H0nCtj/2HO3gvI4eheMRKfhqfC7vyPfCFfLiM/kv/tDCOXzj8P8k8EMKrNbncJCfS1cqpkE7prutWC9kbVn/9tpqIy01bXYTFCO1CVcTXaX1/nm8MX1U4v3GW9/furAnBsSXhaE5WFBw2CqPT9q7XcsjpSTxXbqKeRJTrOdPi9+UqlMlRbvYBfHMuzJKzbu4ntTsiL6JqWoqvX9h+7VbNT7s74X9vfDto97Bi/lDGFwJ7gR/BTtBGDwODoK/g2FwGIjOZudh51ln0N3sPuw+7T6bh17oLDi3g6XVf0LxvVGg=</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

σ ✏ Inv ∧ σ00 ✏ Inv ∧ reachableσi(σ) ∧ reachableσi(σ00) ∧

<latexit sha1_base64="qlTzsUOuTJpWtUFU5aQdFuG20qw=">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</latexit>

∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">AGi3icjVThbts2EJbzmuzZUvbn/3DzhiSAmlgpQVaFDUQpC3QAvhDUtSwDICijrJRChSIE9xXUbP05fpC/RtelKUYJadbQMnr+7j8cS4UNLhcPitd+v2nR/6P969t/HTz5u/Lp1/8GxM6UVcCSMvZjzB0oqeEIJSr4WFjgeazgJD57U/tPzsE6afTfuChgmvNMy1QKjgSdbn1hLEqN5UpFLHIy/lu29fG9u70QZjOx39hMWnb/lbsYiUhRGJ5MoB5tBOqVUc0gyqGktoY9urIp/C+ZzZBba+ZXIva9LRLfX6NRjuDZvFVo2wNQZBu8an9+98jRIjyhw0CsWdm4TDAqeW5RCQbURlQ4KLs54BhMyNc/BTX3T0Yr9TkjCqDH08ga9J8Mz3PnFnlMkTnHmev6anCdb1Ji+nLqpS5KBC0uhdJSMTSsvh6WSOolqgUZXFhJtTIx45YLpEtcUqlzF/Vp7NJZvMOc24VNltF6lKTO3DL6fK4SxjKs89PRYdeOHyqTUJt24iUQwfo6W7BCV6A1E4mMPIHF5W/OKh2yaG4zkrijcYLnBldQzSkUjhcKBjRwBnUBsHJzxAhpjyXalEHuZmZO9IEN0q5clBjwuT1HbUpmCfWQD9+PnwcayoQJKs69Iwr2O5TnxkispHoF1poe6Tj5qbSD3hVbUcSsO1PvYDObrBeR09CacTUvDV9EbeqR+EK+T2M/kv/tjCDXzj8P8k8GMKrNbncJDfSFcqpkE7o7ut2CBkbX/7tpqIy07XYzFCN1CVcTXZX1/nH0eH1c4g3GWD/SerAnBsSPhaE5WFBw2CpOr9q7XcsjpSbxRbq6KySmWM+fF/9Sqk6VFN1iW+K1d2WUmndxPanZEX0TU9VUev/C7mu3ahzv74XDvfDP54ODw/YlvBs8Cn4LdoIweBEcBO+DcXAUiN5m71nvdW/U3+w/67/qv74MvdVrOQ+DpdV/9x1klPr</latexit>

Premerge = wp(merge(σ, σ00), Inv)

<latexit sha1_base64="BxFoLBEin136hsqngLpFwLx40cE=">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</latexit>

✘ ✘

slide-74
SLIDE 74

INVARIANTS FOR SB-CRDTS

  • Invariant constraints
  • Operations preserve the invariant


… … … …

∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒ σ0 ✏ Inv

<latexit 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slide-75
SLIDE 75

INVARIANTS FOR SB-CRDTS

  • Invariant constraints
  • Operations preserve the invariant

  • merge preserves the invariant

… … … …

∀ op, σ, σ0, σ ✏ Preop ∧ (σ, σ0) ∈ JopK ⇒ σ0 ✏ Inv

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∀ σ, σ0, σ00, (σ, σ00) ✏ Premerge ∧

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>

∧ merge(σ, σ00) = σ0 ⇒ σ0 ✏ Inv

<latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="hJQuF2O9Szy+oT5XtUz1hFQA=">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</latexit><latexit sha1_base64="3R9a+Cv8wmpCOhFBaiy+JRSPs=">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</latexit>
slide-76
SLIDE 76

… … … …

INVARIANTS FOR SB-CRDTS

  • merge Pre constraints
  • Initial state satisfies merge Pre


Premerge(σi, σi)

<latexit sha1_base64="eY+2poLSCEiPWLCwMh7WPJjeqH4=">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</latexit>
slide-77
SLIDE 77

… … … …

INVARIANTS FOR SB-CRDTS

  • merge Pre constraints
  • Initial state satisfies merge Pre

  • Operations preserve the merge Pre


∀ op, σ, σ0, σ00,   σ ✏ Preop ∧ (σ, σ00) ✏ Premerge ∧ (σ, σ0) ∈ JopK   ⇒ (σ0, σ00) ✏ Premerge

<latexit 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sha1_base64="S/oZhOcI/jU4ETAprLEktBNknzc=">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</latexit><latexit sha1_base64="2u5PqlVTthOZjptSjVw3gl7nKSQ=">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</latexit>

Premerge(σi, σi)

<latexit sha1_base64="eY+2poLSCEiPWLCwMh7WPJjeqH4=">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</latexit>
slide-78
SLIDE 78

CONCURRENCY CONTROL

110 100 100

INVARIANTS TO AVOID CONCURRENCY PROBLEMS

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental
slide-79
SLIDE 79

CONCURRENCY CONTROL

110 100 100

INVARIANTS TO AVOID CONCURRENCY PROBLEMS

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental

100 110

TOKENS FOR CC: TAKEN Released

slide-80
SLIDE 80

CONCURRENCY CONTROL

110 100 100

INVARIANTS TO AVOID CONCURRENCY PROBLEMS

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental

100 110

TOKENS FOR CC: TAKEN Released

100 110

release
 token

slide-81
SLIDE 81

CONCURRENCY CONTROL

110 100 100

INVARIANTS TO AVOID CONCURRENCY PROBLEMS

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental
  • Tokes go from taken ⟼ released (by owner)

100 110

TOKENS FOR CC: TAKEN Released

100 110

release
 token

slide-82
SLIDE 82

CONCURRENCY CONTROL

110 100 100

INVARIANTS TO AVOID CONCURRENCY PROBLEMS

  • Auction cannot be closed while bids are being placed
  • Status can only go from: inital ⟼ open ⟼ closed
  • Winner is highest bid when auction is closed
  • Bids are strictly incremental
  • Tokes go from taken ⟼ released (by owner)
  • Close when all tokens are released

100 110

TOKENS FOR CC: TAKEN Released

100 110

release
 token

slide-83
SLIDE 83

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS: TOKENS:

slide-84
SLIDE 84

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

TOKENS:

slide-85
SLIDE 85

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

100

TOKENS:

slide-86
SLIDE 86

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 100

TOKENS:

slide-87
SLIDE 87

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 100 100

✘ ✘

TOKENS:

slide-88
SLIDE 88

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 100 100

✘ ✘

100

✘ ✘

TOKENS:

slide-89
SLIDE 89

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 100 100

100 110

✘ ✘

100

✘ ✘

TOKENS:

slide-90
SLIDE 90

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 110

100 100

100 110

✘ ✘

100

✘ ✘

TOKENS:

slide-91
SLIDE 91

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 110

100 100

100 110

✘ ✘

100

✘ ✘

100 110

✘ ✘ ✘

TOKENS:

slide-92
SLIDE 92

AUCTION + CC STATE EVOLUTION

AUCTION STATUS: AUCTION RESULT:

100

AUCTION BIDDERS:

110 110

100 100

100 110

✘ ✘

100

✘ ✘

100 110

✘ ✘ ✘

100 110

✘ ✘ ✘

100 110

✘ ✘ ✘

TOKENS:

slide-93
SLIDE 93

AUCTION IN BOOGIE

slide-94
SLIDE 94

TOOL SUPPORT

  • Inputs:
  • Operations
  • Ordering relation ⩽ for semi-lattice
  • Invariant Inv
  • Derive Premerge from Inv
  • Global invariants: Inv and Premerge
  • Check semi-lattice: convergence
  • Proofs are local to each operation
  • Boogie for (sequential) verification
  • https://github.com/sreeja/soteria_tool
slide-95
SLIDE 95

CONCLUSION

  • Modular verification of State-based CRDT applications
  • SOTERIA: Tool support based on Boogie
  • WIP: Concurrency Control synthesis (recommendations)
slide-96
SLIDE 96

OPERATION-BASED CRDTS

  • Operation-based CRDTs
  • Each operation is delivered to each replica
slide-97
SLIDE 97

OPERATION-BASED CRDTS

  • Operation-based CRDTs
  • Each operation is delivered to each replica
  • rigin replica

client replica replica

u u u u u u u u v v v v

slide-98
SLIDE 98

OPERATION-BASED CRDTS

  • Operation-based CRDTs
  • Each operation is delivered to each replica
  • Invariant Checking (CISE)
  • Requires causal delivery