[PPT] - The Ramify Rule of Separation Logic Compositional Reasoning for PowerPoint Presentation

SLIDE 1

The Ramify Rule of Separation Logic

Compositional Reasoning for Sharing Jules Villard1

Joint work with Aquinas Hobor2

1University College London 2National University of Singapore

SLIDE 2

Programs with Sharing in the Wild

Graphs Acyclic graphs (DAGs) Overlaid data structures (threaded tree)

Sharing ④

SLIDE 3

Programs with Sharing

✌ Everywhere ✌ Many variations over a few core principles (traversal, marking, copying, . . . ) ✌ Short programs, intricate reasoning ✌ Lots of pointer swinging (tree rotation, Schorr-Waite, . . . ) ✌ Challenge for compositionality

Sharing ④

SLIDE 4

Compositional Formal Verification

✌ Reasoning about a system by reasoning about its parts in isolation ✌ System = Program ✌ Parts = Functions ✌ Reasoning = tP✉ c tQ✉

Sharing ④

SLIDE 5

Compositionality for Pointer Programs

Success: Separation Logic

✌ The frame rule provides compositional reasoning: Frame tP✉ c tQ✉ tP ✎ F✉ c tQ ✎ F✉ t ✉ t ✉ t ✉ t ✉ ✏ ✌ Data structures without sharing (lists, trees, . . . ) ✌ Compositionality based on disjointness of memory accesses An answer to the frame problem: “Describing what does not change as a result of an action”

Sharing ④

SLIDE 6

Compositionality for Pointer Programs

Success: Separation Logic

✌ The frame rule provides compositional reasoning: Frame tP✉ c tQ✉ tP ✎ F✉ c tQ ✎ F✉ t ✉ c t ✉ t ✉ c t ✉ F ✏ ✌ Data structures without sharing (lists, trees, . . . ) ✌ Compositionality based on disjointness of memory accesses An answer to the frame problem: “Describing what does not change as a result of an action”

Sharing ④

SLIDE 7

Framing vs Data Structures with Sharing

Frame

t ✉ c t ✉ t ✉ c t ✉ F ✏ ✌ ✌ ✌

Sharing ④

SLIDE 8

Framing vs Data Structures with Sharing

Frame

t ✉ c t ✉ t ✉ c t ✉ F ✏ ✌ ✌ ✌

Sharing ④

SLIDE 9

Framing vs Data Structures with Sharing

Frame

t ✉ c t ✉ t ✉ c t ✉ F ✏

Previous Attempts

✌ Contrived predicates that circumvent the sharing ✌ Leads to compositional, but ad-hoc reasoning ✌ No general solution

Sharing ④

SLIDE 10

This Talk: Ramification

Ramification Problem in AI: “The ramification problem is concerned with indirect consequences of an action.”

Ramification Rule of Separation Logic

✌ Embrace sharing ✌ Concise, compositional proofs ✌ Expose and resolve global effects of local actions uniformly ✌ All within vanilla separation logic

Sharing ④

SLIDE 11

Separation, Frame, and Trees

SLIDE 12

The Frame Rule of Separation Logic

Frame tP✉ c tQ✉ tP ✎ F✉ c tQ ✎ F✉ ✌ σ1 ✌ σ2 is the disjoint union of σ1 and σ2 ✌ σ ✭ P1 ✎ P2 iff ❉σ1, σ2. σ ✏ σ1 ✌ σ2 & σ1 ✭ P1 & σ2 ✭ P2 P1 ✎ P2 ô P1 P2

Separation, Frame, and Trees ④

SLIDE 13

The Frame Rule of Separation Logic

Frame tP✉ c tQ✉ tP ✎ F✉ c tQ ✎ F✉ P

c

Q ✌ σ1 ✌ σ2 is the disjoint union of σ1 and σ2 ✌ σ ✭ P1 ✎ P2 iff ❉σ1, σ2. σ ✏ σ1 ✌ σ2 & σ1 ✭ P1 & σ2 ✭ P2 P1 ✎ P2 ô P1 P2

Separation, Frame, and Trees ④

SLIDE 14

The Frame Rule of Separation Logic

Frame tP✉ c tQ✉ tP ✎ F✉ c tQ ✎ F✉ P F

c

Q F ✌ σ1 ✌ σ2 is the disjoint union of σ1 and σ2 ✌ σ ✭ P1 ✎ P2 iff ❉σ1, σ2. σ ✏ σ1 ✌ σ2 & σ1 ✭ P1 & σ2 ✭ P2 P1 ✎ P2 ô P1 P2

Separation, Frame, and Trees ④

SLIDE 15

Binary Trees in SL

tree♣x, τq

def

✏ ♣x ✏ 0 ❫ emp ❫ τ ✏ ❍q ❍ ❴ ❉L, R, M, τL, τR. x ÞÑ m : M, ℓ : L, r : R ✎ tree♣L, τLq ✎ tree♣R, τRq ❫ τ ✏ node♣x, M, τL, τRq

Separation, Frame, and Trees ④

SLIDE 16

Marking a Tree

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴tr❡❡✭ str✉❝t ♥♦❞❡ ✯t✮ ④ ✴✴ ttree♣t, τq✉

4

✐❢ ✭✦t ⑤⑤ t✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ t✲❃❧✱ ✯r ❂ t✲❃r❀

6 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ q ✎ ♣r q ❫ ✏ ♣ q ✯

7

♠❛r❦❴tr❡❡✭❧✮❀

8 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r q ❫ ✏ ♣ q ✯

9

♠❛r❦❴tr❡❡✭r✮❀

10 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

11

t✲❃♠ ❂ ✶❀

12 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ ttree♣t, m♣τqq✉

Separation, Frame, and Trees ④

SLIDE 17

Marking a Tree

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴tr❡❡✭ str✉❝t ♥♦❞❡ ✯t✮ ④ ✴✴ ttree♣t, τq✉

4

✐❢ ✭✦t ⑤⑤ t✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ t✲❃❧✱ ✯r ❂ t✲❃r❀

6 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, τℓq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

7

♠❛r❦❴tr❡❡✭❧✮❀

8 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r q ❫ ✏ ♣ q ✯

9

♠❛r❦❴tr❡❡✭r✮❀

10 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

11

t✲❃♠ ❂ ✶❀

12 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ ttree♣t, m♣τqq✉

Separation, Frame, and Trees ④

SLIDE 18

Marking a Tree

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴tr❡❡✭ str✉❝t ♥♦❞❡ ✯t✮ ④ ✴✴ ttree♣t, τq✉

4

✐❢ ✭✦t ⑤⑤ t✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ t✲❃❧✱ ✯r ❂ t✲❃r❀

6 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, τℓq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

7

♠❛r❦❴tr❡❡✭❧✮❀

8 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

9

♠❛r❦❴tr❡❡✭r✮❀

10 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

11

t✲❃♠ ❂ ✶❀

12 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ ttree♣t, m♣τqq✉

Separation, Frame, and Trees ④

SLIDE 19

Marking a Tree

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴tr❡❡✭ str✉❝t ♥♦❞❡ ✯t✮ ④ ✴✴ ttree♣t, τq✉

4

✐❢ ✭✦t ⑤⑤ t✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ t✲❃❧✱ ✯r ❂ t✲❃r❀

6 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, τℓq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

7

♠❛r❦❴tr❡❡✭❧✮❀

8 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

9

♠❛r❦❴tr❡❡✭r✮❀

10 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, m♣τrqq ❫ τ ✏ node♣0, τℓ, τrq ✯

11

t✲❃♠ ❂ ✶❀

12 ✴✴

✧ t ÞÑ ♠ ❧ r r ✎ ♣❧ ♣ qq ✎ ♣r ♣ qq ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ ttree♣t, m♣τqq✉

Separation, Frame, and Trees ④

SLIDE 20

Marking a Tree

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴tr❡❡✭ str✉❝t ♥♦❞❡ ✯t✮ ④ ✴✴ ttree♣t, τq✉

4

✐❢ ✭✦t ⑤⑤ t✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ t✲❃❧✱ ✯r ❂ t✲❃r❀

6 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, τℓq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

7

♠❛r❦❴tr❡❡✭❧✮❀

8 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, τrq ❫ τ ✏ node♣0, τℓ, τrq ✯

9

♠❛r❦❴tr❡❡✭r✮❀

10 ✴✴

✧ t ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, m♣τrqq ❫ τ ✏ node♣0, τℓ, τrq ✯

11

t✲❃♠ ❂ ✶❀

12 ✴✴

✧ t ÞÑ ♠ : 1, ℓ : ❧, r : r ✎ tree♣❧, m♣τℓqq ✎ tree♣r, m♣τrqq ❫ τ ✏ node♣0, τℓ, τrq ✯

13 ⑥ ✴✴ ttree♣t, m♣τqq✉

Separation, Frame, and Trees ④

SLIDE 21

Program Proofs without Sharing

1. Define inductive predicates for recursive data structures
2. Express pre- and post-conditions of the program
3. Apply logic rules to the program

Separation, Frame, and Trees ④

SLIDE 22

Overlap, Ramification, and DAGs

SLIDE 23

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣dag♣ℓ, δℓq ? dag♣r, δrqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ ✎ ✌ ❫ ✌ ❫ ✎ ✌

Overlap, Ramification, and DAGs ④

SLIDE 24

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣dag♣ℓ, δℓq ✎ dag♣r, δrqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ With “✎”: a tree ✌ ❫ ✌ ❫ ✎ ✌

Overlap, Ramification, and DAGs ④

SLIDE 25

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣dag♣ℓ, δℓq❫dag♣r, δrqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ With “✎”: a tree ✌ With “❫”: ✌ ❫ ✎ ✌

Overlap, Ramification, and DAGs ④

SLIDE 26

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣dag♣ℓ, δℓq❫dag♣r, δrqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ With “✎”: a tree ✌ With “❫”: a list ✌ ❫ ✎ ✌

Overlap, Ramification, and DAGs ④

SLIDE 27

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣♣dag♣ℓ, δℓq✎trueq❫♣dag♣r, δrq✎trueqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ With “✎”: a tree ✌ With “❫”: a list ✌ With “❫” and “ ✎ true”: a DAG + anything ✌

Overlap, Ramification, and DAGs ④

SLIDE 28

Describing DAGs in SL?

✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣♣dag♣ℓ, δℓq✎trueq❫♣dag♣r, δrq✎trueqq ❫ δ ✏ node♣x, m, δℓ, δrq ✌ With “✎”: a tree ✌ With “❫”: a list ✌ With “❫” and “ ✎ true”: a DAG + anything ✌ We need something else. . .

Overlap, Ramification, and DAGs ④

SLIDE 29

Overlapping Conjunction

✌ Separating vs Overlapping conjunction: P1 ✎ P2 ô P1 P2 P1 ❨ ✎ P2 ô P1 P2 ✌ σ ✭ P1 ❨ ✎ P2 iff ❉σ1, σ2, σ3. σ ✏ σ1 ✌ σ2 ✌ σ3 & σ1 ✌ σ2 ✭ P1 & σ2 ✌ σ3 ✭ P2

Overlap, Ramification, and DAGs ④

SLIDE 30

Overlapping Conjunction

✌ Separating vs Overlapping conjunction: P1 ✎ P2 ô P1 P2 P1 ❨ ✎ P2 ô P1 P2 ✌ DAG predicate: dag♣x, δq

def

✏ ♣x ✏ 0 ❫ ❡♠♣ ❫ δ ✏ ❍q ❴ ❉ℓ, r, m, δℓ, δr. x ÞÑ ℓ : ℓ, r : r, m : m ✎ ♣dag♣ℓ, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣x, m, δℓ, δrq

Overlap, Ramification, and DAGs ④

SLIDE 31

A Failed Attempt at Framing

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ q ❨ ✎ ♣r qq ❫ ✏ ♣ q ✯

7 ✴✴

t ♣ q ✎ ✉

8

♠❛r❦❴❞❛❣✭❧✮❀

9 ✴✴ 10

♠❛r❦❴❞❛❣✭r✮❀

11 ✴✴ 12

❞✲❃♠ ❂ ✶❀

13 ✴✴ 14 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 32

A Failed Attempt at Framing

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7 ✴✴

t ♣ q ✎ ✉

8

♠❛r❦❴❞❛❣✭❧✮❀

9 ✴✴ 10

♠❛r❦❴❞❛❣✭r✮❀

11 ✴✴ 12

❞✲❃♠ ❂ ✶❀

13 ✴✴ 14 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 33

A Failed Attempt at Framing

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7 ✴✴ tdag♣l, δlq ✎ ???✉ 8

♠❛r❦❴❞❛❣✭❧✮❀

9 ✴✴ 10

♠❛r❦❴❞❛❣✭r✮❀

11 ✴✴ 12

❞✲❃♠ ❂ ✶❀

13 ✴✴ 14 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 34

A Failed Attempt at Framing

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7 ✴✴ tdag♣l, δlq ✎ ???✉ 8

♠❛r❦❴❞❛❣✭❧✮❀

9 ✴✴ stuck! 10

♠❛r❦❴❞❛❣✭r✮❀

11 ✴✴ 12

❞✲❃♠ ❂ ✶❀

13 ✴✴ 14 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 35

The Ramify Rule of Separation Logic

Ramify tP✉ c tQ✉ ramify♣P Q, Rq ✏ R✶ tR✉ c tR✶✉ ✌ ramify♣P Q, Rq ✏ R✶

def

✏ R ✩ P ✎ ♣Q ✁ ✁ ✎ R✶q ✌ σ ✭ P1 ✁ ✁ ✎ P2 iff ❅σ✶ ✭ P1. σ ✌ σ✶ ✭ P2

Overlap, Ramification, and DAGs ④

SLIDE 36

Program Proofs with Sharing

1. Define inductive predicates for recursive data structures
2. Express pre- and post-conditions of the program
3. Apply logic rules to the program
4. Prove ramification conditions

Overlap, Ramification, and DAGs ④

SLIDE 37

Marking a DAG

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ q ❨ ✎ ♣r qq ❫ ✏ ♣ q ✯

7

♠❛r❦❴❞❛❣✭❧✮❀

8 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r

✶qq

❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

9

♠❛r❦❴❞❛❣✭r✮❀

10 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

11

❞✲❃♠ ❂ ✶❀

12 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 38

Marking a DAG

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7

♠❛r❦❴❞❛❣✭❧✮❀

8 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r

✶qq

❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

9

♠❛r❦❴❞❛❣✭r✮❀

10 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

11

❞✲❃♠ ❂ ✶❀

12 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 39

Marking a DAG

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7

♠❛r❦❴❞❛❣✭❧✮❀

8 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, δ✶

rqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

9

♠❛r❦❴❞❛❣✭r✮❀

10 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

11

❞✲❃♠ ❂ ✶❀

12 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 40

Marking a DAG

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7

♠❛r❦❴❞❛❣✭❧✮❀

8 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, δ✶

rqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

9

♠❛r❦❴❞❛❣✭r✮❀

10 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, m♣δ✶

rqqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

11

❞✲❃♠ ❂ ✶❀

12 ✴✴

✧ ❞ ÞÑ ♠ ❧ r r ✎ ♣ ♣❧ ♣ qq ❨ ✎ ♣r ♣ ✶qqq ❫ ♣ q ✏ ♣ ✶q ❫ ✏ ♣ q ✯

13 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 41

Marking a DAG

1 str✉❝t

♥♦❞❡ ④s❤♦rt ♠❀ str✉❝t ♥♦❞❡ ✯❧✱✯r❀⑥❀

2 3 ✈♦✐❞

♠❛r❦❴❞❛❣✭str✉❝t ♥♦❞❡ ✯❞✮ ④ ✴✴ tdag♣❞, δq✉

4

✐❢ ✭✦❞ ⑤⑤ ❞✲❃♠✮ r❡t✉r♥❀

5

str✉❝t ♥♦❞❡ ✯❧ ❂ ❞✲❃❧✱ ✯r ❂ ❞✲❃r❀

6 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, δℓq ❨ ✎ dag♣r, δrqq ❫ δ ✏ node♣0, δℓ, δrq ✯

7

♠❛r❦❴❞❛❣✭❧✮❀

8 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, δ✶

rqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

9

♠❛r❦❴❞❛❣✭r✮❀

10 ✴✴

✧ ❞ ÞÑ ♠ : 0, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, m♣δ✶

rqqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

11

❞✲❃♠ ❂ ✶❀

12 ✴✴

✧ ❞ ÞÑ ♠ : 1, ℓ : ❧, r : r ✎ ♣dag♣❧, m♣δℓqq ❨ ✎ dag♣r, m♣δ✶

rqqq

❫ m♣δrq ✏ m♣δ✶

rq ❫ δ ✏ node♣0, δℓ, δrq

✯

13 ⑥ ✴✴ tdag♣❞, m♣δqq✉

Overlap, Ramification, and DAGs ④

SLIDE 42

Ramification Conditions

dag♣ℓ, δℓq ❨ ✎ dag♣r, δrq ✩ dag♣ℓ, δℓq ✎ ♣dag♣ℓ, m♣δℓqq ✁ ✁ ✎ dag♣ℓ, m♣δℓqq ❨ ✎ dag♣r, δ✶

rq ❫ m♣δrq ✏ m♣δ✶ rqq

(1) dag♣ℓ, δ✶

ℓq ❨

✎ dag♣r, δ✶

rq

✩ dag♣r, δ✶

rq ✎ ♣dag♣r, m♣δ✶ rqq ✁

✁ ✎ dag♣ℓ, δ✷

ℓ q ❨

✎ dag♣r, m♣δ✶

rqq ❫ m♣δ✶ ℓq ✏ m♣δ✷ ℓ qq

(2)

Overlap, Ramification, and DAGs ④

SLIDE 43

Overlaid Data Structures

SLIDE 44

Threaded Tree

list♣sq ❫ tree♣tq t s

Overlaid Data Structures ④

SLIDE 45

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴

t♣❉ ÞÑ ❫ ✏ ✎ ♣ qq ❫ ♣ q✉

7

s ❂ s✲❃♥❡①t❀

8 ✴✴

t♣ ÞÑ ✎ ♣ qq ❫ ♣ q✉

9

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

10 ✴✴

t ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

11

r❡t✉r♥ ❝❀

12 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With ttree♣tq✉ tree_remove(t,c) ttree♣retq ✎ c ÞÑ ✁, ✁, ✁✉

Overlaid Data Structures ④

SLIDE 46

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴

t♣ ÞÑ ✎ ♣ qq ❫ ♣ q✉

9

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

10 ✴✴

t ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

11

r❡t✉r♥ ❝❀

12 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With ttree♣tq✉ tree_remove(t,c) ttree♣retq ✎ c ÞÑ ✁, ✁, ✁✉

Overlaid Data Structures ④

SLIDE 47

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

10 ✴✴

t ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

11

r❡t✉r♥ ❝❀

12 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With ttree♣tq✉ tree_remove(t,c) ttree♣retq ✎ c ÞÑ ✁, ✁, ✁✉

Overlaid Data Structures ④

SLIDE 48

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

10 ✴✴ t??? ❫ ♣tree♣tq ✎ c ÞÑ ✁, ✁, ✁q✉ 11

r❡t✉r♥ ❝❀

12 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With ttree♣tq✉ tree_remove(t,c) ttree♣retq ✎ c ÞÑ ✁, ✁, ✁✉

Overlaid Data Structures ④

SLIDE 49

Skeleton Trees and Lists

list♣sq ❫ tree♣tq t s

Overlaid Data Structures ④

SLIDE 50

Skeleton Trees and Lists

tree♣tq t s tree♣xq

def

✏ ♣x ✏ 0❫empq❴❉L, R, N. x ÞÑ L, R, N✎tree♣Lq✎tree♣Rq

Overlaid Data Structures ④

SLIDE 51

Skeleton Trees and Lists

sktree♣tq t s sktree♣xq

def

✏ ♣x ✏ 0❫empq❴❉L, R, N. x ÞÑ L, R✎sktree♣Lq✎sktree♣Rq

Overlaid Data Structures ④

SLIDE 52

Skeleton Trees and Lists

sklist♣sq t s sklist♣xq

def

✏ ♣x ✏ 0 ❫ empq ❴ ❉N. x 2 ÞÑ N ✎ sklist♣Nq

Overlaid Data Structures ④

SLIDE 53

Skeleton Trees and Lists

tree♣tq ô sktree♣t, πq ✎ sklist♣t, πq t s

Overlaid Data Structures ④

SLIDE 54

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9 ✴✴

t♣ ÞÑ ✎ ♣ qq ❫ ♣ ♣ ❩ t ✉q ✎ ♣ ❩ t ✉qq✉

10

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

11 ✴✴

t♣ ÞÑ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✎ ♣ ❩ t ✉qq✉

12 ✴✴

t♣ ÞÑ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✎ ♣ q ✎

ÞÑ ✁q✉

13 ✴✴

t♣ ÞÑ ✁ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

14

r❡t✉r♥ ❝❀

15 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With tsktree♣t, π ❩ tc✉q✉ tree_remove(t,c) tsktree♣ret, πq ✎ c ÞÑ ✁, ✁✉

Overlaid Data Structures ④

SLIDE 55

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ ♣sktree♣t, π ❩ tc✉q ✎ sklist♣c, π ❩ tc✉qq✉ 10

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

11 ✴✴

t♣ ÞÑ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✎ ♣ ❩ t ✉qq✉

12 ✴✴

t♣ ÞÑ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✎ ♣ q ✎

ÞÑ ✁q✉

13 ✴✴

t♣ ÞÑ ✁ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

14

r❡t✉r♥ ❝❀

15 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With tsktree♣t, π ❩ tc✉q✉ tree_remove(t,c) tsktree♣ret, πq ✎ c ÞÑ ✁, ✁✉

Overlaid Data Structures ④

SLIDE 56

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ ♣sktree♣t, π ❩ tc✉q ✎ sklist♣c, π ❩ tc✉qq✉ 10

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

11 ✴✴ t♣c ÞÑ ✁, ✁, s ✎ list♣sqq ❫ ♣sktree♣t, πq ✎ c ÞÑ ✁, ✁ ✎ sklist♣s, π ❩ tc✉qq✉ 12 ✴✴

t♣ ÞÑ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✎ ♣ q ✎

ÞÑ ✁q✉

13 ✴✴

t♣ ÞÑ ✁ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

14

r❡t✉r♥ ❝❀

15 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With tsktree♣t, π ❩ tc✉q✉ tree_remove(t,c) tsktree♣ret, πq ✎ c ÞÑ ✁, ✁✉

Overlaid Data Structures ④

SLIDE 57

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ ♣sktree♣t, π ❩ tc✉q ✎ sklist♣c, π ❩ tc✉qq✉ 10

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

11 ✴✴ t♣c ÞÑ ✁, ✁, s ✎ list♣sqq ❫ ♣sktree♣t, πq ✎ c ÞÑ ✁, ✁ ✎ sklist♣s, π ❩ tc✉qq✉ 12 ✴✴ t♣c ÞÑ ✁, ✁, s ✎ list♣sqq ❫ ♣sktree♣t, πq ✎ c ÞÑ ✁, ✁ ✎ sklist♣s, πq ✎ c 2 ÞÑ ✁q✉ 13 ✴✴

t♣ ÞÑ ✁ ✁ ✁ ✎ ♣ qq ❫ ♣ ♣ q ✎ ÞÑ ✁ ✁ ✁q✉

14

r❡t✉r♥ ❝❀

15 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With tsktree♣t, π ❩ tc✉q✉ tree_remove(t,c) tsktree♣ret, πq ✎ c ÞÑ ✁, ✁✉

Overlaid Data Structures ④

SLIDE 58

Removal from a Threaded Tree

1 str✉❝t

♥♦❞❡ ④ str✉❝t ♥♦❞❡ ✯❧✱✯r❀

2

str✉❝t ♥♦❞❡ ✯♥❡①t❀ ⑥❀

3 str✉❝t

♥♦❞❡ ✯ ♣♦♣✭✈♦✐❞✮ ④ ✴✴ tlist♣sq ❫ tree♣tq✉

4

✐❢ ✭✦s✮ r❡t✉r♥ ✵❀

5

str✉❝t ♥♦❞❡ ✯ ❝ ❂ s❀

6 ✴✴ t♣❉n. s ÞÑ ℓ, r, n ❫ s ✏ c ✎ list♣nqq ❫ tree♣tq✉ 7

s ❂ s✲❃♥❡①t❀

8 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ tree♣tq✉ 9 ✴✴ t♣c ÞÑ ℓ, r, s ✎ list♣sqq ❫ ♣sktree♣t, π ❩ tc✉q ✎ sklist♣c, π ❩ tc✉qq✉ 10

t ❂ tr❡❡❴r❡♠♦✈❡✭t✱❝✮❀

11 ✴✴ t♣c ÞÑ ✁, ✁, s ✎ list♣sqq ❫ ♣sktree♣t, πq ✎ c ÞÑ ✁, ✁ ✎ sklist♣s, π ❩ tc✉qq✉ 12 ✴✴ t♣c ÞÑ ✁, ✁, s ✎ list♣sqq ❫ ♣sktree♣t, πq ✎ c ÞÑ ✁, ✁ ✎ sklist♣s, πq ✎ c 2 ÞÑ ✁q✉ 13 ✴✴ t♣c ÞÑ ✁, ✁, ✁ ✎ list♣sqq ❫ ♣tree♣tq ✎ c ÞÑ ✁, ✁, ✁q✉ 14

r❡t✉r♥ ❝❀

15 ⑥ ✴✴ t♣list♣sqq ❫ tree♣tqq ✎ r❡t ÞÑ ✁, ✁, ✁✉

With tsktree♣t, π ❩ tc✉q✉ tree_remove(t,c) tsktree♣ret, πq ✎ c ÞÑ ✁, ✁✉

Overlaid Data Structures ④

SLIDE 59

Ramification Conditions

♣c ÞÑ ℓ, r, n ✎ list♣sqq ❫ ♣sktree♣t, π ❩ tc✉q ✎ ptrs♣π ❩ tc✉qq ✩ sktree♣t, π ❩ tc✉q ✎ ♣sktree♣t✶, πq ✎ c ÞÑ ✁, ✁ ✁ ✁ ✎ ♣c ÞÑ ✁, ✁, n ✎ list♣sqq ❫ ♣sktree♣t✶, πq ✎ c ÞÑ ✁, ✁ ✎ ptrs♣πqq

Overlaid Data Structures ④

SLIDE 60

Towards Tool Support

SLIDE 61

Program Proofs with Ramification

✌ Meta-theory validated in Coq ✌ Programs proved by hand ✌ Ramification conditions proved in Coq (work in progress)

Towards Tool Support ④

SLIDE 62

Ramification Library

Collection of lemmas to simplify ramification conditions, e.g. ✌ ❅P, Q, R, R✶, F R ✩ P ✎ ♣Q ✁ ✁ ✎ R✶q R ✩ P ✎ F ✎ true F ✁ ✁

✎ R✶ ✩ F ✁

✁ ✎ R✶ R ✩ P ✎ F ✎ ♣Q ✎ F ✁ ✁ ✎ R✶q ✌ ❅P, Q, R, R✶, F precise♣Pq precise♣Qq P ❨ ✎ R ✩ P ✎ ♣Q ✁ ✁ ✎ Q ❨ ✎ R✶q ♣P ✎ Fq ❨ ✎ R ✩ P ✎ ♣Q ✁ ✁ ✎ ♣Q ✎ Fq ❨ ✎ R✶q ✌ ❅P, Q, R, R✶, F P ❨ ✎ R ✩ P ✎ ♣Q ✁ ✁ ✎ Q ❨ ✎ R✶q P ❨ ✎ ♣R ✎ Fq ✩ P ✎ ♣Q ✁ ✁ ✎ Q ❨ ✎ ♣R✶ ✎ Fqq

Towards Tool Support ④

SLIDE 63

Benchmark: Cheney’s GC

SLIDE 64

Cheney’s Copying Garbage Collector

1 ✈♦✐❞ ❝♦❧❧❡❝t✭✈♦✐❞ ✯✯r✮ ④ 2 ✈♦✐❞ ✯t♠♣ ❂ ❢r♦♠❙♣❛❝❡❀ 3 ❢r♦♠❙♣❛❝❡ ❂ t♦❙♣❛❝❡❀ 4 t♦❙♣❛❝❡ ❂ t♠♣❀ 5 ❢r❡❡ ❂ t♦❙♣❛❝❡❀ 6 s❝❛♥ ❂ ❢r❡❡❀ 7 ❝♦♣②❴r❡❢✭r✮❀ 8 ✇❤✐❧❡ ✭s❝❛♥ ✦❂ ❢r❡❡✮ ④ 9 ❝♦♣②❴r❡❢ ✭✭ ✈♦✐❞ ✯✯✮ s❝❛♥ ✮❀ 10 ❝♦♣②❴r❡❢ ✭✭ ✈♦✐❞ ✯✯✮✭ s❝❛♥ ✰ ✹✮✮❀ 11 s❝❛♥ ❂ s❝❛♥ ✰ ✽❀ 12 ⑥ 13 ⑥ 1 ✈♦✐❞ ❝♦♣②❴r❡❢✭✈♦✐❞ ✯✯♣✮ ④ 2 ✐❢ ✭♣ ✫✫ ✯♣✮ ④ 3 ✈♦✐❞ ✯♦❜❥ ❂ ✯♣❀ 4 ✐♥t ❢✇❞ ❂ ✯✭ ✐♥t ✯✮ ♦❜❥❀ 5 ✐❢ ✭❢✇❞ ✫✫ 6 t♦❙♣❛❝❡ ❁❂ ✭✈♦✐❞ ✯✮❢✇❞ ✫✫ 7 ✭✈♦✐❞ ✯✮ ❢✇❞ ❁ t♦❙♣❛❝❡✰s♣❛❝❡❙③ ✮④ 8 ✯✭ ✈♦✐❞ ✯✯✮♣ ❂ ✭✈♦✐❞ ✯✮ ❢✇❞❀ 9 ⑥ ❡❧s❡ ④ 10 ✈♦✐❞ ✯♥❡✇❖❜❥ ❂ ❢r❡❡❀ 11 ❢r❡❡ ❂ ❢r❡❡ ✰ ✽❀ 12 ✯✭✐♥t✯✮ ♥❡✇❖❜❥ ❂ ✯✭✐♥t✯✮ ♦❜❥❀ 13 ✯✭✐♥t ✯✮✭ ♥❡✇❖❜❥ ✰ ✹✮ ❂ 14 ✯✭ ✐♥t ✯✮✭ ♦❜❥ ✰ ✹✮❀ 15 ✯✭ ✈♦✐❞ ✯✯✮ ♦❜❥ ❂ ♥❡✇❖❜❥❀ 16 ✯✭ ✈♦✐❞ ✯✯✮♣ ❂ ♥❡✇❖❜❥❀ 17 ⑥ ⑥ ⑥

Benchmark: Cheney’s GC ④

SLIDE 65

With Framing

Loop Invariant

iso♣φ, FORW, BUSYq ❫ ♣ALIVE ✏ FORW ❨ UNFORWq ❫ Reachable♣head, tail, ALIVE, rootq❫♣ALIVE❑NEWq❫PtrRg♣head, ALIVEq❫ PtrRg♣tail, ALIVEq ❫ Tfun♣head, ALIVEq ❫ Tfun♣tail, ALIVEq ❫ ♣✼ALIVE ↕ ✼NEWq ❫ ♣root P FORWq ❫ ♣scan ↕ freeq ❫ Ptr♣freeq ❫ Ptr♣scanq ❫ Ptr♣offsetq ❫ Ptr♣maxFreeq ❫ ❅✎y P UNFORW.♣♣❉z. ♣y, zq P head ❫ y ÞÑ zq ✎ ♣❉z✶. ♣y, z✶q P tail ❫ y 4 ÞÑ zqq ✎ ❅✎y P FORW.♣❉z. ♣y, zq P φ ❫ y ÞÑ z, ✁q ✎ ❅✎y P UNFIN.♣♣❉z. ♣y, zq P head ✆ φ✿ ❫ y ÞÑ zq ✎ ♣❉z✶. ♣y, z✶q P tail ✆ φ✿ ❫ y 4 ÞÑ z✶qq ✎ ❅✎y P FIN.♣♣❉z. ♣y, zq P φ ✆ ♣head ✆ φ✿q ❫ y ÞÑ zq ✎ ♣❉z✶. ♣y, z✶q P φ ✆ ♣tail ✆ φ✿q ❫ y 4 ÞÑ z✶qq ✎ ❅✎y P FREE.y ÞÑ ✁, ✁

Benchmark: Cheney’s GC ④

SLIDE 66

With Ramification

Loop Invariant

to ↕ scan ↕ free ➔ to size❫ cheney♣✯to, scan, freeq ❨ ✎ cheney♣✯scan, scan, freeq

In-Copy Graph Predicate

cheney♣g, scan, freeq

def

✏ ♣g ✏ 0 ❫ ❡♠♣q ❴ ♣g ÞÑ a, b ❫ ♣to ↕ g ↕ scan ñ to ↕ a, b ↕ to sizeq❫ ♣scan ↕ g ↕ free ñ from ↕ a, b ↕ from sizeqq ❨ ✎ cheney♣a, scan, freeq ❨ ✎ cheney♣b, scan, freeq

Benchmark: Cheney’s GC ④

SLIDE 67

Conclusion

SLIDE 68

Summary

Ramify Rule

✌ Small and intricate programs with sharing ✌ Exposes the essence of the proofs ✌ Concise and compositional proofs ✌ Valid in any separation logic

Ramification Conditions R ✩ P ✎ ♣Q ✁ ✁ ✎ R✶q

✌ Beyond the reach of today’s automatic theorem provers ✌ Simplification lemmas provided for Coq ✌ Expressed as SL entailments ✌ ❨ ✎ is a useful connective!

Conclusion ④

SLIDE 69

Prospects for Automation

Current Tools

✌ Automatic shape analysis tools cannot deal with sharing ✌ Have separation baked in

Automatic Proofs of Programs with Sharing

✌ Extend classic shape domains to express sharing ✌ Automate checks of ramification conditions ✌ More proved programs to come!

Conclusion ④