HAL Id: hal-00992524https://hal.archives-ouvertes.fr/hal-00992524

Submitted on 18 May 2014

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A Weakest Oracle for Symmetric Consensus inPopulation Protocols

Joffroy Beauquier, Peva Blanchard, Janna Burman

To cite this version:Joffroy Beauquier, Peva Blanchard, Janna Burman. A Weakest Oracle for Symmetric Consensus inPopulation Protocols. 2014. hal-00992524

❲st r ♦r ②♠♠tr ♦♥s♥ss

♥ P♦♣t♦♥ Pr♦t♦♦s

❬ rr s♠ss♦♥ ♦r t st st♥t ♣♣r r ❪

♦r♦② qr P ♥r⋆ ♥ ♥♥ r♠♥

Prs♦t ❯♥rst② rs② r♥ ④♥rr♠♥⑥rr

strt ❲ ♥stt t ♥♠♥t ♣r♦♠ ♦ tr♠♥t♥ strt ♦♥s♥ss ♥ ♥♥ ♥r♦♥♠♥t ♦ r rs♦r♠t ♠♦ s♥s♦r ♥t♦rs ♠② ♦♥sr t ♠♦♦ ♣♦♣t♦♥ ♣r♦t♦♦s ♥ ts ♠♦ ♥ ♥♥♦♥ ♥♠r ♦ s②♥r♦♥♦s ♥♦♥②♠♦s ♥ ♥t stt♠♦ ♥ts ♥trt ♥ ♣rs ♥ts r ♥ t♦ tt ♦tr ♥ts r ♣rs♥t t ♥♦t♥trt♥ ♥ ♥♦ rs t♥ rs ♣♣♥ s s t ♠♥ rs♦♥ ♦r t ♠♣♦sst② ♦♦♥s♥ss ♥ ts ♠♦ tr ♣r♦♥ ts ♠♣♦sst② rst ♥stt t ♦♥t♦♥s t♦ t♦t ♦r♥ ♠♦ ♦r ♦t♥♥ s♦t♦♥s ❲ ♦♣t t ♥♦♥ t♥q t♦ ♥♣st ts ♦♥t♦♥s ♥ ♥ ♦r strt ①tr♥ ♠♦ t♦ ♣r♦ s♦♠ ♣ ♥♦r♠t♦♥ ♦r s♦♥♣r♦♠s ♦♠ r② ♥♦♥ ♦rs ♦ ♥♦t t t ♠♦ ♦ ♣♦♣t♦♥ ♣r♦t♦♦s s♦♠ ♦trs r♥♦t r♥t t♦ t ♦♥s♥ss ♣r♦♠ ❲ ♣r♦♣♦s ♥ ♥tr② ♥ t♦r② ♦ ♦rs ❲ s♦ ♦ s♣ ♦r ♥ ts t♦r② ♦s t♦ s♦ s②♠♠tr ♦♥s♥ss str♦♥r t ♥tr rs♦♥♦ ♦♥s♥ss ♥ t ♦♥sr ♠♦ s s♦t♦♥ t♦rts ♥② ♥♠r ♦ rs rs ♥② ♣r♦ tt t ♣r♦♣♦s ♦r s t st ♥ ts t♦r②

②♦rs ♥t♦rs ♦ ♠♦ ♥ts ♣♦♣t♦♥ ♣r♦t♦♦s tt♦r♥ ♦♥s♥ss ♦rs st♦r rs rs

⋆ P st♥t ♦♥tt t♦r ât ❯♥rsté Prs rs② ① r♥ t

♥tr♦t♦♥ ♥ t ❲♦r

♦♥s♥ss s s♦♥ ♣r♦♠ ss ♥ tt♦r♥t strt ♦♠♣t♥ ♥ ts ♣r♦♠ ♣r♦ss ♥t ♥ ♦r s s ♥ ♥t② ♥ ♥♦♥t② s t♦ t ♥t② ♥ rrrss♦♥ tr♠♥t♦♥ ♦♥t♦♥ Pr♦sss ♠st ♦♥ ♦♠♠♦♥ ♣♥♥ ♦♥ t ♥♣t s♦r♥ t♦ r♠♥t ♥ t② ♥♦♥trt② ♦♥t♦♥s s t ♣rs ♥t♦♥ ♥ ♥❬❪ ♦r ♠♦r ts ♦♥ ♦♥s♥ss

♦♥s♥ssrt ♣r♦♠s r r♥t t♦ ♠♦ s♥s♦r ♥t♦rs ♥ ♠♥② r♥t ♦♥t①ts ♦r①♠♣ ♦♥ s ❬❪ sr♠ ♦r♠t♦♥ ♦♥tr♦ s ❬❪ strt s♥s♦r s♦♥ s ❬❪ ♥ ttt ♥♠♥t s ❬❪ s♦ ❬❪ ♦r sr②s ♥ rr♥s ♦♥ ♦♥s♥ssrt♣r♦♠s ♥ ♠♦ rss ♥t♦rs

tt♦r♥ s r t♦r ♥ ♥ t ♠♦ ♥ts ② r ② ♥tr r ♥ ♣r♦♥t♦ rs s ❬❪ ♦r sr② ♦♥ tt♦r♥ ♥ ♠♦ rss ♥t♦rs ❲ ♠♥t♦♥ r s♦♠ ♠♦str♥t ♦rs ♦♥ tt♦r♥t ♦♥s♥ss st③♥ rs♦♥ ♦ ♦♥s♥ss ♦♥trr② t♦ t tr♠♥t♥♦♥ s ♥ st ♥ ❬❪ ♦r t ♠♦ ♦ ♠♦ ♥ts q♣♣ t ♥q ♥trs ♥ ♣r♦♥t♦ rs ♥ ②③♥t♥ rs ♥♦tr ♦♥r♥ rs♦♥ ♦ ♣♣r♦①♠t ♦♥s♥ss ♥ t r♥♦♠③♦♠♠♥t♦♥ ♠♦ ♦ ♣♦♣t♦♥ ♣r♦t♦♦s ♣r♦♥ t♦ ②③♥t♥ rs s ♥ st ♥ ❬❪ r ♦♥sr ss tr♠♥t♥ rs♦♥ ♦ ♦♥s♥ss ♦♥trr② t♦ ts ♦rs

♥♠♥t rst ② sr ②♥ ♥ Ptrs♦♥ ❬❪ stts tt ♥ t ss s②♥r♦♥♦s ♠ss♣ss♥ ♠♦ ♥♦ tr♠♥st ♦rt♠ ♦r ♦♥s♥ss ①sts ♥ ♥ t s ♦ ♥q ♣♦ss rst♥ r t s ♥♦t sr♣rs♥ tt t s♠ ♠♣♦sst② ♦s ♥ t ♠♦ ♦ ♣♦♣t♦♥ ♣r♦t♦♦sPP ♥tr♦ ♥ ② ♥♥ t ❬❪ s ♠♦ s ♥♠♥t② s②♥r♦♥♦s s ♦♥♦ t ♠♥ rs♦♥s ♦r t rst ♥ ❬❪ ♦r s♦♠ ♥r♥t rtrsts ♦ ♣♦♣t♦♥ ♣r♦t♦♦s♠ ♦♥s♥ss ♥ ♠♦r t ♥ ts ♠♦ ♥ts r ♥♦r♠ ♥st♥s ♥ ①t♥ ts♠ ♦ ② ♦♥st♥t ♠♠♦r② s③ ♥ ts ♥♥♦t ♥tr ♦t♥ ♥♦r st♦r s ♦r ♥② ♦tr♥♦r♠t♦♥ ♣♥♥ ♦♥ t ♥t♦r s③ ♥ts ♦♠♠♥t ② s②♥r♦♥♦s ♥trt♦♥s s tt ♥trt♦♥ s t♥ ♦♣ ♦ ♥ts ♦ r♦st ♦♠♠♥t♦♥ s t♦ ts♠tt♦♥s t ♥ts r ♥ t♦ tt ♦tr ♥ts r ♣rs♥t t ♥♦t ♥trt♥ ♥ ♥♦rs r s ♣♦ss ♥ ♥ ♣♦♣t♦♥ ♣r♦t♦♦s ♥ t t ss♠♣t♦♥ ♦ s♥ ♦ rs♦♥s♥ss s ♠♣♦ss s ♦r♦r tr s ♥♦ r♥♦♠③ s♦t♦♥ tr

♥ t ♠ss ♣ss♥ ♠♦ t s♠s ♥trst♥ t♦ st② t s ♠ss♥ ♦r s♦♥ ♦♥s♥ss♥ PP ❲ ♦♣t t ♣♦♥t ♦ ♦ ♥r ♥ ♦ ❬❪ ♦r ♥♥ t ♣♦ss ♠ss♥ ♥♦r♠t♦♥♥r t ♦r♠ ♦ ♦rs r s strt ①tr♥ ♠♦ t♦ ♣r♦ ♣r♦ss t s♦♠♥♦r♠t♦♥ ♦♣② s t♦ s♦ ♥ ♣r♦♠ ♦rs ♥tr♦ ♥ ❬❪ r tt♦rs♣r♦ ♣r♦ss t rrt ♥♦r♠t♦♥ ♣② t r tt♦rs ♦ ❬❪ ♥♥♦t s♥ ♦r s s t② r♥s sts ♦ ♣r♦ss ♥trs st♠t t♦ rs s r② ♠♥t♦♥♥trs r s♥t ♥ t PP ♠♦ t♦ t ♦♥st♥t s③ ♠♠♦r② rqr♠♥t

rtss sr ♥tt②r ♦rs ①st ♥ t trtr r tt♦r ♥tr♦ ♥ ❬❪♦t♣ts ♦♦♥ t r② ♣r♦sss ♥ s♦s t (n−1)st r♠♥t ♣r♦♠ ♥ n♣r♦ss ♠ss♣ss♥ s②st♠ s r tt♦r s ♥ s♦♥ t♦ t st t♦ s♦ ts ♣r♦♠ s rt♥ ♦♥s♥ss s ♥♦s tt s ♥ ♦r ♥♥♦t ♣ t♦ s♦ ♦♥s♥ss ♥ PP ♥♦tr t②♣♦ ♦rs ♣r♦♣♦s ♥ ❬❪ ♥ s ♥ ❬❪ t♦ t ♥♦♥②♠t② ♣r♦ ♥♦r♠t♦♥ ♦♥ t♥♠r ♦ rs ♣r♦sss ♦♥ ② f < n ♥ ♦r t s♠ rs♦♥ ♦ ♦♥st♥t ♥t stt s♣♥♥♦t s ♥ t PP r♠♦r s♦ rtt r tt♦r ♣r♦♣♦s ♥ ❬❪ rqrs t♦♠♥t♥ ♥♦♥ ♦♥trs t r② ♣r♦ss ♥ ts ♥ s ♥♦t st ♥ ts st② ♦♠ ♦trr tt♦rs tt r s t♦ s♦ ♦♥s♥ss ♥ ♣t t♦ ♥♦♥②♠♦s s②st♠s AΩ ❬❪ AL ♥AΣ′ ❬❪ ♥ s ♥ PP t♦ s♦ ♦tr tss t♥ ♦♥s♥ss ♦t tt ♥ ♠ss♣ss♥ s②st♠ts ♦rs r s ♥ ♦♠♥t♦♥ t ♦tr ♣♦r ♠♦ ss♠♣t♦♥s ♥ ♣ts s ♣♦sstr♠♥t♥ r♦st ♥ ♥♦♥ ♣r♦ss ♠♠♦r② r ♥ ♥ PP

s s t rs♦♥ ② ♥tr♦ ♥ t♦r② ♦ ♦rs ♦♥str♥ts ♥ ♠♥ ♥s♥♥ ts ♦rs r s② t♦ ♠ t♠ ♠♣♠♥t t ♠♥♠♠ ①tr♥ ss♠♣t♦♥s♦♥ t s②st♠ rst ♥t tt t ♦t♣t ♦ ♦rs ♣♥ ♦♥② ♦♥ t ♣st s sq♥ ♦

♥trt♦♥s ♥ rss ♥ t s ♦ rs rs s rqr♠♥t ♣ts s t ♦rs s t♦s♥tr♦ ♥ ❬❪ ♥ ♥r③ ♥ ❬❪ ♦r s♦♥ t r t♦♥ ♣r♦♠ ♥ PP ♥ t♦s ♦rs♦sr t ♦♥rt♦♥s r ② t s②st♠ ♥ ♥ ♣♥ ♦♥ t ①t ♣r♦t♦♦ ♥ ♥♦t ♦♥②♦♥ t s

♦♥ ♥ ♣st s ♥ t♦ ♥t ♥ts ♥t t st ♦ ♣♦ss ♦ ♦t♣t sq♥s♥ ② t ♦r t♦ ♦♥ ♦ t ♥ts t♦ t s♠ s ♦r t ♦tr s②♠♠tr②

♥② s♦ ♥t tt r t♦r ♦r ♦ ♦♦s ♦r ♥st♥ ♦♥ ♦ t ♥ts①t♥ t r② rst ♥trt♦♥ ♦r tr♠♥♥ t ♦t♣t ♦♥s♥ss ♦s ♥♦t ♥tr ♥t♦ t♠② ♦ ♦♥sr ♦rs ♥ ♥ ♠♦ s ♥♦r♠ ♥ s②♠♠tr s PP s r② ♣♦r ♦rs♠s r② r t♦ ♠♣♠♥t s♣② ♥ ♣rs♥ ♦ rs rs s♣♣♠♥tr② ss♠♣t♦♥s♦ ♥♦ tt t ♦r s t♦ st♥s t♥ ♥ts ♥ t PP ♠♦ t② r s②♥st♥s s s ♥ t♦♥ rs♦♥ ② ♥tr♦ s②♠♠tr② ♦♥t♦♥ ♥ t ♥t♦♥ ♦♦r ♦rs s s ♦♥t♦♥ ♥♦t ♦♥② ♠♥ts t r t♦r ♦r t s♦ ♥② ♦rs ♦♥ s♥♥ ♣rtr ♥q ♥t t s♦♠ ♣♦♥t ♦ ♦♠♣tt♦♥

♦t tt t ♦♠♥t♦♥ sr ♦♥str♥ts s♦ ① t s♥ tt♦r ♦r ♣r♦♣♦s♥ ❬❪ ♥ q♣♣ t ♣♦r st♥s ♦r♥♦ ♣ ♦ tt♥ t ♣rs♥ ♦r s♥♦ ♥② stt r♦♠ t ♣♦♣t♦♥ ♦sr♥ t ♦♥rt♦♥s ♦ t ①t ♣r♦t♦♦

♦ s♠♠r③ t t♦r② ♦ ♦rs ♣r♦♣♦s ♥ t ♣♣r ♣r♦ ♥♦r♠t♦♥ rt ♦♥② t♦ t ♣stss ♦ ♣r♦ ts ♥♦r♠t♦♥ ♥ ♦r ♦t♣ts ♦♦♥ t r② ♥t s♠r② t♦ tr tt♦r ♥ ❬❪ ♦r s ♥♦t rqr t♦ ♣r♦ ts ♥♦r♠t♦♥ t t ♣rs t♠ ♥ t♣♣rs t ♦♥② ♥t② t st ♦♥ ♥ t s♦♠ ♥t t s ♥r ♥ ts s♥s ♥② t♣r♦♣♦s ♦rs r s②♠♠tr s ①♣♥ ♦

♦♥trt♦♥s rst ♦r♠② ♣r♦ t ♠♣♦sst② rsts ♠♥t♦♥ ♦ ♥♠② t ♠♣♦sst②♦ ♦♥s♥ss ♥ PP s rst s ♥♦t ♥ ♣tt♦♥ ♦ t ♠♣♦sst② ♦ ♦♥s♥ss ♥ ♦tr ♠♦ss ♥ ♦r s ♦♥s♥ss s ♠♣♦ss ♥ tr r ♥♦ rs

♦♥ ♣r♦♣♦s ♦r♠ ♠♦ ♦r ♦rs ♥ PP s ♠♦ s qt ♥r ♥ ♥ s ♦r ♦tr ♦♥t①ts t♥ ♦♥s♥ss

r t♦ ♦r♦♠ t ♠♣♦sst② ♦ ♦♥s♥ss ♥ PP ♦s ♥ ♦r tt DejaV u r♦♠t t♦r② ♦ ♦rs sss ♦ s② ts ♦r ♥ ♥♦r♠ ♥ ♥t tt t s ♥trt♥rt② t t ♥♦♥t② ♥ts ♦r t ♦r♠ ♥t♦♥ s ♥ ♥ t s♣rt ♦ ❬❪ ♦♥sr s②♠♠tr ♦♠♣t ♥t♦♥s ② PP ♦♥sr str♦♥r s②♠♠tr ♦r♠ ♦ ♦♥s♥ss s♠s ttr ♣t t♦ PP s②♠♠tr ♦♥s♥ss s ♦♥s♥ss ♥ ♣r♠tt♦♥ ♦ t♥♣t s ♦ t ♥♦♥t② ♥ts ♦s ♥♦t ♥ t s♦♥ ♦r t ♦r♠ ♥t♦♥ s ❲ s♦ ♦ DejaV u ♦s s♦♥ t s②♠♠tr ♦♥s♥ss ♣r♦♠ ♦t t ♥ t♦t rs rs

♦rt ♣r♦ tt t DejaV u ♦r s t st ♥ t t♦r② ♥ t s♥s ♦ ❬❪ s ♥t♦♥♥ ♦r s♦♥ t ♣r♦♠ t ♦r t♦t rss ♦t tt ts rst ss ♦♠♣t②r♥t t♥qs t♥ t♦s ♦ ❬❪

♥ ♦♥s♦♥ sss t ♠♣♠♥tt♦♥ ♦ t ♦rs ♥tr♦

♦ ♥ ♥t♦♥s

P♦♣t♦♥ Pr♦t♦♦

❲ s t s♠ ♥t♦♥s s ♥ ❬❪ t s♦♠ st ♠♦t♦♥s ♥t♦r s r♣rs♥t ② rtr♣ G = (V, E) t n rts ♥ ♥♦ ♠ts ♥♦r s♦♦♣s rt① r♣rs♥ts ♥tstts♥s♥ ♥ ♥t ♥ ♥ (u, v) ∈ E ♥ts t ♣♦sst② ♦ ♦♠♠♥t♦♥ ♠t♥♥trt♦♥ t♥ t♦ st♥t ♥♦s u ♥ v ♥ u ♣②s t r♦ ♦ t ♥tt♦r ♥ v ♦ trs♣♦♥r ♦r♥tt♦♥ ♦ ♥ ♦rrs♣♦♥s t♦ ts s②♠♠tr② ♥ r♦s ❲ ♦t♥ rt G ♥st ♦ Vt♦ rr t♦ t rts ♦ G r♣ s s②♠♠tr ♥ (u, v) ∈ E ♥ ♦♥② (v, u) ∈ E ♥ ts ♣♣r ♦♥sr ♦♥② s②♠♠tr ♦♠♣t r♣s

♣♦♣t♦♥ ♣r♦t♦♦ A(Q,V, Init, Input, δ) ♦♥ssts ♦ ♥t stt s♣ Q st V ♦ ♥t s st Input ♦ ♥♣t s ♥ ♥t③t♦♥ ♠♣ Init : V → Q ♥ tr♥st♦♥ ♥t♦♥ δ : (Q×Input)2 → Q2

tt ♠♣s ♥② t♣ (q1, i1, q2, i2) t♦ ♥ ♠♥t δ(q1, i1, q2, i2) ♥ Q2 tr♥st♦♥ r ♦ t ♣r♦t♦♦ s t♣ (q1, i1, q2, i2, q

′1, q′

2) r (q′

1, q′

2) = δ(q1, i1, q2, i2) ♥ s ♥♦t ② (q1, i1)(q2, i2) → (q′

1, q′

2) ❲

rr t♦ (q1, i1)(q2, i2) rs♣ (q′1, q′

2) s t t rs♣ rt ♣rt ♦ t r ♦t tt ♦♥② ♦♥sr

tr♠♥st ♣♦♣t♦♥ ♣r♦t♦♦s t rt ♣rt s ♥q② tr♠♥ ② t t ♣rt ♦ t r♥tt② t ♥♣t s ♥ Input r ♣r♦ t♦ t ♥ts ② s♦♠ ①tr♥ ♥ ♦r

s ♦ ♦r♦r t t ♥♥♥ r② ♥t s ss♥ ♥ ♥t r♦♠ t st V ♥ t♥t③t♦♥ ♠♣ ts t t ♦rrs♣♦♥♥ stt s ♥ ts ♣♣r t ♥t s r♣rs♥t t ♥ts ♥ t ♦♥s♥ss ♣r♦♠

s st♦rs

♦♥sr r♣ G ♠t♥ ♥t s r♣rs♥t ② ♥ ♦ G rs ♥t s r♣rs♥t ② rt①♥ G ♥ ♥t s tr ♠t♥ ♥t ♦r rs ♥t ♥ ♥t e ♥♦s ♥ ♥t x ♥ tr e st rs ♥t ♦ t ♥t x ♦r e s ♠t♥ ♥t t x s ♦♥ ♦ t s ♥♣♦♥ts ♦ ♥ts e1♥ e2 r ♥♣♥♥t ♥ ♥♦ ♥t s ♥♦ ♥ ♦t e1 ♥ e2

s s sq♥ ♦ ♠t♥ ♥ rs ♥ts e1e2 . . . s tt ei = x s t rs ♥t ♦♥t x t♥ ♥♦ ♥t ej t j > i ♥♦s x ♠♣t② s s ♥♦t ② ǫ s ♥ ♥t♦r ♥♥t

♥ st♦r② H = (S, h) t s ♥ t st Input s s S t♦tr t ♥t♦♥ h ttss♦ts t r② ♦rr♥ ♦ ♠t♥ ♥t (x, y) ♦♣ (ix, iy) ♦ s r♦♠ Input ix rs♣ iy s t ♦t♣t ♦ t st♦r② t x rs♣ t y ♥ ♥t e s S s t ♥r②♥ s♦ t st♦r②

s ♦ s S ♥♦t ② base(S) s t sst ♦ ♥ts ♥ G ♣♣r♥ ♥ t s S ❲♥♦t ② crashed(S) rs♣ correct(S) t st ♦ ♥ts tt rs ♥ S rs♣ tt ♦ ♥♦t rs ♥ S ♥♥ st♦r② H = (S, h) ♥ base(H) = base(S) crashed(H) = crashed(S) correct(H) = correct(S)

♦♥sr ♥t s S ♥ s S′ ♥t ♦r ♥♥t s tt base(S′) ♦s ♥♦t ♦♥t♥ ♥②rts ♥ crashed(S) ♥ t s ♦♥t♥t♦♥ ♦ ♦rs ②s s ♥♦t ② S · S′ ♦r SS′❲ ①t♥ ts ♦♣rt♦♥s ♦♥ st♦rs s♠r②

♥ s S = (ei)0≤i<T ♥t ♦r ♥♥t t P(S) = (i, ei), 0 ≤ i < T t ♦rr♥s ♦♥ts ♥ S ♣rt② ♦rr st ♣♦st t t ♣rt ♦rr ♥rt ② (i, ei) (j, ej) ♥♦♥② i ≤ j ♥ ei, ej ♥♦ ♦♠♠♦♥ rt①

♠♥t (i, ei) ♥ P(S) s ♥tr② t t ♥t ei ♥ ♦t♥ rr t♦ t ♠♥ts♦ P(S) s ♥ts s ♦ ss S ♥ S′ r q♥t ♥♦t ② S ≃ S′ ♥ tr s ♣♦sts♦♠♦r♣s♠ t♥ P(S) ♥ P(S′) tt ♣rsrs t ♥ ♦ ♥ ♦tr ♦rs t s S′ sq♥t t♦ S t s ♦t♥ r♦♠ S ② ♣r♠t♥ sss② ♥♣♥♥t ♥ts ♥tt② P(S) ♥ s s r♠ s ♥ t♦ ss r q♥t ♥ tr s r♠s♦♦ t s♠ ♦t tt ② ♦♥strt♦♥ (i, ei) s rs ♥t t♥ tr s ♥♦ ♥t (j, ej) 6= (i, ei)s tt (i, ei) (j, ej) (i, ei) s ♠①♠

s r♠s ♦r S1 = (a, b)(c, d)(b, c) S2 = (c, d)(a, b)(b, c) ♥ S1 ≃ S2

s s s ♦♠♠t♥ t♦r ♦ s S tr ①st ss s1 ♥ s2 ♣♦ss② ♠♣t②s tt S ≃ s1 ·s ·s2 q♥t② s s ♦♠♠t♥ t♦r ♦ S ♥ t ♣♦st P(s) s s♣♦st ♦ P(S)

♥ ♥ ♥t x r rs♣ ♦rr ♦♥ t x s ♥t s c s tt P(c) s ♠①♠♠rs♣ ♠♥♠♠ ♠♥t (i, ei) ♥ ei ♥♦s x ♥ ♦tr ♦rs ♦r r② (j, ej) ∈ P(c) (j, ej) (i, ei)rs♣ (i, ei) (j, ej)

♥ s S ♥ ♥ ♥t p = (i, ei) ♥ P(S) ♣st rs♣ tr ♦♥ ♦ p ♥ S s t s♣♦stPastCone(p) = p′ ∈ P(S), p′ p rs♣ FutureCone(p) = p′ ∈ P(S), p p′ ② ♦♥strt♦♥ ♦ ts rs ♥t ♥♥♦t ♣♣r ♥ t ♣st ♦♥ ♦ ♥② ♦tr ♥t

♥ ♥t s S s② tt ♥t x s ♥rt② ♠t t ♥t y ♥ S tr ①sts ♥♥t p ♥ S ♥♦♥ x ♥ ♥ ♥t p′ ♥♦♥ y ♥ t ♣st ♦♥ ♥ S ♦ p Prs② tr r ♥tsy = a0, a1, . . . , ak−1, ak ♥ ♠t♥ ♥ts e0 e1 · · · ek−1 s tt ei ♥♦s ai−1 ♥ ai

♥ ♥♥t s S s ② r ♥ ♦♥② r② ♦rrt ♥t ♥ S ♠ts ♥rt② r② ♦rrt♥t ♥♥t② ♦t♥ ❯♥ss stt ♦trs r② ♥♥t s ♥ ts ♣♣r s ss♠ t♦ ②r

①t♦♥s

♦♥sr tr♠♥st ♣♦♣t♦♥ ♣r♦t♦♦ A t stt s♣ Q ♥♣t st Input ♥ ♥t st V

♦♥sr ♦♠♣t r♣ G ♦♥rt♦♥ γ s ♥t♦♥ tt ss♦ts t r② ♥t x sttγ(x) ♥ Q ♦ t rss ss♠ tt Q ♦♥t♥s s♣ s②♠♦ crashed ♦r r② ss♥♠♥tκ ♦ ♥t s r♦♠ V t♦ t ♥ts ♦ t r♣ ♥ t ♦rrs♣♦♥♥ ♥t ♦♥rt♦♥γκ = Init κ r Init s t ♥t③t♦♥ ♠♣ ♦ t ♣r♦t♦♦

t γ, γ′ ♦♥rt♦♥s e = (x, y) s ♠t♥ ♥t ♥ i = (ix, iy) s ♦♣ ♦ s ♥ Inputt♥ s② tt γ ♦s t♦ γ′ e, i ♥ (γ(x), ix)(γ(y), iy) → (γ′(x), γ′(y)) s r ♦ t ♣r♦t♦♦ ♥♦r r② ♥♦ z 6∈ x, y γ(z) = γ′(z) e s t rs ♥t ♦ ♥t x t♥ s② tt γ ♦s t♦ γ′ e ♥ γ′(x) = crashed ♥ ♦r r② ♥♦ z 6= x γ′(z) = γ(z) ♥ ♦tr ♦rs γ′ s t ♦♥rt♦♥ ttrsts r♦♠ γ ② ♣♣②♥ t ♥t e t t ♥♣t s i ♥ s ♦ ♠t♥ ♥t

♥ ①t♦♥ s sq♥ γ0w0−−→ γ1

w1−−→ . . . r t γis r ♦♥rt♦♥s ♥ ♥② wi s tr ♠t♥ ♥t t♦tr t ♦♣ ♦ ♥♣t s ♦r rs ♥t sq♥ w0w1 . . . ♥tr②②s st♦r② H ♥ t tr♠♥st ♣♦♣t♦♥ ♣r♦t♦♦ t ①t♦♥ s ♥tr② tr♠♥② t st♦r② H ♥ t rst ♦♥rt♦♥ γ0 ♥ ♥ ♥♦t ts ①t♦♥ ② H[γ0] st♦r②H s rrr t♦ s t ♥♣t st♦r② ♦ t ①t♦♥ H[γ]

rs

♥ t ♦♦♥ ♥ ♥♦t♦♥s s♠r t♦ t♦s rt t♦ st♦rs ♦ ss r tt♦rs ❬❪

♥ ♦r O s ♥t♦♥ tt ss♦ts t r♣ G ♥ ② r s S ♦♥ G stO(G,S) ♦ st♦rs ♦♥ G t s ♥ s♦♠ st Range(O) tt S s tr ♥r②♥ sRange(O) s ♦r s s ♦r Os ♥ st♦r② H = (S, h) ♦♥ G s st♦r② ♦ ♥ ♦rO H ∈ O(G,S)

❲ s② tt ♣r♦t♦♦ A ss ♥ ♦r O ♥ t ♦♥② ①t♦♥s ♦ A ♦♥sr r t♦s ♦s♥♣t st♦rs r st♦rs ♦ O

♥tt② ♥ ♦r O1 s r t♥ ♥ ♦r O2 tr ①sts ♣♦♣t♦♥ ♣r♦t♦♦ tt ss O2

♥ ♣r♦s st♦rs ♦ O1

♦r ♦r♠② ♥ ♦r O1 s r t♥ ♥ ♦r O2 tr ①sts ♣♦♣t♦♥ ♣r♦t♦♦ A ♥O2s s ♥♣t s s tt ♦r r② st♦r② H ♦ O2 ♦r r② ♥t ♦♥rt♦♥ γ ♦A t ①t♦♥ H[γ] ②s st♦r② ♦ O1 ♥ t ♦♦♥ s♥s r ①sts ♥ ♦t♣t ♠♣

Out : Q → Range(O1) s tt H[γ] = γ0w0−−→ γ1 . . . t st♦r② H ′ = (S, h′) r h′ ss♦ts t

r② ♠t♥ ♥t ei = (xi, yi) t ♦♣ (Out(γi(x)), Out(γi(y))) ♦ O2s s st♦r② ♦ O2

♥ ♠② F ♦ ♦rs st ♦r ♥ F s ♥ ♦r O tt s r t♥ r② ♦tr ♦rs♥ F

♣ Pr♦♠s ♥ rs

♦♥s♥ss ♥ ②♠♠tr ♦♥s♥ss

♦♥sr ♣♦♣t♦♥ ♣r♦t♦♦ A t ♥t s V ❲ ss♠ tt t ♥ts ♥ ♥strt♦♥ decide

ss t♠ t♦ rrrs② ♦♥ s♦♠ ♥ V ♣♦♣t♦♥ ♣r♦t♦♦ A ♣♦ss② s♥ ♥ ♦r s s t♦ s♦ t ♦♥s♥ss ♣r♦♠ ♦r

♦♠♣t r♣ G ♦r ♥t ♦♥rt♦♥ γ ♦r ♥② ② r ①t♦♥H[γ] t stss t ♦♦♥ tr♠♥t♦♥ r② ♦rrt ♥t ♥t② s ♥ t ①t♦♥ H[γ] r♠♥t t♦ ♦rrt♥ts ♥♥♦t ♦♥ r♥t s t② t ♥ts t s♠ ♥t v t♥ ♦rrt ♥t ♥ ♦♥② ♦♥ v

♣r♦t♦♦ A s s t♦ s♦ t s②♠♠tr ♦♥s♥ss ♣r♦♠ t s♦s t ♦♥s♥ss ♣r♦♠ ♥♦r ♦♠♣t r♣ G t stss t ♦♦♥ t♦♥ ♦♥t♦♥ s②♠♠tr② ♦r ♥② ②r ①t♦♥ H[γ] ♥ ♥② ♥t ♦♥rt♦♥ γ′ ♦t♥ r♦♠ γ ② ♣r♠t♥ t ♥t s ♦ t♦rrt ♥ts ♥② ♦rrt ♥t s ♦♥ t s♠ ♥ t ①t♦♥ H[γ] ♥ ♥ H[γ′] ♥tt②t s♦♥ ♦ ♦rrt ♥t ♥ ♥ ①t♦♥ ♦s ♥♦t ♣♥ ♦♥ t strt♦♥ ♦ t ♥ts t♥ t ♦rrt ♥ts

Mnemosyne ss

❲ ♥ s♣ ss ♦ ♦rs t s ♥ 0, 1 ♥♠② t Mnemosyne ss ♦r O ♥Mnemosyne s ♥ ② ♠② ♦ sts Scheds(O,G, x) ♦ ♥t ss ♦r r② ♦♠♣t r♣ G

♥ r② ♥t x ♥ G s tt rss ♥ ♦♥s ♦r r② s S ♥ Scheds(O,G, x) tr①sts s k · c r k s s ♦♠♣rs♥ rss ♦♥② ♥ c s r ♦♥ t x stt S ≃ k · c ♦♠♠t♥ s②♠♠tr② S ∈ Scheds(O,G, x) ♥ S ≃ S′ t♥ S′ ∈ Scheds(O,G, x) ♥ts s②♠♠tr② ♦r ♥② ♣r♠tt♦♥ σ ♦ t rts ♦ G t s S = e1 . . . er ♦♥s t♦Scheds(O,G, x) ♥ ♦♥② t s σ(S) = σ(e1) . . . σ(er) ♦♥s t♦ Scheds(O,G, σ(x))

st♦rs H = (S, h) ♦♥ ♥② ♦♠♣t r♣ G ♦ t Mnemosyne ♦r O r ♥ ② t♦♦♥ ♣r♦♣rts st② ♦r ♥② ♣r① u ♦ S ♦r ♥② ♥t x ♥ G t st♦r② H s ♦t♣t 1 tx ♥ s♦♠ ♥t ♥ u t♥ s♦♠ s s ∈ Scheds(O,G, x) s ♦♠♠t♥ t♦r ♦ u u ≃ v · s · w♦r s♦♠ ss v w ♥ss t A t st ♦ ♦rrt ♥ts x r♥ S s tt S s s♦♠s s ∈ Scheds(O,G, x) s ♦♠♠t♥ t♦r A s ♥♦t ♠♣t② t♥ ♦r t st ♦♥ ♥t x ♥ Atr ①sts ♥ ♦rr♥ ♦ ♥ ♥t e ♥ S ♥♦♥ x s tt t st♦r② ss♦t t x s 1

♥tt② t Mnemosyne ♦r O s ♦r t ♣rs♥ ♦ ss r♦♠ Scheds(O,G, x) ♥ t♣st ♦♥ ♦ x st② ♣r♦♣rt② ♥srs tt O ♦t♣ts 1 t x t♥ t ♦rrs♣♦♥♥ ♣r① t②♦♥t♥s s ♦♠♠t♥ t♦r s r♦♠ Scheds(O,G, x) ♥ss ♣r♦♣rt② ♥srs tt tst ♦♥ ♦rrt ♥t s ♥t② ♥♦t ♦t ts t ♦♠♠t♥ s②♠♠tr② ♦♥t♦♥ ♠♣stt ♥ ♦r ♥ t Mnemosyne ss s ♥♦t t♦ st♥s ss S1 = e1e2 ♥ S2 = e2e1r e1 ♥ e2 r ♥♣♥♥t ♥ts s S1 rs♣ S2 ♠♥s tt ♥ rt♠ t ♥t e1rs♣ e2 ♦rs ♦r e2 rs♣ e1 ♥ t ♦♠♠t♥ s②♠♠tr② ①♣rsss t t tt t ♦r s♥♦ ss t♦ rt♠ ♦ ♥ts s②♠♠tr② ♦♥t♦♥ ①♣rsss tt t ♦r s ♥♦ ss t♦ t♥tts ♦ t ♥ts ♦ t s②st♠ s ♦♥t♦♥s r ♦t t ♣♦sst② ♦ r ♦r ♥♦r tt ♦ ♥t② ♦t♣t 1 t ♥q ♥t ♣♦ss② t♠♣♦rr②

DejaV u ♦r

♦r DejaV u ♥ t Mnemosyne ss s ♥ ♦r t ♦♠♣t r♣s G s ♦♦s sS ♦♥s t♦ Scheds(DejaV u,G, x) ♥ ♦♥② S ≃ k · c r k ♦♠♣rss rss ♦♥② ♥ c s r ♦♥ t x s tt base(c) = G− base(k)

♥tt② t♥s t♦ t ♣r♦♣rts ♦ t Mnemosyne ss t DejaV u ♦r ♥ ♦t♣t 1 t ♥♥t x ♥ x s ♥rt② ♠t t ♥ts tt ♥♦t rs s♦ r

♦r ①♠♣ ♦♥sr t s (x, a)(x, b)(x, c)(x, y)(x, y)(crash y)(a, b)(crash x)(a, c)(a, c) ♦♥ t♦♠♣t r♣ x, y, a, b, c s r♠ ♦ t s s ♣t ♥ ♦t tt tr s ♦♥ t t (x, a)(x, b)(x, c)(x, y) ♦♥s t♦ Scheds(DejaV u,G, x) s♥ t s ♥♦ rss♥ ts s ♦♥t♥s t ♥ts ♥ ♥♦t tt t r s ♦♥ t rt s q♥t t♦(crash x)(crash y)(a, b)(a, c) ♥ ts ♦♥s t♦ Scheds(DejaV u,G, a) ♦t tt S s ② rs t♥ S ♦♥t♥s ♥♥t② ♠♥② ss r♦♠ Scheds(DejaV u,G, x) ♦r r② ♦rrt ♥t x

r s ♦♥ t t s ♥ Scheds(DejaV u,G, x) t r s ♦♥ t rt s ♥Scheds(DejaV u,G, a)

♠♣♦sst② ♦ ♦♥s♥ss t♦t r

♦r♠ ❯♥r r♥ss ♥ t♦t rss t ♦♥s♥ss ♣r♦♠ ♦r ♦♠♣t r♣s s♠♣♦ss t♦t ♥ ♦r

Pr♦♦ ss♠ tt tr ①sts ♣♦♣t♦♥ ♣r♦t♦♦ A tt s♦s t ♦♥s♥ss ♣r♦♠ ♦r ♦♠♣tr♣s t♦t rss P ♦♠♣t r♣ G ♦ 2 ·n ♥ts rts ♥ st t♦ ♦♠♣t sr♣sG0 G1 ♦ n ♥ts t γ t ♥t ♦♥rt♦♥ ♦ A ♦rrs♣♦♥♥ t♦ t ♥ts ♥ G0 rs♣G1 ♥ t ♥t ♦♥s♥ss 0 rs♣ 1 t Sv ② r s rs r ♦r Gv② t t② ♦♥t♦♥ ♦ t ♦♥s♥ss ♣r♦♠ ♥ t ①t♦♥ Sv[γ] ♥ts ♥ Gv ♦♥ t v t S′

v ♥t ♣r① ♦ Sv s tt t ♥ts ♥ Gv ♦♥ v ♥ t ♥t ①t♦♥S′v[γ] t S′′ ♥② ② r rsr ①t♥s♦♥ ♦ t ♦♥t♥t s S′

0· S′

1 ♥ ♥ t

①t♦♥ S′′[γ] t ♥ts ♥ G0 ♦♥ 0 ♥ t ♥ts ♥ G1 ♦♥ 1 ♥ ♦♥trt♦♥ tt r♠♥t ♦♥t♦♥ ⊓⊔

♦t tt ts ♣r♦♦ s s② ①t♥ t♦ t ♠♦ t ♦ r♥ss ❬❪ ♦r ♦r ♥♦♥tr♠♥st♣♦♣t♦♥ ♣r♦t♦♦s ♥ t ♣♣ r s ♦s♥ r♥♦♠② r♥♦♠③ ♣♦♣t♦♥ ♣r♦t♦♦s

②♠♠tr ♦♥s♥ss t DejaV u

♥ ts st♦♥ ♣rs♥t ♣r♦t♦♦ tt s♦s t s②♠♠tr ♦♥s♥ss s♥ t ♦r DejaV u ss♦t♦♥ s ♣rs♥t ♥r t ♦r♠ ♦ ♣s♦♦ s q♥t t♦ t r♣rs♥tt♦♥ s♥tr♥st♦♥ rs ♦rt♠ s s♠♣ t ts ♦rrt♥ss ♣r♦♦ s ♥♦ s♣② s ♦ rsrs

❲ ♥♦t ② V t st ♦ ♥t ♦♥s♥ss s r② ♥t x s t ♦♦♥ rs ♥ st♠t♦ t ♦♥s♥ss valx ♥t② st t♦ ♥ V ♦♦♥ decidedx ♥t② false ♥ r♦♥② ♦♦♥ r doneDV

x s ♦t♣t ② t ♦r DejaV u ❲ ss♠ tt t st ♦ ♦♥s♥sss s t♦t② ♦rr ❲♥ t♦ ♥ts x ♥ y ♠t t② ♦t st t ♠♥♠♠ ♦ valx ♥ valy s

♦r s ♦ rt② t r ♦s ♥♦t ♣t r② rt♦♥

♥ st♠t ♦ t ♦♥s♥ss ♥ ♥t x s ♦♥ ts st♠t ♥ tr t ♦r DejaV u

♦t♣ts tr ♦r t ♠ts t ♥ ♥t tt s r② t ♥t t♥ sts ts decidedx t♦true

♦rt♠ ②♠♠tr ♦♥s♥ss t DejaV u

doneDVx ♦t♣t ♦ t ♦r DejaV u t x

♥t③t♦♥ valx ← ♥ V decidedx ← false

♥ ♠t♥ ♥t (x, y) ♦ t ♥ts x ♥ y valx ← min(valx, valy)

¬decidedx ∧ (doneDVx ∨ decidedy) t♥

♦♥ valx decidedx ← true

♠♠ r♠♥t♦♥ ♥ ❱t② ♦♥sr s♥ DejaV u t H st♦r② ♦ t♦r DejaV u ♦♥ ♦♠♣t r♣ G ♦s ♥r②♥ s S s ② r ♥ γ ♥ ♥t♦♥rt♦♥ ♥ ♥ t ①t♦♥ H[γ] ♦r ♦rrt ♥t x x ♥t② s ♦♥ s♦♠ ♥t ♣rs♥t ♥ γ

Pr♦♦ ♥ ♥ ♥t x ♥ ♦♥② ♦♥ ts st♠t valx ♥ s♥ r② ♣t ♦ valx ss♥s ♦ s♦♠ ♥t x ♥ ♦♥② ♦♥ ♣rs♥t ♥ γ ♦r♦r ♦r ♥② ♦rrt ♥t x s♥ S s② r x ♥t② ♥rt② ♠t t r② ♦rrt ♥t ♥ ♦tr ♦rs S s ♥t♣r① u s tt P(u) ♦♥t♥s P(v) ♦r s♦♠ v = k · c r k s t s ♦♠♣rs♥ t rs♥ts ♦ t ♥ts tt rs r♥ S base(k) = crashed(S) ♥ c s r ♦♥ t x stt base(C) = G− base(k) s r♠r ♦s ♦r r② ♦rrt ♥t ♥ss ♣r♦♣rt② ♦ t ♦rDejaV u ♠♣s tt t ♦r ♥t② ♦t♣ts true t s♦♠ ♦rrt ♥t x ts x ♥t② s♥s t♦ r♥ss r② ♦rrt ♥t ♥t② ♥rt② ♠t t x ♥ s t♦♦ ts ♥♦t r② ⊓⊔

♠♠ ♦♥sr s♥ DejaV u t H = (S, h) ♥② st♦r② t s ♥ 0, 1 ♦♥ t♦♠♣t r♣ G ♥ γ ♥ ♥t ♦♥rt♦♥ ♦♥sr ♠t♥ ♥t p ♥ S ♥♦♥ ♥ ♥t x♥ t u t ♣r① ♦ S ♥♥ t p ♥ ♥ t ①t♦♥ H[γ] rt tr u t ♦ valx t xs q t♦ t ♠♥♠♠ ♦ t ♥t s ♦ t ♥ts ♥ t s ♦ t ♣st ♦♥ ♦ p

Pr♦♦ ♦r ♥② ♦rr♥ ♦ ♠t♥ ♥t p ♥ S ♦r ♥② ♥t z ♥♦ ♥ p ♥♦t ② val(p, z)t ♦ valz rt tr p ❲ ♥♦t ② val(ǫ, z) t ♥t ♦ t ♥t z

t x, y t ♥ts ♥♦ ♥ t ♥t p t px rs♣ py t ♠♠t ♣rss♦r ♦ p♥ PastCone(p) tt ♥♦s t ♥t x rs♣ t ♥t y s ♥ ♠♠t ♣rss♦r ♦s ♥♦t①st p s t rst ♥t ♥♦♥ x rs♣ y t♥ st px = ǫ rs♣ py = ǫ ② ♥ ♥ val(p, x) = min(val(px, x), val(py, y)) ② trt♥ t val(p, x) = minvz, z ∈ base(PastCone(p)) ⊓⊔

♠♠ ♦♥sr s♥ DejaV u t H = (S, h) st♦r② ♦ t ♦r DejaV u ♦rs♦♠ ② r s S γ ♥ ♥t ♦♥rt♦♥ ♥ x′

1 x′

2 t♦ ♦rrt ♥ts ♥ t ①t♦♥

H[γ] ♦ t ♥ts x′1♥ x′

2 t ♥t p′

1♥ p′

2rs♣t② t♥ base(PastCone(p′

1)) =

base(PastCone(p′2)) ⊃ correct(S)

Pr♦♦ ❲♥ x′i s t s tr s ♦ t ♠t♥ t ♥ ♥t s r② ♦r

s t ♦r s ♦t♣t 1 t x′i ♥ ♥ tr s ♥ ♥t pi ♥ S ♥♦♥ s♦♠ ♥t

p′ ♥♦s x ♥ px p′ p t♥ p′ = px ♦r p′ = p

xi s tt pi p′i ♥ t ♦r s ♦t♣t 1 t xi r♥ pi ♥♦t tt pi ♥ p′i ♠② t s♠ ♥t♥ ♣rtr ts ♠♥s tt t ♣r① ui ♦ S tt ♥s t t ♥t pi ♥ rtt♥ ui ≃ fikicigir fi gi r ♥t ss ki s ♥t s ♦♠♣rs♥ rs ♥ts ♦♥② ♥ ci s r♦♥ t xi s tt base(ci) = G− base(ki) ♦t tt t ♣st ♦♥ ♦ p

′i ♦♠♣rss t ♣st ♦♥ ♦ pi ♥

ts t ♥ts ♦rr♥ ♥ cit G′

i = base(PastCone(p′i)) ❲ s♦ tt G′1= G′

2② ♦♥trt♦♥ ♥ ss♠ tr s ♥ ♥t

a ∈ G′1 s ♥♦t ♥ G′

2 a 6∈ G′

2 t ea t rst ♥t ♥♦♥ a ♥ S ♥ ② ♥t♦♥ t

tr ♦♥ ♦ ea ♥ t ♣st ♦♥ ♦ p′2♥♥♦t sr ♦♠♠♦♥ ♥t

t Fa t s ♦ t tr ♦♥ ♦ ea ❲ ♠ tt t ♥ts ♥ Fa rs r♥ k2 Fa ⊂base(k2) = G − base(c2) t s ♥♦t t s t♥ tr ①sts ♥ ♥t b ∈ Fa ∩ base(c2) t ttb ∈ Fa ♠♣s tt tr ①sts sq♥ ♦ ♥ts a = b0 b1 bt−1 bt = b s tt ea = (a, b1) (b1, b2) · · · (bt−1, b) r♥ S ♥ ♣rtr r② ♥t bi ♦♥s t♦ Fa

❲ s♦ b ∈ base(c2) ♥ tr s ♥ ♥t eb ♥♦♥ b tt ♦rs ♥ c2 ♥ eb ♥ (bt−1, b)♥♦ ♦♠♠♦♥ ♥t t♥ tr eb (bt−1, b) ♦r (bt−1, b) eb (bt−1, b) eb t♥ eb ♦ ♦r ♥t tr ♦♥ ♦ ea ts s ♠♣♦ss s♥ eb s♦ ♦rs ♥ t ♣st ♦♥ ♦ p′

2 r♦r eb (bt−1, b)

♥ tt s ♥ rt u2 ≃ f2k2 . . . eb . . . (bt−1, b) . . . s ♠♣s tt bt−1 ♦s ♥♦t rs r♥ k2♥ ts bt−1 ∈ base(c2) ♥ ♦tr ♦rs bt−1 ∈ Fa ∩ base(c2) b

trt♦♥ ♦ t ♣r♦s r♠♥t s♦s tt t ♥ts bi t ≥ i ≥ 0 ♦♥ t♦ Fa ∩ base(c2)♥ ♣rtr a = b0 ∈ base(c2) ⊂ G′

2 ♥ ♦♥trt♦♥ ❲ ♦♥ tt Fa ⊂ base(k2) t

a ∈ G′1= base(PastCone(p′

1)) ♥ p′

1♥♦s x′

1 ♥ x′

1∈ Fa ♥ x′

1rss r♥ k2 s s

♦♥trt♦♥ s♥ x′1s ♦rrt ♥t r♥ S s s♦s tt G′

1⊂ G′

2 ② s②♠♠tr② ♦ t ♣r♦s

rs♦♥♥ G′2⊂ G′

1♥ ts G′

1= G′

2 ♦r♦r s♥ t ♣st ♦♥ ♦ p′

1♦♥t♥s t ♦♥ c1

G′2= G′

1⊃ base(c1) = G− base(k1) ⊃ correct(S) ♥♦ ♦rrt ♥ts rss r♥ k1 ⊓⊔

♦r♠ ss DejaV u t♥ t s♦s t s②♠♠tr ♦♥s♥ss

Pr♦♦ tr♠♥t♦♥ ♥ t② ♦♥t♦♥s r sts t♥s t♦ ♠ r♠♥t ♥ s②♠♠tr②♦♥t♦♥s r sts t♥s t♦ ♠ ♥ ⊓⊔

❲st r ♦r ②♠♠tr ♦♥s♥ss

♥ ts st♦♥ ♣r♦ tt ♥② ♦r ♥ t ♥♠♦s②♥ ss ♦♥ t♦ s♦ s②♠♠tr ♦♥s♥ss♥ s t♦ ♠♣♠♥t DejaV u s t♦tr t t rsts ♦ t ♣r♦s st♦♥ ts ♣r♦s ttDejaV u s t st ♦r ♥ Mnemosyne ♥ t s♥s ♦ ❬❪ t♦ s♦ s②♠♠tr ♦♥s♥ss

♦♦♥ ♠♠ s♦s tt ♥ ♦r O ♥ t Mnemosyne ss ♥ s t♦ s♦ s②♠♠tr♦♥s♥ss t♥ t sts Scheds ♥♥ O r ssts ♦ t♦s ♥♥ DejaV u rs♣t②

♠♠ t A ♣♦♣t♦♥ ♣r♦t♦♦ tt s♦s t s②♠♠tr ♦♥s♥ss ♣r♦♠ ♦r ♦♠♣tr♣s s♥ ♥ ♦r O ♥ t Mnemosyne ss ♥ ♦r r② ♦♠♣t r♣ G ♥ r② ♥t x ♥G Scheds(O,G, x) ⊂ Scheds(DejaV u,G, x)

Pr♦♦ ♥ ts ♣r♦♦ ♦r s ♦ rt② s t s♠ ♥♦tt♦♥ ♦r t ♥t ♦ ♥ ♥t ♥t ♦rrs♣♦♥♥ ♥t stt ss♠ tt tr s s♦♠ s K ∈ Scheds(O,G, x) tt s ♥♦t ♥Scheds(DejaV u,G, x) K = k · c k ♦♠♣rss rss ♦♥② c s r ♦♥ t x ♥ base(c) s strt sr♣ ♦ G− base(k) t D = base(K) = base(k) ∪ base(c) ♥ c ♥♥♦t ♥♦ ♥ts ttrs ♥ k base(c) ♥ base(k) r s♦♥t r♣ D s strt sr♣ ♦ G t S = K · S′ ♥②② r ①t♥s♦♥ ♦ K ♦♥ D s tt crashed(S) = base(k) q♥t② correct(S) = base(c) ♥H ♥② st♦r② ♦ O t s S

♦r ♥② ♥t ♦♥rt♦♥ γ ♦♥ D ♥ ①t♦♥ H[γ] ♦ A ♥ r② ♥t ♥ correct(S) =base(c) s ② t t② ♣r♦♣rt② ♦ t ♦♥s♥ss t ♥ts t s♠ ♥t 0 rs♣1 t♥ ♦rrt ♥ts ♦♥ 0 rs♣ 1 ♥ tr ①st t♦ ♥t ♦♥rt♦♥s γ0 ♥ γ1 ♦♥D s tt ♦r s♦♠ ♥t a ♥ D γ0(a) = 0 γ1(a) = 1 ♥ ♦r r② z ∈ D − a γ0(z) = γ1(z) ♥

t ♦rrt ♥ts ♦♥ t 0 rs♣ 1 ♥ t ①t♦♥ H[γ0] rs♣ H[γ1] ♥t a ♥♥♦t♦♥ t♦ base(k) ♦trs a rss t t ♥♥♥ ♦ S ♥ ♥ ♦ ♥t t a ♦ ♥♦t♥ t s♦♥ ♦ t ♦rrt ♥ts ♥ a ∈ D − base(k) = base(c) = correct(S)

♥t x s♦ ♦♥s t♦ base(c) = correct(S) c s r ♦♥ t x s x s ♦♥ 0 ♥H[γ0] tr s♦♠ ♥t p0 ♥ S ♥ s ♦♥ 1 ♥ H[γ1] tr s♦♠ ♥t p1 ♥ S t L ♣r① ♦ Stt ♦♥t♥s p0 ♥ p1 ♥ HL t rstrt♦♥ ♦ t st♦r② H t♦ t ♣r① L ② t ♥ ♦ t ♥t①t♦♥s HL[γ0] HL[γ1] x s ♦♥ t s 0 ♥ 1 rs♣t② ❲ ♥ ①t♥ HL t♦ t ②r st♦r② H ′ ♦ O ♦♥ t r♣ G st ♦♥sr t s L · S′′ ♦r s♦♠ ② r rs rs S′′ ♦r G − crashed(L) = G − base(k) ♥ ss♦t S′′ t st♦r② ♦ O ♥ ♣rtrcorrect(H ′) = G− base(k)

♦r v ∈ 0, 1 t gv t ♥t ♦♥rt♦♥ ♦♥ G s tt gv s q t♦ γv ♦♥ D ♥ 1 sr♥ H ′[g0] ♥t x s ② t ♥ ♦ HL s ♦ t ♣st ♦♥ ♦ t ♥t ♣r♥ ts s♦♥s base(c) ♥ s♥ g0 ♥ γ0 r q ♦♥ base(c) ⊂ D x s 0 ♥ H ′[g0] ♦r s♠r rs♦♥s xs 1 ♥ H ′[g1] ♦ ♣ ♥ ♥t y ♥ G−D ♥ t g t ♥t ♦♥rt♦♥ ♦t♥ r♦♠ g0 ②♣r♠t♥ t s ♦ a ♥ y ♥ ♦tr ♦rs g(a) = g0(y) = 1 g(y) = g0(a) = γ0(a) = 0 ♥ ♦r r②b ∈ G − a, y g(b) = g0(b) ♦t t ♥t a ♥ y r ♦rrt ♥ts r♥ H ′ ♥ y ∈ G − D =G − (base(k) ∪ base(c)) ⊂ G − base(k) = correct(HL) ♥ a ∈ base(c) ⊂ G − base(k) = correct(HL) rstrt♦♥ ♦ g t♦ D s q t♦ γ1 ♥ ♥ H ′[g] t ♥t x s ♦♥ t 1 ♥ t ♦tr ♥s♥ t ♣r♦t♦♦ s♦s t s②♠♠tr ♦♥s♥ss x s ♦♥ t 0 ♥ ♦♥trt♦♥ ⊓⊔

♦♦♥ ♠♠ s♦s tt ♥ ♦r O ♥ t Mnemosyne ss ♥ s t♦ s♦ s②♠♠tr♦♥s♥ss t♥ ♥ ♥② ② r s O ♥t② ♦t♣ts 1 t s♦♠ ♦rrt ♥t

♠♠ tr ①sts ♣r♦t♦♦ A tt s♦s t s②♠♠tr ♦♥s♥ss ♣r♦♠ s♥ ♥ ♦r O ♥t Mnemosyne ss t♥ ♦r ♥② ② r s S ♦♥ r♣ G ♦r s♦♠ ♥t x ∈ correct(S) Ss s♦♠ s ♥ Scheds(O,G, x) s ♦♠♠t♥ t♦r

Pr♦♦ ❲ ♣r♦ t rst ② ♦♥trt♦♥ ss♠ tr ①sts ② r s S ♦♥ G s ttS ♦♥t♥s ♥♦ s ♥ Scheds(O,G, x) s ♦♠♠t♥ t♦r ♦r r② ♦rrt ♥t x r♦r tst♦r② H = (S, h) ②s ♦t♣ts 0 t r② ♥t s st♦r② ♦ O ♦♥ G t γv v ∈ 0, 1 t ♥t ♦♥rt♦♥ ♦♥ G ♥ t ♥ts t ♥t v ♥ ♥ t ①t♦♥ H[γv] t ♦rrt ♥ts ♦♥ t v t Hv t s♠st ♣r① ♦ H s tt t ♦rrt♥ts ♥ t ♥t ①t♦♥ Hv[γv] ♥ Sv t ♥r②♥ s

t n t ♥♠r ♦ ♥ts rts ♥ G ♥ ♦♥sr r♣ Γ ♦ 2 · n ♥ts t G0 ♥ G1 t♦ sr♣s ♦ Γ ♦ n rts s tt G0 ♥ G1 s♦♥t sts ♦ rts r♣ Gv ss♦♠♦r♣ t♦ t r♣ G ♥ ♥ tr♥st t s Sv ♦♥ G t♦ t s ♦♥ Gv ♥♦t ②S′v ♦♥sr ♥② ② r rsr s S′′ ♦♥ Γ s S′ = S′

0·S′

1·S′′ s t♥ ② r

s ♦♥ t r♣ Γ ♦♥t♦♥s ♦ t Mnemosyne ss ♦s t ♦r O t♦ rtrr② ② ts♦t♣t ♥ tr ①sts st♦r② H ′ = (S′, h) ♦ O ♦♥ Γ s tt t ♦r ♦t♣ts 0 r②rr♥ S′

0· S′

1 ♦♥sr t ♥t ♦♥rt♦♥ g ♦♥ Γ s tt r② ♥t ♥ G0 rs♣ G1 s ♥ t

♥t 0 rs♣ 1 ♥ ② ♦♥strt♦♥ ♥ t ①t♦♥ H ′[g] r② ♦rrt ♥t ♥ G0 rs♣ G1s ♦♥ t 0 rs♣ 1 s ♦♥trts t r♠♥t ♦♥t♦♥ ♦ t ♦♥s♥ss ⊓⊔

♦r♠ ❲st r DejaV u ♦r s t st ♦r ♥ t Mnemosyne ss tt♥ s t♦ s♦ s②♠♠tr ♦♥s♥ss

Pr♦♦ t O ♥ ♦r ♥ t Mnemosyne ss s tt tr s ♣♦♣t♦♥ ♣r♦t♦♦ A tt s♦st s②♠♠tr ♦♥s♥ss s♥ O t B t ♦♦♥ ♣r♦t♦♦ ts stt s♣ s 0, 1 t ♥t stts 0 ts ♥♣t s r 0, 1 ♥ t rs r (∗, ix)(∗, iy) → (ix, iy) r ∗ ♠♥s ♥② ♣♦ss stt♥tt② t ♣r♦t♦♦ B st ♦♣s t ♥♣t t♦ ts stt

t Hin = (S, hin) st♦r② ♦ O ♦♥ r♣ G ♥ γ t ♥q ♥t ♦♥rt♦♥ ♦B ♦♥ G ①t♦♥ Hin[γ] ②s st♦r② Hout = (S, hout) s ♦♦s ♦r ♥② ♦rr♥ ♦ ♠t♥♥t e = (x, y) ♥ S hout(e) = (jx, jy) r jx rs♣ jy s t st ♥♣t ♦t♣t ② O t ♥t xrs♣ ♥t y ♦r e

❲ ♠ tt Hout s st♦r② ♦ DejaV u ♦♥ G ♦r t st② ♦♥t♦♥ ss♠ tr ①sts ♥♥t e′ ♥♦♥ ♥ ♥t x ♥ S s tt hout ♦t♣ts 1 t x s ♥♦s tt tr s ♥ ♥t e

♥♦♥ x ♦r e′ s tt hin ♦t♣ts 1 t x s ♠♥s tt t ♣r① ♥♥ t e ♦♥t♥s s r♦♠ Scheds(O,G, x) s ♦♠♠t♥ t♦r ② ♠ Scheds(O,G, x) ⊂ Scheds(DejaV u,G, x)♥ t ♣r① ♥♥ t e′ ♦♥t♥s s r♦♠ Scheds(DejaV u,G, x) s ♦♠♠t♥ t♦r

❲ ♥♦ ♣r♦ tt t ♥ss ♦♥t♦♥ ♦s ♥ Hout ♦t tt t s S ♥ ② r♦r r② ♦rrt ♥t x S ♦♥t♥s ♥♥t② ♠♥② ss r♦♠ Scheds(DejaV u,G, x) s ♦♠♠t♥t♦rs s t ♥ss ♦♥t♦♥ s sts hout ♦t♣ts 1 t ♦♥ ♦rrt ♥t t st t A t st ♦ ♦rrt ♥ts x s tt S ♦♥t♥s s r♦♠ Scheds(O,G, x) s ♦♠♠t♥ t♦r ②♠ t st A s ♥♦t ♠♣t② ♥ ② t ♥ss ♦♥t♦♥ ♦r Hin tr s s♦♠ ♦rrt ♥t x

♥ A s tt t st♦r② Hin ♦t♣ts 1 t x ♥ s♦♠ ♥t e ♥ S r♦r ② t ♥t♦♥ ♦ B tst♦r② ♦t♣ts 1 t x ♥ s♦♠ ♥t ♥ S t ♥①t ♥t tr e ♥♦♥ x s t st♦r② Hout s st♦r② ♦ DejaV u ⊓⊔

sss♦♥ ♦♥ rs ♠♣♠♥tt♦♥

♥ ts st♦♥ sss t ♠♣♠♥tt♦♥ ♦ t ♦rs ♥ t t♦r② ♥ ♥ ♥ ♣rtr ♦t DejaV u ♦r s ss ♦♥r♥s t rqr♠♥ts ①tr♥ t♦ t PP ♠♦ r s②st♠ ♦♠♦ ♥ts s♦ sts② ♥ ♦rr t♦ ♠ ♦rs

♥ ♥ ♦♥sr t♦ t②♣s ♦ rqr♠♥ts ♦s tt ♥♦ ♥ ♥trt♦♥ t t ①t ♣r♦t♦♦♥ t♦s r ♥♣♥♥t ❲ t ♦r♠r str♦♥ ♥ t ttr

♦t rst tt t ♦r ♥ ❬❪ Ω? ♥s str♦♥ rqr♠♥t ♥ ♦♥ ♠st t♦♦sr ♦② t ①t♦♥s ♦ ♣r♦t♦♦ s tr s ♥♦ ♥tr ♦srt♦r② ts rqr♠♥t ♥ssttst st t rr t ♦ s♥♣s♦ts ② ♦ ♣r♦r t s♠s qt t ♥ s②st♠ ♥ t ♥ts r ♥st♥s ♥ ♥♦ ① ♦③t♦♥ ①t s♠ r♠r ♦s ♦r t s♥tt♦r ♦ ❬❪ r ♥ ♦r s♥s s♦♠ ♣rtr ♥t stt s ♠ss♥ ♥ ♦♥rt♦♥ ② t② t ♥♦② t♥ t s♥ tt♦r ♥ Ω? s ♠♥t♦♥ ♥ ❬❪

rqr♠♥ts ♥♦♥ s♣ ♥t ♥trt♥ t t ♥ts ①t♥ ♣r♦t♦♦ s s♦ str♦♥ rqr♠♥t ♥ ts t♦r② ♦♥ ♦ t♥ ♦ s♣ ♣♦r ♥t s stt♦♥ ♥ ♠♥②sts s t♦ tt t ♥♥♥ ♦ ♥ ①t♦♥ ♥ r♦♠ ts ♣♦♥t t♦ s t ♥♣t ♦♥② ♣rtr ♥t ♥ ♠ t ♦trs ♦♥ ts s rqr♠♥t s str♦♥ s t sstt♦♥ s t♦ ♥trt t t ♣r♦t♦♦ ♦r♦r t rqrs t ♥tr♦t♦♥ ♦ st♥s ♥t ♥t ♥t♦r

♦ ♦♥sr t DejaV u ♦r ♥tr♦ s② t tts tr ♦r ♥♦t s♦♠ st②s♠ ♦ ♥trt♦♥s s ♣rs♥t ♥ t ♣st s s t s ♥r tr s ♥♦ ♥ t♦ tt s s♠ t t ♣rs t♠ ♥ t ♣♣rs ♦r♦r t r♥ss ss♠♣t♦♥ ♥♦s tt t ♥ssr②♥t② ♣♣r ♥ ♥② s s t tt♦♥ ♥ ♦♥ t r② s♠♣ ♥ rst ♠♥s tss t♦ t s♥t② ♦♥ t♠ t♦ sr tt rqr st② s♠ s ♣♣♥ ♦r ①♠♣♦r♥ t♦ t ♥t♦♥ ♦ DejaV u t s♥t t♦ t ♥t t ♣♦ss ♥trt♦♥s ♣♣♥ rqr♠♥t ♥ ♠♣♠♥t ♥ s②st♠s ♥ tr s s♦♠ ♦ s②♥r♦♥② ♥ t♥trt♦♥s t♥ ♥ts ②♥ ♥ ♣♣r ♦♥ ♦♥ t t♠ ♦r ♥♦♥t② ♥t t♦ ♦♠♠♥tt r② ♦tr ♥♦♥t② ♥t s ♣♣r ♦♥ ♥ ♦t♥ r♦♠ t ♣r♠trs ♦ r♥♦♠ t ♥ts ♠♦ r♥♦♠② s ♣♣r ♦♥ ♥s t ♥ ♣s ♥ stt♦♥♦ r♦st s♥ t t ♥ ♦ ♣s ♥ ♣♣r ♦♥ t♦ t ①♣t ♥♠r ♦ rss s s♦rqr ♥ ♥ ♥ ♥t rs rt♥ ♥♠r ♦ t♠s t s♥ ♣♥♥ ♦♥ t ♣r♠trs♦ t r♥♦♠ t ♣♣r ♦♥ ♦ t ①♣t ♥♠r ♦ rss ♥ ♠② ♦tr rtrsts ♦t s②st♠ t ♥

r♦st stt♦♥ ♥ ♠♣♠♥t ♦r ♥st♥ ♦♥ stt ♥ t s♥s ♥ s② r♥t ♣r♦t♦♦s ♦♥ r♥t ♥t♦rs ♥♣♥♥t② ♦t tt t r♦st stt♦♥ r s r②r♥t r♦♠ t s stt♦♥ ♥ t ♣r♥ ♣rr♣ t s ♥♦t t♦ ♥trt ♥ ♥② ② t t ♣r♦t♦♦♥ ②s t♥ s♠♣ ♠♣♠♥tt♦♥

r♥s

r ❲ ♥ ♥ ♦ ❯s♥ t rtt r tt♦r ♦r qs♥t r ♦♠♠♥t♦♥♥ ♦♥s♥ss ♥ ♣rtt♦♥ ♥t♦rs ♦r ♦♠♣t

♥♥ s♣♥s ❩ ♠ sr ♥ Prt ♦♠♣tt♦♥ ♥ ♥t♦rs ♦ ♣ss② ♠♦♥tstt s♥s♦rs ♥ P ♣s

♥♥ s♣♥s ❩ ♠ sr ♥ Prt ♦♠♣tt♦♥ ♥ ♥t♦rs ♦ ♣ss② ♠♦♥tstt s♥s♦rs strt ♦♠♣t♥

♥♥ s♣♥s ♥ s♥stt s♠♣ ♣♦♣t♦♥ ♣r♦t♦♦ ♦r st r♦st ♣♣r♦①♠t ♠♦rt②strt ♦♠♣t♥

♥♥ s♣♥s s♥stt ♥ ♣♣rt ♦♠♣tt♦♥ ♣♦r ♦ ♣♦♣t♦♥ ♣r♦t♦♦s strt ♦♠♣t♥

♥♥ sr ♥ ♥ t③♥ ♦♥s♥ss ♥ ♠♦ ♥t♦rs ♥ ♣s

tt② ♥ ❲ strt ♦♠♣t♥ r s ♥ ♥ P♦ sr② ♦ ♣♥t② sss ♥ ♠♦ rss ♥t♦rs ♥

r♣♦rt qr P ♥r ♥ r♠♥ st③♥ r t♦♥ ♥ ♣♦♣t♦♥ ♣r♦t♦♦s ♦r rtrr②

♦♠♠♥t♦♥ r♣s ♥ P ♣s ♦♥♥t ♥ ②♥ ♣r ♦ ♥♦♥②♠t② ♣t♠ ♦♥s♥ss s♣t s②♥r♦♥② rs ♥ ♥♦♥②♠t②

♥ ♣s ♦♥♥t ♥ ②♥ ♥♦♥②♠♦s s②♥r♦♥♦s s②st♠s s ♦ r tt♦rs ♥ ♣s

❩ ♦③ ♥ rrs ♥♦♥②♠t② rs tt♦rs ♥ ♦♥s♥ss ♥ r♣♦rt ❩ ♦③ ♥ rrs r ♥♥♦♥♠♥t ♥♦♥②♠t② rs tt♦rs ♥ ♦♥s♥ss ♥ ♣s

♥r ❱ ③♦s ♥ ♦ st r tt♦r ♦r s♦♥ ♦♥s♥ss

♥r ♥ ♦ ❯♥r r tt♦rs ♦r r strt s②st♠s

♦rtés rtí♥③ rts ♥ ♦ ♦r ♦♥tr♦ ♦r ♠♦ s♥s♥ ♥t♦rs ♦♦ts

♥ t♦♠t♦♥ ♣♦rtt ♦♥♥r rr♦ ♥ ♠♥♥ st r tt♦r ♦r ♠ss

♣ss♥ str♠♥t ♥ ♣s sr ♥ ♥ st③♥ r t♦♥ ♥ ♥t♦rs ♦ ♥tstt ♥♦♥②♠♦s ♥ts ♥ P

♣s sr ②♥ ♥ Ptrs♦♥ ♠♣♦sst② ♦ ♦♥s♥ss t ♦♥ t② ♣r♦ss ♦r♥ ♦

t ♣r t♦♥ ♥ ❲ r ②♥r♦♥③ ♠t♣ s♣rt r♦tt♦♥s t♦♠t ②♥ strt ♦rt♠s ♦r♥ ♠♥♥ t③♥♥s ♥ P ♣rs r♠♥t♥ ♣♦♣t♦♥ ♣r♦t♦♦s s♦♠ ♠♥♠ ♦

♥♦ ss♠♣t♦♥s ♥ ♣s ♦sté♦ s♠ ②♥ ♥ rrs ♦♠♥ ♣♦r ♦ ♦♥t♦♥s ♥ ♥♦r♠t♦♥ ♦♥

rs t♦ s♦ s②♥r♦♥♦s st r♠♥t ♦♠♣t ♦sté♦ s♠ ②♥ ♥ rrs ♥ t ♦♠♣tt② ♣♦r ♥ t r♦st♥ss ♦ st

r♠♥t♦r♥t r tt♦r sss strt ♦♠♣t♥ tr ① ♥ rr② ♣② t♦ ♦♠♠♥ts ♦♥ ♦♥s♥ss ♥ ♦♦♣rt♦♥ ♥ ♥t♦r

♠t♥t s②st♠s Pr♦♥s ♦ t tr ♥ ♠♠ ♦♥s♥ss trs ♦r ♥s♦r t♦rs ♥ strt ♥s♦r s♦♥ t

♦♥ s♦♥ ♥ ♦♥tr♦ ♥ r ♦♥tr♦ ♦♥ ♣s

s♠♥ strt ♦♠♣tt♦♥ ♥ ❲rss ♥ ②♥♠ t♦rs P tss ♣rt♠♥t ♦tr ♥♥r♥ ♥ ♦♠♣tr ♥

❲ ♥ ❲ r ♥ t♥s sr② ♦ ♦♥s♥ss ♣r♦♠s ♥ ♠t♥t ♦♦r♥t♦♥ ♥♠r♥ ♦♥tr♦ ♦♥r♥ ♣s

♥tr♦t♦♥ t♦ strt ♦rt♠s ♥ ♠r ❯♥rst② Prss ❲ ❳ ❳ ♥ ♥ rs st♦st ♦rt♠ ♦r s♦r♥③t♦♥ ♦ t♦♥♦♠♦s sr♠s ♥ Pr♦

t ♦♥ s♦♥ ♥ ♦♥tr♦ ♥ r ♦♥tr♦ ♦♥ ♣s