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Table of Contents

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  (EURNH) 2 3DXO ǫ← (WKH YDVH)Y  ǫ →  X I I22 I21 F ² SDJH  3DXO ERXJKW D QHZ YDVH ǫ← (D QHZ YDVH)Y  ǫ → 3DXOX  (ERXJKW)I2 I22 I21

D ² SDJH  1JDUUNXQWX DUL NDQ\LUU ǫ→ NDQ\LUUY  WX→ 1JDUUNXQX  DULI2 I22 I21

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E ² SDJH  1JDUUNXQ DULOL NDQ\LUUNX NX→ 2

I2 > E

NDQ\LUUO  ǫ ← 1 1JDUUNXQX  (DULOL)I2−1 I1

F ² SDJH  ,Q~ JD KDVtWWH LPiVX JD← 1 ,Q~X 

I1

ǫ→ LPiVXE  KDVtWWHI1 I1

G ² SDJH  ,Q~ ZD KDVtWWH LPiVX ZD→ 1 ,Q~X 

I1

LPiVXE  KDVtWWHI1 ǫ→ I1

H ² SDJH  <XNt ZD VLUyL GHVX GHVXE  ZD→ <XNtX  VLUyLA ǫ← I11 I11 I ² SDJH  6RQy EjUD ZD DNiL GHVX ǫ← GHVXE  ZD→ ǫ ← 6RQyA  EDUjX  DNiLA I11 I11 X

D ² SDJH  , VZHDW ǫ→ ,  (VZHDW)I1 I11 X

E ² SDJH  6DODNVKD ǫ← 6DX  ODNVKDI1 I11

F ² SDJH  LR KR VXGDWR ǫ→ LR X  (KR VXGDWR)I2 I21

D ² SDJH  3DROR KD FRUVR SHU XQ¶RUD   FRUV→ RA  ǫ ←  ǫ→ ǫ ← XQA  D←   RUO  ǫ ← A O O I22 Y I22 A > Yaux E > I22 O →    2   ǫ 2 ( 3DROR)X   FRUU Yaux  Imean I1 → ← ǫ 2 R X  3DROX  KD 2 I I1 X

SHU←



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E ² SDJH  3DROR q FRUVR D FDVD D←

E > I22

D←   FDVO  ǫ →  ǫ→ FRUV→ RA  O O A I22 A > Yaux → ←   )   ǫ 2 Y  ǫ 2 ( 3DROR X FRUUI2  Yaux  mean

I2 I1 ← ǫ← 2 RX X  3DROX  qI2 I1

mean

F ² SDJH  3DROR FRUVH ǫ→ R←   3DROX  FRUVHI2 I11 X X

D ² SDJH  EDOO WDEOH ǫ← WDEOHE  EDOOI2 I22 1

E ² SDJH  WKH EDOO LV RQ WKH WDEOH → → RQ→ )  ǫ ← (   WDEOH)O  (  EDOO)X   WKH D EDOO LV 2 E > A D  (
ǫ I2 
ǫ Y  Y I2 WKH I 2

D ² SDJH  PRUH PLON ǫ← PRUHA  PLONY I22

E ² SDJH  , ZDQW PRUH PLON

 2 ǫ← ǫ ← PRUHA  PLONY 
ǫ → ,   ZDQW I I21 X I22 Y D ² SDJH  PRUH JUDSHV ǫ← PRUHA  JUDSHVY I22

E ² SDJH  , ZDQW PRUH JUDSHV

 2 ,   ǫ ← PRUHA  JUDSHVY 
ǫ → ǫ← ZDQW I I21 X I22 Y D ² SDJH  DQFRUD PHOD ǫ← DQFRUDA  PHODY I22

1

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E ² SDJH  LR YRJOLR DQFRUD PHOD  2 ǫ← ǫ ← DQFRUDA  PHODY 
ǫ → LR  YRJOLR I I21 X  I22 Y D ² SDJH  DQFRUD SDVWD ǫ← DQFRUDA  SDVWDY I22

E ² SDJH  LR YRJOLR DQFRUD SDVWD  2 LR  YRJOLR ǫ← ǫ ← DQFRUDA  SDVWDY 
ǫ → I I21 X  I22 Y D ² SDJH  MXLFH JRQH ǫ← ǫ → JRQHA  MXLFHX  govI1 I1 X

E ² SDJH  WKH MXLFH LV JRQH

  MXLFHX   WKH ǫ← ǫ → (JRQH)I > A  ǫ ← LV 1 X  A I I11 X D ² SDJH  R QRIR SHSH ǫ← SHSHX  R QRIRI1 I11

E ² SDJH  R QRIR LD SHSH LD→ 1 SHSHX  R QRIRI1

I1

D ² SDJH  IRUWHV YHQWRV ǫ ← (IRUWHV)A  (YHQWRV)O O

E ² SDJH  YHQWRV IRUWHV ǫ → (IRUWHV)A  (YHQWRV)O O

D ² SDJH  QRVWUD PHPRULD D← (QRVWU)A  (PHPRULD)O

O

E ² SDJH  PHPRULD QRVWUL L→ (QRVWU)A  (PHPRULD)O

O





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D ² SDJH  DPRU SDWULV LV↔ SDWUA  DPRUO

O

E ² SDJH  DPRU SDWULV LV↔ SDWUA  DPRUO

O

D ² SDJH  WKH HOHSKDQW ǫ ← WKHA  HOHSKDQWO O

E ² SDJH  WKDW HOHSKDQW ǫ → WKDWA  HOHSKDQWO O

F ² SDJH  ,¶YH VHHQ WKH RQH ǫ ← WKHA  RQHO O

H ² SDJH  ,¶YH VHHQ WKDW RQH ǫ → WKDWA  RQHO O

I ² SDJH  ,¶YH VHHQ WKDW ǫ → WKDWA  O O

D ² SDJH  'RPDQL YHQJR ǫ ← GRPDQLE  ǫ → LR X  YHQJRI1 I I11

E ² SDJH  9HQJR GRPDQL LR X  YHQJRI1 ǫ ← GRPDQLE  ǫ → I I11 F ² SDJH  4XDQGR YHUUDL" ǫ→ (TXDQGR)E  ǫ → WX X  YHUUDLI1 I11 I11

 ² SDJH  WKDW KLJKO\ VHQVLWLYH WRSLF ǫ → WKDWA  ǫ ← ǫ → O\→ D  KLJKA  (VHQVLWLYH). . . > A  O O A E WRSLFO

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 ² SDJH  XQ YUDLPHQW EHDX SD\VDJH ǫ ← XQA  ǫ ← ǫ ← PHQW→ D  YUDLA  EHDXA  O O A E

(SD\VDJH). . . > O

$ 7UDQVIRUPDWLRQV D ² SDJH  /L]D ZDON /L]DX  ZDONI1 ǫ→ I11 E ² SDJH  /L]D ZDONV ǫ→ /L]DX  (ZDONV)I1 I11

F ² SDJH  /L]D >noun: proper, singular@ ZDONV >verb: [tense: simple present: [3rd person, singular]]@ ǫ→ /L]DX >noun@ >singular@  ǫ ← V  ZDONI1 rd I11 I1 I >3 person@ >singular@

D ² SDJH  /L]D ZDONV DQG 3DXO ZDONV DQG↔ 

X

V  ZDONI1 V  ZDONI1  ǫ → /L]DX  ǫ ← ǫ→ 3DXOX  ǫ ← I11 I1 I I11 I1 I

E ² SDJH  /L]D DQG 3DXO ZDON ǫ→ DQG↔ 3DXOO  /L]DO  ZDONI1 X I11

D ² SDJH  /L]D LV D TXLHW JLUO /L]DX  LVI2 ǫ→  ǫ ← D  ǫ → TXLHWA  JLUOY  ǫ ← I21 I22 Y A Y E ² SDJH  /L]D LV TXLHW ǫ→  ǫ → TXLHWA  Y  ǫ ← /L]DX  LVI2 I22 Y I21

F ² SDJH  %H D TXLHW JLUO → 2 

I2

ǫ ← DA  ǫ → TXLHWA  JLUOY  ǫ ←   EHI2 Y Y I21 X



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G ² SDJH  %H TXLHW → 2 

ǫ → TXLHWA  Y  ǫ ←   EHI2 Y I21 X

I2

D ² SDJH  3DXO LV VWXG\LQJ PDWKV

   ǫ ← PDWKVYmean  ǫ →  3DXO ǫ→  LQJ→ Xmean  VWXG\I2  Yaux D I22 I21 I22 ǫ← 3DXOXaux  LVI2 I21 aux E ² SDJH  3DXO LV JRLQJ WR VWXG\ PDWKV

  VWXG\ 2  3DXO  LQJ→   WR← ǫ ←   ǫ ← PDWKVY  ǫ → ǫ→ X Yaux D I22 Yaux+ D I22 I I21 I22 → ←   ǫ 2 3DXOX  JRI2  ǫ 2 3DXOXaux  LVI2 aux+ I1 I1 aux+ aux F ² SDJH  0DWKV LV SUHSDUHG E\ 3DXO E\→

E>I

→ 3DXOXact  ǫ → 2  ǫA > Y I2

aux

G→ 2

I2 > A

  PDWKV Yact 

 ǫ→ 3DXO  Xact  SUHSDUHI2  I22 ←  PDWKV  Yact  ǫ I2 PDWKVYact { Xpass  LVI2aux 1

G ² SDJH  0DWKV LV SUHSDUHG →   ǫ→  ǫ→ G→  PDWKV PDWKV Yact  ǫ I2 Xact  SUHSDUHI2   Yact  I22 A > Yaux I22 > A 2 ǫ← 2 PDWKVYact { Xpass  LVI2 I1

aux

H ² SDJH  3DXO LV SUHSDUHG

 3DXO ǫ→  ǫ→ G→   ǫ →  Xact  SUHSDUHI2  Yact  I22 A > Yaux I22 > A Yact I22 ǫ← 3DXOXact  LVI2 I21 aux I ² SDJH  %H SUHSDUHG

  SUHSDUHI2  Yact  ǫ→  ǫ→ G→   ǫ → \RX I22 A > Yaux I22 > A Yact I22  Xact ← X  EH 2 \RX ǫ 2  act Iaux I1

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D ² SDJH  7KH VKDK VOHSW LQ WKH EHGV LQ→ 2 (WKH EHGV)Y 

I2

ǫ→ (WKH VKDK)X  VOHSWI2 I21

E ² SDJH  7KH EHGV ZHUH VOHSW LQ E\ WKH VKDK

(() WKH(EHGV VOHSW LQ→ (( (WKH VKDK)Xact  ǫ →  ǫ→ Yact  I22 > A I22 A > Yaux (()  2    WKH(VKDK VOHHS ǫ→ (( Xact   Yaux  Imean I21 ← ǫ 2 (WKH EHGV)Y  ZHUH 2 act I

E\→

E>I

I1

aux

F ² SDJH  6KDK¶V VOHHSLQJ ·V→

A>O

6KDKO  LQJ→   VOHHSI O D

D ² SDJH  $ FDPHO VWHSSHG RQ WKH SDFNDJH ((()Y  ǫ → ($ FDPHO)X  VWHSI2 SHG RQ→ (WKH(SDFNDJH I2 I2 ( 1

2

E ² SDJH  7KH SDFNDJH ZDV VWHSSHG RQ E\ D FDPHO

((()Y  SHG RQ→ (WKH(SDFNDJH (D FDPHO)Xact  ǫ →  ǫ→ act I22 > A ( I22 A > Yaux (() WKH(FDPHO  VWHS    ǫ→ 2 2 (( Xact Yaux Imean I1 ǫ←  ( WKH SDFNDJH )  ZDV 2 2 Yact I

E\→

E>I

I1

aux

F ² SDJH  7KH FDPHO¶V VWHSSLQJ ·V→ (7KH FDPHO)O  SLQJ→ D  VWHSI

O

O

ǫ ← 7KH ·V→ FDPHOO  SLQJ→ D  VWHSI O O O

D ² SDJH  7KH LFH FXEH PHOWHG LQ WKH RYHQ → WKH  RYHQ  LQ← 2 ǫ A O I2

O

(PHOWHG)I2

ǫ← ǫ → 7KHA  ǫ → LFHO > A  FXEHO  O I21 O

F ² SDJH  7KH PHOWLQJ RI WKH LFH FXEH RI← ǫ → WKHA 

O

O

  PHOWI WKHA  LQJ→ ǫ→ LFHO  FXEHO  ǫ ← O D O A>O





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D ² SDJH  7KH WRRWKSDVWH RR]HG LQWR WKH VLQN ← WKH  VLQN  LQ← 2 ǫ A O I2

O

ǫ← ǫ ← 7KHA  (WRRWKSDVWH)O  (RR]HG)I2 I21 O

F ² SDJH  7KH RR]H RI WKH WRRWKSDVWH RI→ ǫ → WKHA  (WRRWKSDVWH)O 

O

O

ǫ ← 7KHA  ǫ → D  RR]HI O O

D ² SDJH  /LWWOH 5HG 5LGLQJ +RRG YDQLVKHG LQ WKH ZRRGV LQ← ǫ ← WKHA  (ZRRGV)O 

I

O

ǫ→ (/LWWOH 5HG 5LGLQJ +RRG)O  (YDQLVKHG)I1 I11

F ² SDJH  7KH YDQLVKLQJ RI /LWWOH 5HG 5LGLQJ +RRG RI← (/LWWOH 5HG 5LGLQJ +RRG)O 

O

ǫ ← WKHA  LQJ← D  YDQLVKI O O

D ² SDJH  ,W¶V D FDU ǫ ← D  FDUO  ·→ ,W  LV 2 ǫ← I21 O I I22 O A E ² SDJH  7KDW¶V WKH HOHSKDQW WKDWO  LVI2 ǫ ← WKHA  HOHSKDQWO  ·→ ǫ← I21 I22 O D ² SDJH  7KDW¶V P\ PXP WKDWO  LVI2 ǫ→ ǫ → P\A  PXPO  ·← I21 I22 O E ² SDJH  ,W¶V 3DXO¶V EURWKHU ǫ→ ·V→ 3DXOO  EURWKHUO  ·← ,W  LV 2 I22 O I21 O I

F ² SDJH  7KHUH DUH P\ VKRHV ǫ→ (P\ VKRHV)Y  ǫ ←   WK← 2 HUHE  DUHI I22 I21 X E>I

G ² SDJH  7KHUH LV /L]D (/L]D)Y  ǫ ←   WK← 2 HUHE  LVI ǫ→ I22 I21 X E>I

([DPSOHV

H ² SDJH  +HUH¶V D JLIW ǫ→ (D JLIW)Y  ǫ ←   K← HUHE  LVI I22 I21 X E > I2

D ² SDJH  0\ VKRHV DUH WKHUH WK→

E > I22

← HUHE  ǫ → 2 Y  ǫ 2 (0\ VKRHV)X  DUHI2 I2

I1

E ² SDJH  /L]D LV WKHUH WK→

E > I22

← HUHE  ǫ → 2 Y  ǫ 2 (/L]D)X  LVI2 I2

I1

F ² SDJH  3DXO LV KHUH K→

E > I22

← HUHE  ǫ → 2 Y  ǫ 2 3DXOX  LVI2 I2

I1

 ² SDJH  NHGLP YDU PY  ǫ ← NHGLX  YDUI2 ǫ→ I22 I21 D ² SDJH  3DXO KDV D %HQWOH\ ǫ← (D %HQWOH\)O  ǫ → 3DXOO  KDVI2 I22 I21

E ² SDJH  3DXO KDV JRW D %HQWOH\ ǫ←  ǫ → ǫ → (D %HQWOH\)Ymean  I22 Yaux I22 > A →  ǫ 2  3DXOX  JRWI2  Yaux  mean I1 mean ǫ→ 2 3DXOXaux  KDVI2 I1 aux

F ² SDJH  3DXO KDV JRW D )RUG (D )RUG)Ymean   ǫ → ǫ → ǫ← I22 Yaux I22 > A →  ǫ 2  3DXO Xmean  JRWI2mean  Yaux  I1 ǫ→ 3DXOXaux  KDVI2 I21 aux





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G ² SDJH  3DXO KDV ZDVKHG KLV FDU ǫ←  ǫ → HG→ (KLV FDU)Ymean  I22 Yaux I22 > A →  3DXO ǫ 2  Xmean  ZDVKI2mean  Yaux  I1 ǫ→ 3DXOXaux  KDVI2 I21 aux

H ² SDJH  3DXO JRW D )RUG ǫ← (D )RUG)O  ǫ → 3DXOO  JRWI2 I22 I21

I ² SDJH  3DXO ZDVKHG KLV FDU ǫ← (KLV FDU)O  ǫ → 3DXOO  (ZDVKHG)I2 I22 I21

D ² SDJH  3DROD KD FRUVR OD PDUDWRQD HQWUR TXDWWUR RUH   RUO  ǫ ← TXDWWURA  H← O O E > I22 O ǫ←  ǫ → FRUV→ RA  ǫ ← (OD PDUDWRQD)Y  A I22 Yaux I22 )  FRU   2  Yaux  ǫ→ ( 3DROD UH X Imean I21 → ← ǫ 2 D X  3DROX  KD 2 I I1 X

HQWUR←

E ² SDJH  3DROD q FRUVD D FDVD D←

D←   FDVO  ǫ →  ǫ → V→ D  O O I22 Yaux A A )  FRU   2  Yaux  ǫ→   ǫ ← ( 3DROD UH X Imean I22 Ymean I21 ← ← ǫ 2 D X  3DROX  q 2 X I

E > I22

I1

D ² SDJH  0DULR KD ODYDWR OD FDPLFLD (3DROR)X  KDI2  ǫ ← (ODYDWR OD FDPLFLD)I2 > A  Yaux  ǫ → ǫ← I21 I22 Yaux E ² SDJH  0DULR KD OD FDPLFLD ODYDWD ǫ←  ǫ → (ODYDWD)I2 > A  (OD FDPLFLD)Yaux  ǫ → (3DROR)X  KDI2 I22 Yaux I21

([DPSOHV



 ² SDJH  0HHWLQJ SHRSOH LV HDV\ (  ǫ ← LQJ→   ǫ ← SHRSOH ((( VRPHWKLQJ ǫ→ ǫ → HDV\A  ( Ybe Ymeet  Xbe D I22 I21 I22 Yaux → ǫ 2 X  PHHWI2  LVI2 meet I1

meet

be

D ² SDJH  7KH PDFKLQH QHHGV IL[LQJ () → VRPHERG\ ((() ((( WKH(PDFKLQH ǫ← LQJ→ (( Xfix  IL[I2  Yfix  ǫ I2 (( I22 I22 > Yneed fix 1 → ǫ 2 (WKH PDFKLQH)X  (QHHGV)I2 need I1

need

E ² SDJH  0\ KRXVH UHTXLUHV SDLQWLQJ () → VRPHERG\ ) ((( ǫ← LQJ→ (P\KRXVH Xpaint  SDLQWI2 Ypaint  ǫ I2 (( I2 I2 > Yreq  2

2



paint

1

ǫ→ (0\ KRXVH)Xreq  (UHTXLUHV)I2 I21 req

 ² SDJH  3DXO LV D ZRUNLQJ FODVV KHUR

) ǫ→ 3DXOO1  ǫ ← ǫ ← D  ǫ ← ǫ ← LQJ→ ( D ZRUN Ywork  I22 I21 Ybe A A > Ybe O I22 > O  →  ) ǫ 2 ( ZRUNHU Xwork  ZRUNI2  FODVVO  KHURYbe  (LV)I2 I1 work D ² SDJH  3DXO LV D GHDG PDQ ZDONLQJ

→  LQJ→  ZK ǫ→ 3DXOO1  ǫ ← ǫ ← D   RX  ZDONI1  A O walk I22 I21 Ybe A walk → GHDGI > A  PDQYbe  (LV)I2 ǫ Ybe  ² SDJH  :LWKRXW WHOOLQJ KLP /L]D FDQQRW EH KDSS\ (  ǫ ←   ((( VRPHWKLQJ ( KLPZ  ǫ ← ZLWKRXW← LQJ→ Y I3 /L]DX  WHOOI3  («)I E>I I3 I3 > O 3

2

1

D ² SDJH  $YHQGR WHOHIRQDWR /LVD 3DROR SLDQVH     3DROR H←  ǫ → D{DWR→ QGR→ Y I22 > A I22 > E > I I22 Yaux
→ ǫ 2 /LVDX  WHOHIRQI2  Yaux  mean I1 mean   /LVD  ǫ→  K D { DY   2  Xaux Iaux (3DROR SLDQVH)I I21

3



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F ² SDJH  (VVHQGR SDUWLWD /LVD 3DROR SLDQVH QGR→ 2

H→  ǫ → LW→ D  ǫ → /LVDXmean  SDUWI1  Yaux  I2 > E > I I22 Yaux I11 > A A I11 mean  ǫ← 2 /LVDXaux  HVVI2  (3DROR SLDQVH)I I1 aux

G ² SDJH  3DUWLWD /LVD 3DROR SLDQVH

→  QGR 2

D  ǫ → /LVDXmean  SDUWI1  Yaux  H→  ǫ → LW→ I11 > A A I11 I2 > E > I I22 Yaux mean  2  (3DROR SLDQVH)I  HVV ǫ← 2 /LVDXaux   I I1 aux

D ² SDJH  $ /LVD SLDFH EDOODUH D→ 2 /LVDY 

I2

  EDOO 2  SLDFH 2 /LVD   ǫ →  ǫ← DUH→ X Xpiace Y I21 I I I21

 ² SDJH  3DROR YXROH SURYDUH DG DQGDUH D IDUH XQD YDFDQ]D ǫ←  DUH→  DG DUH→  D DUH→ (XQD YDFDQ]D)Y  I22 I22 > Y I22 > Y I22 > Y   DQG 2  ǫ →    SURY 2    ǫ→ 3DRORX  II2  ǫ → 3DROR  3DROR X X I I I21 I21 I21 → ǫ 2 3DRORX  YXROH 2 I I1

 ² SDJH  /DPHQWR HOHV WHUHP SHUGLGR R FRPERLR ǫ → LGR→ (R FRPERLR)Yperd  ǫ→ HP→ I22 > A I22 I2 > Ylam. Yter →  ǫ→  HOHV Xperd  SHUGI2  Yter  ǫ I2 HOHVXter  WHUI2ter  (ODPHQWR)I21 I21 perd 1 D ² SDJH  0DULR KD XQD FDPLFLD ODYDELOH D VHFFR

((()  (FDPLFLD ǫ←  ǫ → D→ (VHFFR)I > A  ELO→ H  ǫ → DI  ǫ → XQD (( Y A A I22 I22 Y A I22 → → ǫ 2 0DULRX  ODY 2  (XQD FDPLFLD)Y  ǫ 2 0DULRX  KD 2 I I I1 I1  ² SDJH  /H OLYUH GH 3LHUUH ǫ → GH→ D  3LHUUHO  (OH OLYUH)O O A

 ² SDJH  8QH WDVVH GH WKp ǫ → GH→ D  WKpO  (XQH WDVVH)O O A

([DPSOHV

 ² SDJH  8QH VDOOH GH FODVVH ǫ → GH→ D  FODVVHO  (XQH VDOOH)O O A

 ² SDJH  -H pFULUH GH OD PDLQ JDXFKH GH← (OD PDLQ JDXFKH)O  (MH pFULUH)I

E

 ² SDJH  -H UpSqWH GH PpPRLUH GH← (PpPRLUH)O  (MH UpSqWH)I

E

 ² SDJH  /H WUDLQ SDUWL GH 3DULV«

) ǫ → GH← 3DULVO  ( SDUWL I > A  (OH WUDLQ)O E O  ² SDJH  /L]D ZLSHG WKH WDEOH FOHDQ

  ǫ→   FOHDQ 2  (WKH WDEOH)  →  → D  ǫ → ǫ←  ZK HG  LFK VKH Y I2  X Y Y A I I22 I22 1 → ǫ 2 /L]DX  (ZLSHG) 2 I I1 D ² SDJH  3DXO KDV JRW KLV FDU ZDVKHG

 → HG→ D   ǫ→ ǫ →  3DXO ZK ǫ← ǫ →  Xgot  JRWI2got   Ygot A I22 Yaux Yaux I22 > A I21 → →  ǫ 2  LFKY  ǫ 2 KHX  ZDVK 2  (KLV FDU)Y  got I I2 I1 → ǫ 2 3DXOX  KDV 2 aux Iaux I1 E ² SDJH  +H GURYH KHU PDG

→ KHUY  ZK ǫ← ǫ → +HXmean  GURYHI2   Ymean mean I22 I21 mean → ← ←   D  ǫ 2 PDGA > Y  ǫ 2  HG VKHX   LV 2  A I I2 I1 D ² SDJH  $OIUHG FDQ SD\ EHFDXVH KH LV ULFK EHFDXVH← 

ǫ→ ǫ → ULFKA  Y  ǫ → KHX  LVI2  I I22 Y I21 dep → $OIUHG  FDQ  SD\  ǫ ǫ← 2 X I > O I2 Igov I22 1





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D ² SDJH  $OIUHG SHXW SD\HU SDUFH TXH LO HVW ULFKH SDUFH TXH← 

ǫ→ ǫ → ULFKHA  Y  ǫ → LO  HVWI2  I22 Y I21 X dep → ǫ← 2 SD\HUI > O  ǫ 2 $OIUHGX  SXHWI2 I2 I1 gov I

D ² SDJH  $OIUHG SXz SDJDUH SHUFKp q ULFFR

  q 2  ǫ→ ǫ → ULFFRA  Y  ǫ → HJOL I I22 Y I21  X Idep ǫ← SDJDUHI > O  ǫ → $OIUHGX  SXzI2 I22 I21 gov

SHUFKp← 

E ² SDJH  $OIUHG FDQ SD\ KHQFH KH LV ULFK KHQFH→ 

ǫ→ ǫ → ULFKA  Y  ǫ → KHX  LVI2  I I22 Y I21 dep → ǫ← 2 SD\I > O  ǫ 2 $OIUHGX  FDQI2 I2 I1 gov

E ² SDJH  $OIUHG SHXW SD\HU GRQF LO HVW ULFKH GRQF→ 

ǫ→ ǫ → ULFKHA  Y  ǫ → LO  HVWI2  I I22 Y I21 X dep → $OIUHG  SXHW ǫ←  SD\HU  ǫ 2 I > O I2 X Igov I22 1

E ² SDJH  $OIUHG SXz SDJDUH GXQTXH q ULFFR

  q 2  ǫ→ ǫ → ULFFRA  Y  ǫ → HJOL I I22 Y I21  X Idep ǫ← SDJDUHI > O  ǫ → $OIUHGX  SXzI2 I22 I21 gov

GXQTXH→ 

D ² SDJH  $OIUHG ZKR LV ULFK FDQ SD\ ǫ← SD\I > Y  ǫ → ZK← ǫ → ULFKA > Ydep  I > A > X I22 I22 I21 ǫ→ RXdep  LVI2  $OIUHGX  FDQI2 I21 dep

E ² SDJH  $OIUHG ZKR FDQ SD\ LV ULFK ǫ← ULFKA > Y  ǫ → ZK← ǫ → SD\I > Ydep  I > A > X I22 I22 I21 ǫ→ RXdep  FDQI2  $OIUHGX  LVI2 I21 dep

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D ² SDJH  WKH PDQ ZKR ERXJKW WKH ERRN IRU WKH JLUO ZK→

IRU→ (WKH JLUO)O  ǫ ← (WKH ERRN)Y  I>A>O E>I I42 → ǫ 4 RX  (ERXJKW) 4  (WKH PDQ)O Idep I1

E ² SDJH  WKH ERRN ZKLFK WKH PDQ ERXJKW IRU WKH JLUO ZK→

IRU→ (WKH JLUO)O  ǫ ← (LFK)Y  I>A>O E>I I42 ǫ→ (WKH PDQ)X  (ERXJKW)I4  (WKH ERRN)O I41 dep

F ² SDJH  WKH JLUO IRU ZKRP WKH PDQ ERXJKW WKH ERRN ZK→

IRU→ (RP)O  ǫ ← (WKH ERRN)Y  E>I I>A>O I42 ǫ→ WKH PDQX  (ERXJKW)I4  (WKH JLUO)O I41 dep

G ² SDJH  WKH ER\ ZKRVH ERRN WKH PDQ ERXJKW IRU WKH JLUO ǫ → RVHA  ERRNY  (WKH JLUO)O  ǫ ← IRU→ I>A>O E>I I42 Y (WKH PDQ)X  (ERXJKW)I4  (WKH ER\)O ǫ→ I41 dep

ZK→

I ² SDJH  :KR ERXJKW WKH ERRN IRU WKH JLUO" 7KH PDQ "→ (WKH PDQ)O  ZK→

I>A>O

I

IRU→4 (WKH JLUO)O  ǫ ← 4 (WKH ERRN)Y 

ǫ→ RX  (ERXJKW)I4  O I41

E2

I2

D ² SDJH  :KHQ 3DXO ZLOO FRPH EDFN /L]D ZLOO VPLOH ZK→

EDFNE  (ZLOO FRPH)I1   ǫ ← HQE  ǫ → 3DXOX  ǫ → E > I11 I11 I11 I1 ǫ→ /L]DX  (ZLOO VPLOH)I1 I11

E ² SDJH  /L]D ZLOO VPLOH ǫ→ /L]DX  (ZLOO VPLOH)I1 I11



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F ² SDJH  :KHQ ZLOO /L]D VPLOH" :KHQ 3DXO ZLOO FRPH EDFN "→ ZK→

ǫ ← HQE  ǫ → 3DXOX  ǫ → 1 EDFNE  (ZLOO VPLOH)I1  G  I > E I11 I11 E>I ← /L]D  (ZLOO VPLOH)   ZK→ ǫ ←  HQ  ǫ E I1 X G I > E I11 I1

I

D ² SDJH  , ZRQGHU KRZ PDQ\ HUURUV , GLG K→ 2 

I2

ǫ→  ǫ → ǫ → RZE  PDQ\A  (HUURUV)Y  ǫ ← ,  GLGI2  I22 > O Y A I21 X dep → ǫ 2 ,X  ZRQGHU 2 I I1 gov

E ² SDJH  :KDW WLPH LV LW" ZK→ 2 

I2

ǫ ← DWA  WLPHY  ǫ ← LW  LV 2 Y I21 X I

F ² SDJH  +RZ PDQ\ HUURUV , GLG" , ZRQGHU → ← → → "→ ǫ → 1 ,X  ZRQGHUI1  KI > A > O ǫ 2 ǫ Y ǫ A RZE  PDQ\A  I I1

I2

(HUURUV)Y  ǫ → 2 ,X  GLGI2  O I1

D ² SDJH  .DUO JLHVW GLH %OXPHQ ǫ← (GLH %OXPHQ)Y  ǫ → .DUOX  JLHVWI2 I22 I21

E ² SDJH  ,FK ODVVH .DUO GLH %OXPHQ JLH‰HQ JLH‰HQ→ 2+1 (GLH %OXPHQ)Y 

I3

ǫ→ .DUOX  ǫ ← ,FKQ  (ODVVH)I2+1 I2+1 I2+1 cause 2 1

F ² SDJH  ,FK ODVVH GLH %OXPHQ JLH‰HQ JLH‰HQ→ 2+1−1 (GLH %OXPHQ)Y 

I3

  ǫ ← ,FK  (ODVVH) 2+1  .DUO ǫ→ Q X I2+1 Icause I2+1 1 2

 ² SDJH  3DXO PDNHV /L]D EX\ D FDU E\ 5RQ¶V IRU  SRXQGV → ← IRU← 4+1 ( SRXQGV)W  E\ 4+1 (5RQ·V)Z  EX\ 4+1 (D FDU)Y  I5

I4 I3 → (3DXO)  (PDNHV) ǫ→  /L]D  ǫ 4+1 X I4+1 Q Icause I4+1 2 1

D ² SDJH  +LP EH D GRFWRU EH→ 2+1 (D GRFWRU)Y 

I3

Q   \RX  ǫ→ KLPX  ǫ ← O HW 2 Icause I2+1 I2+1 2 1

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E ² SDJH  0\ PRWKHU ULGH D WUDLQ ULGH← 2+1 (D WUDLQ)Y 

I3

Q   \RX O HW 2 ǫ→ (0\ PRWKHU)X  ǫ ←  Icause I2+1 I2+1 2 1

F ² SDJH  +HU ZDVK WKH GLVKHV ZDVK← 2+1 (WKH GLVKHV)Y 

I3

Q   ǫ→ KHUX  ǫ ← O HW 2 \RX  Icause I2+1 I2+1 2 1

D ² SDJH  /HW KHU GR LW GR← 2+1 LWY 

I3

Q   \RX O HW 2 ǫ→ KHUX  ǫ ←  Icause I2+1 I2+1 2 1

E ² SDJH  +HOS KLP SXW KLV VKRHV SXW← 2+1 (KLV VKRHV)Y 

I3

 2 Q   ǫ→ KLPX  ǫ ← KHOS \RX  Icause I2+1 I2+1 2 1

F ² SDJH  0DNH WKLV VWD\ RSHQ VWD\← 2+1 (RSHQ)Y 

I3

 2  Q   \RX PDNH ǫ→ (WKLV)X  ǫ ←  Icause I2+1 I2+1 2 1

$ &RQVWUXFWLYH GLFWLRQDULHV 7KH H[DPSOHV LQ &KDSWHU  DUH QRW UHSRUWHG VLQFH WKH\ DUH QRW µUHIHUHQFH¶ H[DP SOHV ZKLFK FDQ EH H[WUDFWHG IURP WKH FRQWH[W WKH\ KDYH EHHQ SUHVHQWHG DQG XVHG WR LQVWDQFH RWKHU FDVHV RQ WKH VDPH JXLGHOLQHV

$ 'LVFRXUVH DQG EH\RQG D ² SDJH  ,¶YH VHHQ 0DU\ ZLWK 3DXO QR HPSKDVLV ǫ→ ZLWK↔ 3DXOO  0DU\Y  (,·YH VHHQ)I2 A>Y I22 1

E ² SDJH  ,¶YH VHHQ 0DU\ ZLWK 3DXO HJ QRW /L]D ǫ→ ZLWK← 3DXOO  0DU\Y  (,·YH VHHQ)I2 A>Y I22 1



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F ² SDJH  ,¶YH VHHQ 0DU\ ZLWK 3DXO HJ QRW ZLWK -RKQ ǫ→ ZLWK→ 3DXOO  0DU\Y  (,·YH VHHQ)I2 A>Y I22 1

D ² SDJH  , ZRQ WKH UDFH , DP WHOOLQJ WKDW , ZRQ WKH UDFH ǫ ← ǫ ← (WKH UDFH)(Y, S)  ǫ → ,  ZRQ↓ A  (DVVHUW)(↓ A, S) ↓ A I22 I21 S

E ² SDJH  :LQ WKH UDFH , RUGHU \RX WR ZLQ WKH UDFH (WKH UDFH)(Z, R)  ǫ ← \RX R  ZLQ↑ D  (GLUHFW)(↑ D, S) ǫ ← ǫ ← ↑ D I22 I21 F ² SDJH  ,¶OO ZLQ WKH UDFH , SURPLVH P\VHOI WR ZLQ WKH UDFH ǫ ← ǫ ← (WKH UDFH)(Z, S)  ǫ ← ,  (·OO ZLQ)↑ C  (FRPPLW)(↑ C , S) ↑ C I22 I21 S

G ² SDJH  , WKDQN P\ WHDP IRU WKHLU KHOS ← ← ǫ← = E IRUE (WKHLU KHOS)(Y, S)  ǫ I22 (P\ WHDP)R  (, WKDQN)(Y, S) 

(H[SUHVV)(= E, S)

H ² SDJH  DQG WKH ZLQQHU RI WKH UDFH LV DQG← (WKH ZLQQHU RI WKH UDFH LV)l

λ

D  (Declare)l D

 ² SDJH  6SHDNHU DVVHUWV WKDW VSHDNHU¶V EHOLHI H[LVWV ǫ → ·V→ (VSHDNHU)X = S > A  (EHOLHI)Y  (H[LVWV)Ix  ↓A Y Y → (VSHDNHU)X = S  (DVVHUWV)↓ A ↓A

WKDW←

 ² SDJH  , WHOO \RX ZKDW , IRXQG LQ WKH OLEUDU\ ←

↓A

(\RX ZKDW , IRXQG LQ WKH OLEUDU\)↑ D  (I Btell)↓ A

 ² SDJH  6SHDNHU GLUHFWV UHFHLYHU WR GR UHTXHVW ǫ ← ǫ → (UHTXHVW)Z  ǫ ↔ (UHFHLYHU)Y = R  (WRBGR)Ix  Ix ↑ D Ix → (VSHDNHU)X = S  (GLUHFWV)↑ D  ↑D

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 ² SDJH  /L]D RUGHUHG 3DXO WR OHDYH ǫ ← (3DXO WR OHDYH)(Z, R)  (/L]D RUGHUHG)(↑ D, S) ↑D

 ² SDJH  6SHDNHU SURPLVHV WKDW VSHDNHU ZLOO GR WKH SURPLVH ǫ ← ǫ → (SURPLVH)Z  ǫ ↔ (VSHDNHU)Y = S  (ZLOOBGR)Ix  Ix ↑ C Ix →  (VSHDNHU)X = S  (SURPLVHV)↑ C ↑C

 ² SDJH  6SHDNHU H[SUHVVHV VSHDNHU¶V HPRWLRQ RQ UHFHLYHU (VSHDNHU)S  (HPRWLRQ)Ix (UHFHLYHU)R  ·V→ RQ← Ix Ix ǫ→ = E (VSHDNHU)X = S  (H[SUHVVHV)= E

= Y

 ² SDJH  6SHDNHU 'HFODUHV µWKH 5HFHLYHU KDV WKH 'HFODUHG 6WDWXV¶ UHFHLYHUY = R  KDVIx  ǫ ← VWDWXVS  ǫ → Ix l D Ix → VSHDNHUX = S  'HFODUHVl D ǫ lD

←

 ² SDJH  )DWKHU
 ² SDJH  'LG \RX VHH WKH QHZV" "←

('LG \RX VHH WKH QHZV)↓ A  (DVVHUW)↓ A

↓A

 ² SDJH  :K\ KDYH \RX VWLOO WR ILQLVK \RXU KRPHZRUN" "←

↑D

(:K\ KDYH \RX VWLOO WR ILQLVK \RXU KRPHZRUN)↑ D  (GLUHFW)↑ D

 ² SDJH  2Q ZKDW , KDYH P\ PLQG IL[HG WRGD\" "←

↑C

(:K\ KDYH \RX VWLOO WR ILQLVK \RXU KRPHZRUN)↑ C  (FRPPLW)↑ C

 ² SDJH  +RZ VKRXOG , IHHO DERXW WKDW" "←

= E (+RZ VKRXOG , IHHO DERXW WKDW)= E  (H[SUHVV)= E



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 ² SDJH  , SURPLVH \RX¶OO GR LW ǫ → (\RX·OO GR LW)(↑ D, S)  (, SURPLVH)(↑ C , S) ↑D

 ² SDJH  , UHTXHVW P\VHOI WR EH EUDYH ǫ → (WR EH EUDYH)(↑ C , S)  (, UHTXHVW)(↑ D, S) ↑C

 ² SDJH  :K\ VKRXOG , EH DQJU\ ZLWK KLP" "←

↑D

(VKRXOG , EH DQJU\ ZLWK KLP)(= E, S)  (tellBme ZK\)(↑ D, S)

 ² SDJH  , ZRQGHU LI , DP DVKDPHG WR IROORZ KHU LI←

↓A

(, DP DVKDPHG WR IROORZ KHU)(= E, S)  (, ZRQGHU)(↓ A, S)

Appendix B The Formal Model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,Q RUGHU WR H[DFWO\ GHILQH ZKDW ZH LQWHQG IRU FRQVWUXFWLRQV ZH QHHG WKH WHFK QLFDO FRQFHSW RI JUDPPDU SURGXFW 'HILQLWLRQ % *UDPPDU SURGXFW )L[HG D VPDOO FDWHJRU\ C D VHW G = {Gi }i∈GC RI JUDPPDU FKDUDFWHUV ZLWK W ∈ G ZH FDOO JUDPPDU SURGXFW RQ C G W DQ\ SDLU hY1 × · · · × Yn , (z1 , . . . , zn )i ZKHUH n LV D QDWXUDO QXPEHU FDOOHG WKH GHJUHH RI WKH SURGXFW × LV WKH &DUWHVLDQ SURGXFW RI VHWV DQG IRU HYHU\ i LQ 1 . . . n Yi ∈ G \ {W} DQG zi ∈ W 

7KH UHDGHU LV LQYLWHG WR QRWLFH WKDW JUDPPDU SURGXFWV DUH QRW SURGXFWV LQ WKH FDWHJRU\ C²ZH GR QRW HYHQ UHTXLUH WKDW C KDV SURGXFWV ,QVWHDG WKH\ VKRXOG EH WKRXJKW RI DV ILQLWH SURGXFWV RI JUDPPDU FKDUDFWHUV ZKHUH WKH iWK SURMHFWRU LV QDPHG E\ DQ HOHPHQW RI W  ZKLFK GHQRWHV WKH FODVV RI DGSRVLWLRQV &RQVWUXFWLRQV DUH IDPLOLHV RI DUURZV ZLWK WKH VDPH GRPDLQ DQG ZKRVH FR GRPDLQ OLHV LQ D VLQJOH JUDPPDU FKDUDFWHU ZKLFK LV QRW WKH RQH RI DGSRVLWLRQV 7KHVH DUURZV DUH LQGH[HG E\ D JUDPPDU SURGXFW 'HILQLWLRQ % *UDPPDU FRQVWUXFWLRQ *LYHQ D VPDOO FDWHJRU\ C D VHW G = {Gi }i∈GC RI JUDPPDU FKDUDFWHUV ZLWK n oW ∈ G D JUDPPDU FRQVWUXFWLRQ ηP,x RYHU P IURP x LV DQ LQGH[HG FROOHFWLRQ f j ZKHUH ′ j∈P

7KH )RUPDO 0RGHO



‡ P = hP′ , ai LV D JUDPPDU SURGXFW RQ C G W  ‡ x LV DQ REMHFW RI C ‡ WKHUH LV V ∈ G \ {W} VXFK WKDW IRU HYHU\ j ∈ P′  f j ∈ HomC (x, y) ZKHUH y ∈ V :H GHQRWH WKH VHW V E\ cod ηP,x  WKH FRGRPDLQ RI WKH JUDPPDU FRQVWUXFWLRQ 6LPLODUO\ dom ηP,x = x DQG ZH FDOO P WKH SURGXFW RI ηP,x  6RPHWLPHV ZH ZLOO ZULWH η j,x IRU f j   $ IDPLO\ ηP,x x∈U RI JUDPPDU FRQVWUXFWLRQV RYHU P IURP x YDU\LQJ LQ WKH U JUDPPDU FKDUDFWHU IRUPV D JUDPPDU FRQVWUXFWLRQ ηP,U RYHU P IURP U ZKHQ DOO LWV HOHPHQWV VKDUH WKH VDPH FRGRPDLQ :H GHQRWH WKH FRPPRQ FRGRPDLQ ZLWK cod ηP,U  $OVR dom ηP,U = U DQG P LV WKH SURGXFW RI WKH FRQVWUXFWLRQ $JDLQ ZH ZLOO ZULWH η j,x WR LQGLFDWH WKH jWK FRPSRQHQW RI WKH xWK FRPSRQHQW RI ηP,U  :H VD\ WKDW WKH FRQVWUXFWLRQ η RYHU P = hY1 × · · · × Yn , (z1 , . . . , zn )i RQ WKH REMHFW x KDV x DV JRYHUQRU DQG LWV GHSHQGHQWV DUH WKH HOHPHQWV RI Y1 , . . . , Yn  LWV DGSRVLWLRQV DUH z1 , . . . , zn  6R LWV SURGXFW P GHVFULEHV WKH DGSRVLWLRQDO VWUXFWXUH WKDW η LPSRVHV RYHU WKH JRYHUQRU )RU V\PPHWU\ JLYHQ D FRQVWUXFWLRQ η DV DERYH LW LV XVHIXO WR FRQVLGHU WKH FRQVWUXFWLRQV ZKLFK GLIIHU IURP η E\ H[FKDQJLQJ D GHSHQGHQW ZLWK WKH JRYHUQRU 'HILQLWLRQ % &RQMXJDWH FRQVWUXFWLRQ )L[HG D VPDOO FDWHJRU\ C D VHW G = {Gi }i∈GC RI JUDPPDU FKDUDFWHUV ZLWK W ∈ G DQG D JUDPPDU FRQVWUXFWLRQ n o  η = f j : x → c(x, j) ′ j∈P

x∈V

′

RYHU WKH SURGXFW P = hP = P1 × · · · × Pn , (a1 , . . . , an )i IURP WKH V JUDPPDU FKDUDFWHU D JUDPPDU FRQVWUXFWLRQ θ RYHU WKH SURGXFW Q IURP WKH JUDPPDU FKDUDFWHU U LV VDLG WR EH FRQMXJDWH WR η LI IRU VRPH k LQ 1 . . . n U = Pk DQG

Q = hP1 × · · · × Pk−1 × V × Pk+1 × · · · Pn , (ak , a2 . . . , ak−1 , a1 , ak+1 . . . , an )i DQG IRU DOO (x, j1 , . . . , jn ) ∈ C × P′  cod η( j1 ,..., jn ),x = cod θ( j1 ,..., jk−1 ,x, jk+1 ,..., jn ), jk  :H QRWLFH WKDW LQ JHQHUDO D FRQVWUXFWLRQ PD\ KDYH DQ\ QXPEHU RI FRQMXJDWHV IRU HDFK HOHPHQWV LQ LWV SURGXFW HYHQ ]HUR &RQVWUXFWLRQV IURP JUDPPDU FDWHJRULHV FDQ EH FRPSRVHG WR REWDLQ PRUH FRPSOH[ FRQVWUXFWLRQV



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'HILQLWLRQ % &RPSRVLWLRQ RI FRQVWUXFWLRQV *LYHQ D VPDOO FDWHJRU\ C D VHW G = {Gi }i∈GC RI JUDPPDU FKDUDFWHUV ZLWK W ∈ G DQG JUDPPDU FRQVWUXF WLRQV  

η=

n

f(y1 ,...,yn ) : x → c(x, y1 , . . . , yn )

o

y1 ∈P1 ,...,yn ∈Pn x∈V

RYHU WKH SURGXFW P = hP1 × · · · × Pn , (a1 , . . . , an )i IURP WKH V JUDPPDU FKDU DFWHU DQG n o θ = g(z1 ,...zm ) : x → d(x, z1 , . . . , zm ) z1 ∈Q1 ,...,zm ∈Qm x∈U

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