ML L pro o f ne ts a s e rro r-c o rre c ting c o de s Sa to shi - - PowerPoint PPT Presentation

ML L pro o f ne ts a s e rro r-c o rre c ting c o de s

Sa to shi Ma tsuo ka AI ST

T he struc ture o f the ta lk

• 1. I

ntro duc tio n to e rro r-c o rre c ting c o de s

• 2. I

ntro duc tio n to ML L pro o f ne ts

• 3. Ho w to a na lyze ML

L pro o f ne ts using c o ding the o ry

• 4. Our re sults so fa r

Ha mming <7,4> c o de

• A sub se t o f {0,1}^{7} c a lle d c ode

wor ds

• Sa tisfying
• 1. x1 + x2 + x4 + x5 = 0
• 2. x2 + x3 + x4 + x6 = 0
• 3. x1 + x3 + x4 + x7 = 0

whe re xi ¥in {0,1} + is e xc lusive o r (o r pa rity c he c k)

Ha mming <7,4> c o de (c o nt.)

• 1. x1 + x2 + x4 + x5 = 0
• 2. x2 + x3 + x4 + x6 = 0
• 3. x1 + x3 + x4 + x7 = 0

x1 x2 x3 x4 x5 x6 x7

Ha mming <7,4> c o de (c o nt.)

x1 x2 x3 x4 x5 x6 x7 1 1 1 1 1 1 1 1 1

Ha mming <7,4> c o de (c o nt.)

x1 x2 x3 x4 x5 x6 x7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Ha mming <7,4> c o de (c o nt.)

x1 x2 x3 x4 x5 x6 x7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Ha mming <7,4> c o de (c o nt.)

x1 x2 x3 x4 x5 x6 x7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Ha mming <7,4> c o de (c o nt.)

• 2^4 = 16 words are (legitimate) codewords
• Other words (2^7-2^4 = 112) are not

Ha mming <7,4> c o de (c o nt.)

• distance of w1, w2 ¥in {0,1}^{7}

d(w1, w2) = | { i | w1(i) ¥neq w2(i)} |

• Example

d(0101000, 00110011)=4

• The distance of code C, d(C):

the minimum distance of different codewords

• Hamming <7,4> code C has d(C)=3

Ha mming <7,4> c o de (c o nt.)

• So , Ha mming <7,4> c o de is

1-e r r

• r

c or r e c ting

c 1 c 2 w1 w2

c or r e c t c or r e c t

Ha mming <7,4> c o de (c o nt.)

• On the o the r ha nd, Ha mming <7,4> c o de is

2-e r r

• r

de te c ting

c 1 c 2 w1 w2

e r r

• r

de te c t e r r

• r

de te c t

• But, 1-e r

r

• r

c or r e c ting a nd 2-e r r

• r

de te c ting

a re no t c o mpa tib le

T he struc ture o f the ta lk

• 1. I

ntro duc tio n to e rro r-c o rre c ting c o de s

• 2. I

ntro duc tio n to ML L pro o f ne ts

• 3. Ho w to a na lyze ML

L pro o f ne ts using c o ding the o ry

• 4. Our re sults so fa r

L inks

p p A B & A B A B A B

te nsor

• links

par

• links

ID-links

ML L pro o f struc ture (a lso ML L pro o f ne t)

Θ1=

Gra ph-the o re tic c ha ra c te riza tio n the o re m

• T

he o re m (Gira rd, Da no s-Re g nie r)

Θ is ML

L pro o f ne t

iff

fo r a ny DR-switc hing S, the DR-g ra ph

Θ_S is a c yc lic a nd c o nne c te d

DR-g ra ph 1 fo r Θ1

DR-g ra ph 2 fo r Θ1

ML L pro o f struc ture (b ut no t ML L pro o f ne t)

Θ2=

DR-g ra ph 1 fo r Θ2

DR-g ra ph 2 fo r Θ2

T he struc ture o f the ta lk

• 1. I

ntro duc tio n to e rro r-c o rre c ting c o de s

• 2. I

ntro duc tio n to ML L pro o f ne ts

• 3. Ho w to a na lyze ML

L pro o f ne ts using c o ding the o ry

• 4. Our re sults so fa r

T he Ba sic I de a

• PS-family: a se t o f ML

L pro o f struc ture s suc h tha t e a c h me mb e r is re a c ha b le fro m the o the r me mb e rs b y se ve r

al te nsor

• par

e xc hange s

• Pa rtitio n ML

L pro o f struc ture s into PS- fa milie s

• Re g a rd e a c h PS-fa mily a s a c o de

One o f fo ur me mb e rs o f a PS-fa mily

Θ1=

One o f fo ur me mb e rs o f a PS-fa mily

Θ2=

Ha mming dista nc e o n a PS-fa mily

• dista nc e o f Θ1, Θ2 ¥in PS-fa mily F

d(Θ1, Θ2)

= the numb e r o f “lo c a tio ns” whe re multiplic a tive links a re diffe re nt

• F
• r e a c h PS-fa mily F

, d(F ) is the minimum dista nc e o f diffe re nt ML L pro o f ne ts in F

E xa mple d , = 2

T he struc ture o f the ta lk

• 1. I

ntro duc tio n to e rro r-c o rre c ting c o de s

• 2. I

ntro duc tio n to ML L pro o f ne ts

• 3. Ho w to a na lyze ML

L pro o f ne ts using c o ding the o ry

• 4. Our re sults so fa r

F irst Que stio n

• Ho w do we ha ve pro pe rtie s a b o ut

d(F )?

Proposition

• Let F be a PS-family.

If Θ1 and Θ2 are MLL proof nets and both belong to F, then the number of ID- links (tensor-links, and par-links) of Θ1 is the same as that of Θ2.

T he or e m

I f PS-fa mily F ha s mo re tha n two ML L pro o f ne ts, the n d(F )=2. So , suc h a PS-fa mily is just one -e r

r

• r

de te c ting. Ide a of Pr

• of

I f Θ, Θ’ ¥in F , the n we c a n ha ve a se q ue nc e

Θ ⇒ Θ1 ⇒ ・・・ ⇒ Θn ⇒ Θ’

suc h tha t Θ1,…,Θn a re ML L pro o f ne ts whe re Θa ⇒ Θb if Θb is o b ta ine d fro m Θa b y re pla c ing a te nso r-link b y a pa r-link a nd a pa r-link b y a te nso r-link e xa c tly two time s Using g ra ph-the o re tic c ha ra c te riza tio n the o re m, no ntrivia l (a t le a st fo r me ) Using re duc tio n to a b surdity

Summa ry

• We c a n inc o rpo ra te the no tio n o f Ha mming

dista nc e into ML L pro o f ne ts na tura lly

• Go t a n e le me nta ry re sult
• But it’ s o ng o ing wo rk
• Ne e d to g e t mo re re sults (c o mpo sitio n o f

two PS-fa milie s, c ha ra c te riza tio n o f PS- fa milie s with n ML L pro o f ne ts,….)

• T

he ma nusc ript c a n b e fo und in http:/ / a rxiv.o rg / a b s/ c s/ 0703018