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 x2 x5 x6 x4 x3 x1 x7

Ha mming <7,4> c o de (c o nt.) x1 x2 x3 x4 x5 x6 x7 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0

Ha mming <7,4> c o de (c o nt.) x1 x2 x3 x4 x5 x6 x7 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0

Ha mming <7,4> c o de (c o nt.) x1 x2 x3 x4 x5 x6 x7 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1 1

Ha mming <7,4> c o de (c o nt.) x1 x2 x3 x4 x5 x6 x7 0 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 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 or c or r e c ting c 2 c 1 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 or de te c ting c 2 c 1 w1 w2 e r r or de te c t e r r or de te c t • But, 1-e r e c ting a nd 2-e r r or c or r r or 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

inks -links -links ID-links L te nsor par B B B p B & A A A p A

pro o f ne t) L pro o f struc ture (a lso ML Θ 1= L ML

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 Θ 1

DR-g ra ph 2 fo r Θ 1

ML L pro o f struc ture (b ut no t ML L pro o f ne t) Θ 2=

DR-g ra ph 1 fo r Θ 2

DR-g ra ph 2 fo r Θ 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

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 Θ 1=

One o f fo ur me mb e rs o f a PS-fa mily Θ 2=

Ha mming dista nc e o n a PS-fa mily dista nc e o f Θ 1, Θ 2 ¥in PS-fa mily F • d( Θ 1, Θ 2) = the numb e r o f “lo c a tio ns” whe re multiplic a tive links a re diffe re nt • F o 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

= 2 xa mple , E d

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 Θ 1 and Θ 2 are MLL proof nets and both belong to F, then the number of ID- links (tensor-links, and par-links) of Θ 1 is the same as that of Θ 2.

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 de te c ting . r or Ide a of Pr oof f Θ , Θ ’ ¥in F I , the n we c a n ha ve a se q ue nc e Θ ⇒ Θ 1 ⇒ ・・・ ⇒ Θ n ⇒ Θ ’ suc h tha t Θ 1,…, Θ n a re ML L pro o f ne ts whe re Θ a ⇒ Θ b if Θ b is o b ta ine d fro m Θ a 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