Lecture 23 Probabilistically Checkable

Lecture 23

Probabilistically Checkable " " "" "" " E.EE?i - linear feta : Vxiy flxltfly) - flxty)

self - correcting :

if - to linear f is tg close of

- Do Ollogtp) times

Picky randomly . xD

) tflx - output answer ← fly y) , :{ ; }¥YPrf most common answer Output ; self - testing: Given f : Do OLE) times .

' ' ' i'I' .in#a...e..3i:sI.::::ssthen fails fitting , "⇒ Probabilistically Checkable Proofs

want to prove ← Theorem you hp 3. CNF for today: X is O satisfiable randomstring@ Them × is § Profit what is ith bit . ? verifier .is?icxemdgfItj § ' ' !! eddoes If based on TM ↳ past ( time ) g questions poly ) of bits - verifier r random verifiers using to created who knows algorithm & proof by adversary q queries & has unlimited computational ( r if s power det LEPCP ,q ) I r Cp time Tm ) ! [ 1) AXEL F IT SH Pr KIT accepts] -4 v 's random string

' ' L HIT V IT E 44 2) Axel Pr [ , accepts ] v's random strings of all n vars ← SAF E Pcp ( o n proof settings , ) e.g . V doesnt need any randomness

: they ← Nps pep ohh OH ? ( , ) I crazy

Actually : E Pcp ( ) Np Ollogn , OUD

" " : Lets start with a warmup " " =

- { . . inner X. y Hi ya product

" outer = . . . . . x . . . ( , x. yz , ) " Xoy y , ya, , , Xiyj , Xnyn - product Y vector n -bit rectors habit

: if #5 then Pilar # IF ] ' Fact I ta ! " " moda - or gtfo Fared FEfqB } if A. Btc then [ A.B. Ft di ] 242 Prp w - take Obi) to nxn matrices A.%-) compute

I # IF Fact : if I #5 then Pr[ air ] ztz " Efforts if A. Btc then Prp [ A' B. Ft Ef ) 242 .

" " : : - ) Example application setting given rectora-la.az . . an in one : • can ai ⇐ step query • can - specify go try = : to test if a- .in ) query 10,0 , ,o

Do several times : " pick T ELO,B " " if a.F¥o output Fail Output PASS

: if a- = lo . . - o) will PASS behavior , always it then FACT [ a- -toJeY2 it # Lo .no ⇒ Prf a- , )

n- bivector o - for ⇒ Old query testing algorithm model in strange 3 Arithmetical on of SAT :

F ⑦ arithmetic formula Alf) Boolean formula over Zz T I w mod 2 F- ⇒ O Xi ⇐ Xi

I - In. Xi dip d. p

- - - a) LVG I Li Li p)

- r - - ) xvpivr I Li a) Ii pill I as

- - I 4) ) ( l - example : XµVIzVXz ⇒ I ( ( Xa Xz)

iff =L Keyport F satisfied by assignment a LAH)) (a) 1- ⇐ I

- F- . St ) Ace . Ci f- ⇐ o Cgi , vyizvyi Xi ⇐ Xi

" " - - E - Xn In where E Xi XT } - yij , In. I Xi Fieri

- of Clouse = of arithmetical ion Ci .IT#EEiEi::i::m...L Cx) Sf . I. complement satisfies fi ⇒ evaluates to 0 if X F . o ⇒ . . if = (o ) x satifies Ctx) ,

it each Ii is E3 in X Observe deg poly .

(2) V knows coeffs of each £.

)= O . - - - O = . SENDING ( ( . . ( ) WITHOUT Need to convince V that (a) Eilat , Inlay , assignment a High level idea : special encoding of assignment

of F a collection of of Encode satisfiability as polys in vars assignment

⇐ each clause one for

- eval to 0 if assignment satisfies clause - low degree

- clause - coeffs of V knows depend on structure

of clause . & vars

note thatsolving ← concerned that V is time Note : we are only poly , stttwwindpfbithmpress.ve here will not be sublunar

However want # to to be constant , queries proof - ⇐ - T I . St ) F- . Ci Acn Cgi , vyizvyi F- ⇐ O :

- " where E . xn I - - Es ⇐ Xi Ideafor_pro { Xi , Xi Yi; Ii I -Xi

÷:÷÷÷::÷:::c::÷i. arithmetic - ⇒ mod 2 if alto Cca)cr=o) - Fist. Pr [ Iz ⇒ , H ¥÷÷¥ ±""÷÷÷÷÷i÷i:: Pr Caliri )

V doesn't know

What does Ccalar look like?

#\cu v c v u § riffa) = T t § " ## [email protected]; relation in of on his towers does know : toe → gii g. y depend pi; zs.AT !!! Ci 8ijk -7 Zijk High level idea : special encoding of assignment

out fetus of • proof writes all linear assignment dega deg 3

" " " " • for linear possible confusion : symmetric case fxla ) = X. a = Aalxl A inner product

- . • ) - for deg 33 : Daly Goat y (acyl :( aoaoa)-1.2

' all linear fetus ⇒ can test t self -correct Aa Ca are linearity ,Ba , from whatif Au Ba Ca come different assignments , , Proof can cheat ! : ' is a satisfying? all of same as at of (hopefully valves one input coefficients fetns their from are about These care what V computes + we only to 's 's tri det µ corresponding deep poly knows - V linear - = - all fetus → ,z A- I Atx) Eaixi at, 4g If :# , but evaluated at an not a

a assignment "

. - = - oaf i # -2 # ) @ y - 2 fetus B an a Bly aoiajyij B - all deg §, evaluated at a

÷:c.:*:*

we C but i÷÷÷÷""A , check hopefully , need to proof contains : us -

truth BB E all of tables of A , for x. ,z Complete description , inputs y T only need value at

- F- 2 - 2=8 , y B , but extra info helps us check consistency =

- ru x . in ( , X. Kiya , ) Xoy y , ya, , , Xiyj , Xnyn det proof contains : - HUGE②

- linear - = A - all fetus → IF Atx) Eaixi atx ff :#an , of truth tables evaluated at Complete description assignment a

- = a - of A all :# 's# ) @ y ,DB,E for inputs x. gz - fetus B an a aiajyij B - all Bly deg2 §, T evaluated at a " need value at " " Z only ' - → - 2=8 = @ xx C ' ( , y B , C- all 3 fetus #2" HI (g) deg ,§uaiajak2iju= but extra info evaluated ata helps vs check consistency

What does verifier need to check in proof ?

form E- close to linear (1) A in can lest , BYE right

only - correct to access the linear fetus . ← but can self are linear fetus • all

to same a • correspond assignment ⇒ ie = a) Ek)=CaoaoaTz . ATH=aTx ⇒ Bly) @ Ig Te s t consistency of self corrections

(2) a is satisfying assignment

• evaluate to 0 on a all I. 's

cat - . ro recall Cca, . ) ( , a) , Lois ) ftp.kmeut =

- ru x . in ( , X. Xyyz , ) Xoy y , ya, , , Xiyj , Xnyn def - Proof contains : -

- linear - = A - all fetus → I Atx) Eaixi atnx A :#an , of truth tables evaluated at Complete description assignment a

. - = - oaf of all : # # ) @ ) APB,E for inputs x. y,z - fetus B is a aoiajyij B - all Bly deg2 §, T evaluated at a " need value at " " Z only - ' - → - 2=8 - @ xx C ' ) , y B. #2" HI . C-- all 3 fetus Cly ,eaiajak2iju= deg § but extra info evaluated ata helps vs check consistency

Feat Honma! the linear fans . fetus ← GII Yanks:# : all are linear Te s t 41 A. BE in right form

linear PASS in Oct) - to linear lie if all close . • Te s t are all 48 , I BYE - far FAIL ) queries , it one is Yg any to • self corrector From now on use get , of answering ← use f- prob getting wrong - - - F Sc SC E for all 'D Sc B inputs is so small ( , , that Ebitgenoghoonstanl I # # that union bud over all

- a b c to set se - B so E queries , , " " ⇒ Aoa to see error ? aoaoa ? unlikely =

- no ( Xi Yi kids in j , ) def X°Y , Xiyn, , Xiyj , Xnyn - proof contains : -

- linear - = A - all fetus → Atx) Eaixi ATX A :#an Iz of truth tables evaluated at Complete description assignment a

. - = - oaf of A all :# 's# ) @ ) ,DB,E for inputs x. y,z - fetus B an a aoiajyij B - all Bly deg2 §, T evaluated at a " need value at " " Z only ' - → - 2=8 = @ xx C ' ( , y B , - #2" HI C - all 3 fetus (g) deg ,§uaiajak2ijn= but extra info evaluated ata helps vs check consistency

- o are linear faiths . all lest (1) A. BYE in right form a to same a • correspond assignment ⇒ ie = a) Ek)=CaoaoaTz . ATxl=aTx ⇒ Ily) @ Ig Test consistency of self corrections

Goa Pass it sets Sc - to Scoot

- - - - Sc E - Sc Ao Sc B

ai's to # sage Outer Product Tesler : pick random x. * #" Epona is, y

Test so - AH) , .sc#xd=@aiYXiioEajXaj=?jyaiajXiiX2j-i?jbijXlik;]

= Cx sets , ⑤

- - Sc - Atx) so Eaiajakxiy

' ftp.ffaixiof.kbjmyjn-q?,ajbjnXiyjn-- ;D - ④ net so - ) unitary!§µ = Ecxoy ④ def proof contains : -

- linear - = A - all fetus → IF Atx) Eaixi atx If :#an , of truth tables evaluated at Complete description assignment a

. - = - oaf of A all :# 's# ) @ ) ,DB,E for inputs x. y,z - fetus B an a aiajyij B - all Bly deg2 §, T evaluated at a " need value at " " Z only ' - → - 2=8 = @ xx C ' ( , y B. C- all 3 fetus #2" HI (g) deg ,§uaiajak2ijn= but extra info evaluated ata helps vs check consistency

Test so - ACH .sc#xd=@aiYXiioEajXaj=?jyaiajXiiX2j=.?jbijXh-kg) It Picrakedomly = sets cxioxa)

" " test ← since blue hold if b=aoa passes equalities it btaoa : = Ah) . ) B. ( x ) = , bellAha , oxa

"" T.FI#=.eyDDx. " A % ①' 's Da Dam :D. # IF Fact : if I #5 then Pilar ] ztz " if btaoa : trefoils if A. Btc then Prp [ A.B. Ft R 't ] 242 What is prob .

'⇒ @oal.xj-b.x,

- BB - ? Dx. Db # . ⇒ pass with prob 't go Fact ⇒ Pr×{ goal . Xa t b. Xa ) =Y2 Pass with prob ??

if t b - Caoalx , Xz

- ] - then ⇒ xilaoatxatxib.la Ya Fact Pry? so passing test ⇒ fail test 244 Pr [ ) ⇒ safe to assume

- b - aoa ! Test so - ACH Kj .se#lxd--@aiYXiioEajXaj=&jaiaiYiX2j=jbijXh' ) other test similarly passing PYYnddomly-t-sc-BCX.nl# Emma ⇒ safe to assume =

- no ( Xi Yi kids in j , ) def X°Y , Xiyn, , Xiyj , Xnyn - proof contains : -

- linear - = A - all fetus → Atx) Eaixi ATX A :#an Iz of truth tables evaluated at Complete description assignment a

- o are linear faiths . all lest (1) A. BYE in right form a to same a • correspond assignment ⇒ ie = a) Ek)=CaoaoaTz . ATxl=aTx ⇒ Ily) @ Ig Test consistency of self corrections

Goa Pass it sets Sc - to Scoot

- - - - Sc E - Sc Ao Sc B

ai's to # sage Outer Product Tesler : pick random x. * #" Epona is, y

Test so - AH) , .sc#xd=@aiYXiioEajXaj=?jyaiajXiiX2j-i?jbijXlik;]

= Cx sets , ⑤

- - Sc - Atx) so Eaiajakxiy

' ftp.ffaixiof.kbjmyjn-q?,ajbjnXiyjn-- ;D - ④ net so - ) unitary!§µ = Ecxoy ④ =

- ru x . in ( , X. Xyyz , ) Xoy y , ya, , , Xiyj , Xnyn def - Proof contains : -

- linear - = A - all fetus → I Atx) Eaixi atnx A :#an , of truth tables evaluated at Complete description assignment a

Feat Honma! the linear fans . fetus ← GII Yanks:# : all are linear Te s t 41 A. BE in right form

- a b c to set se - B so E queries , , " " ⇒ Aoa to see error ? aoaoa ? unlikely =

- ru x . in ( , X. Kiya , ) Xoy y , ya, , , Xiyj , Xnyn det proof contains : - HUGE②

- linear - = A - all fetus → IF Atx) Eaixi atx If :#an , of truth tables evaluated at Complete description assignment a

. - = - oaf of A all :# 's# ) @ ) ,DB,E for inputs x. gz - fetus B an a aiajyij B - all Bly deg2 §, T evaluated at a " need value at " " Z only ' - → - 2=8 = @ xx C ' ( , y B , C- all 3 fetus #2" HI (g) deg ,§uaiajak2iju= but extra info evaluated ata helps vs check consistency

What does verifier need to check in proof ?

form E- close to linear (1) A in can lest , BYE right

only - correct to access the linear fetus . ← but can self are linear fetus o all # random bits

- ) - OW to same a . correspond assignment * ⇒ Ek)=CaoaoaT2 i.e > . at ⇒ quwiesju, ATH .× ftp.Laoa5.y Te s t consistency of self corrections

(2) a is satisfying assignment

• evaluate to 0 on a all I. 's

- . ro recall Cca, Calista . ) ( ) , Lois ) ftp.kmeut [email protected]÷§ riffa) = T t §

° self - correctors ⇒ recover linear fetus a aoa aoaoa Call , ,

• a but we don't it represents assignment, know

= • . . . . ) a ⇐ CHI ( Ital data) . . ) ( 0,0 ,0 satisfying , , , Satisfiability Te s t : " Pick r Etta #random bits 's ← r 's fthscoeffts 3 - di , , roftdeg - ow compute , Pij Jiji polys Iis Yip Eiji do this ? why = - . :O to SC Aldi - Ln) Wo it ti &.cat # proof get ✓ -

- - . = query - w ' Sc . ) , ) . pm queries , , Bcp Prlpass

se - ) =w if .to " Elk! Inn , fi set . Eilat

Fact ⇒ - D= 's r t Wot w (moda) prkrjc.cat D= , twa so after K times Verify + )=° - means Eri Eula Pr[ ) - ✓ hopefully pass Ik