Today : cutting Planes,

Interpolation+ Automat eoabililq

CUTTING PLANES -ChvatRefutation System

Lines are linearinequalities §ai× ,> K

ai,

K are integers

Axing xizo

1- X.I 1

Ruled D addition Eaixisk ,Eabik.

=k'

Ecautbi )x,

.'s ktk

'

@ multiplication Eauxizk ⇒ C. Eqxi 's c. K

C > o

�3� E 2. AOXIZK ⇒ Eaixizfkg

¥ QX,

t 2×2 = I

X,

t Xz = I

A CP refutation of a set of linear InequalitiesL = El

,lz . . lq } is a sequence 9 inequalities

{ s, { . .

.Sm } st .

each Si .

is either In <

or Ts an axiom,

or follows from z

piluiny lines by a rule,

- final

line Sm-

⇐

Lengthof apndf is the sum of the sites 9 all ofhe colffs written In binary

= the # of lines in the Pf fgmeomgafpggndfgtahdmadf

Lemma CP p- simulates Resolution

⇐= gn . . ncm be an unsat Formula

with a Res refutation R.

then the corresponding family 9 linear ineq 's

has a CP refutation of site IN

Ci = ( ×,

✓ Izvxz ) → x. t ( I - XDTX ,=/

l ×

,vIu)(×wX3u×DCIzyXDCI3

)X. + txzsl Xztxztxpl txtxpl ( -1321

¥×3 )

u±¥¥I¥E||×3

2×3>-1xssl

\$ 031

On the other hand CP is more powerfulCps has really short ref 's of PHP

.

P, .tl?zt..tP.n=1

1 'Put tpzj =L

P.at Put . - + Pail -

tfecn ]

Pna, ,

+ . -. + Payn Put . - . + Pa

, ,jE I÷

is= ntl Ed ; En

Automatiaabililgtfeasibsnpam

InterpolationLet Acp ,§)^ BCF , F) be an UNSAT CNF

A Craig interplant for this Formula is

--

afunction CCF ) st

.

f& Cca )

=D⇒ ACOL

, f) is UNSST

yx can ) = 1 ⇒ BK,

F) is UNSAT

If f occurs only positivity in A,

then CCP ) is monotone

Deft A proof system P has feasible

interpolation propht ) if tuwsst f=A( a g) ^B( EDwith a @ - proof Nste s

,F a crag interplant

circuit for f of size pofc D

P has Monday fed inter . if

f monotone f=A( FF )^B( F. F) with . . s

2 a monotone Cray intep circuit for f

of six poycs)

Thy Let 8 be a prop . pf system

�1� If P has feces intup and NP

§tP1poly

then 8 is not pay bded

�2� If 8 has monotone teas infofrpdatimthe P is not pay bdld

Pf=th .

�1� S pose8 has feas

'

interp .

+ is po} bded

Let Ac PF ) : F encodes

ABCNFformula

and of is a sat . ass . for F

Let BCFT ) : F encodes a CNF formula

and F encodes a P - ref .of F

since 8 is phbded ,A^B has $eyK PM in

n ( * vans under } f) .

Since P is PM bded And has a

8- of Asia pohcn)

Since P has feasint,

ZE a sje pshcn)

that tells Us if AC qf)unsst

or CBCQF ) is UNSOAT

.

.

.C is deciding SAT in PM sjl

-

�2� Spose P has monotone feas interp &

8 - is pm bded .

let Xp , g) : P encode a sigh mn

vertices

of encode a K - cliqueA says

-

if i ,j E K . cliquethen Cyj)Ardis In F

B ( pjr ) : F. of encode ⇐ -1) co cliqueBsaysci.DE

B says

@itsamisdcimassin

the

#§§§§seen ci ;D is not in F Ki

O ,- wdigve

If P -

is pshbdld, partition of V

An B has a fief of into El piecessite PM Cn )

.since P

also has monotone feces . intep ,

this means there is a monotone pohged

avant CCD : 1 If 4 is a K - clique0 it L is a Ckri ) wcliqve

Robby FE,

K= if there is no she znE monotone

circuit C

We will show :

i. Resolution has monotone feas intupalso has

' ' "

z ,

CP also has monotone feas intep& feas intep

aas a corollary 9 the pair this

this implies expl CBs forRes r Cps refutations of chgiycodyue

split formulae

You can't had your cake a eat it too ;

Lennylet P be a pmf system

closed under restrictions.

Then if P '

is automata able then

-P has feces . interpolation

If FDhas feas intep ⇒

it is weak a we can pbhe Lower bds for it

If P doesnt had feces intep ⇒

it is corpllx ,a we cant find the pfs

PfotlemmaAssume P is aut .

Let s be aP ' ref of Acp oil ^B( fir )

i. Run ant . alg for P in ACFQTNBCFF )

to get a P - Prof s'

,

@= PM (

Pggiemsbfp

2. Suppose B ( I. F) is safe 's fable

of NMB

r let BCI,5) be 1

Since P is closed under restrictions

ACE ,§ ) ^ BCI,8) also has a 8- ref

C CI ) : of sie pages' ) =p Bcs )

•

so we will run aut. alg on ACI

, f)for pays

' ) stepsIf it outputs a valid ref → say Acdiq ) UNSAT

If it doesnt ⇒ B ( IT ) Is VNSAT

Onnegakieresults

Thy BPR,

KP

If factoring Blum integers is hard

then Else pwofs do not had

feces.

interpolation

If factoring Blum interns is really hard

threw Ag - Frye pfs . . .

=

PACKING Valiant

Concept class

EEx C = all pdyje in nDNF formulas

@ = all pshge formulas

e = all linear threshold functions

{en } - is ( E. s ) - PK learnable Thing

free. if there is an dlg A ( unknown distribution

gets access to pairs @r4,15 )

( x ,fk ) ) and

after poycn ) stgs ,

A output some hypothecs hn :{ 0,43"→lqD

With prob .

Cl - £ ) Ira @IN = fed ] > 1- E

Known : Dec trees are

DNFS ?

ACO - circuits

my formulas ] muaimntabmu'

under

crypto assumptions