Putting a Junta to the Test Joint work with Eldar Fischer, Dana Ron, Shmuel Safra, and Alex Samorodnitsky Guy Kindler

Property Testing o P – property o f – input o Goal: Distinguish, using the fewest possible queries, between f has P f is  -far from having P d(f,g) = Pr x [ f(x)≠g(x) ]

History o Testing Proofs (PCP): BLR o Combinatorial properties: GGR o PRS: Logic AND, monotonous DNF.

Juntas Boolean Functions: f(f( )= n entries

Juntas Boolean Functions: f(f( )=

Juntas f(f( )= j-junta: depends on at most j coordinates

Juntas f(f( )= j-junta: depends on at most j coordinates.

f(f( )= Definition of j-Junta Test f(f( )= f(f( f(f( f(f( 1 1 1

f(f( )= f(f( )= f(f( f(f( f(f( Accept? Reject? Definition of j-Junta Test

Before we test juntas … Given a set I of coordinates, can we verify that f does not depend on it?

I-independence test

I

I-independence test I w f(f( )= f(f(

I-independence test I f(f( )= w f(f( Claim: If Pr[ I is detected] ≤  then f is at most ”  -dependent on I ”  g independent of I, d(f,g)≤  variation f (I)

Claim: If Pr[ I is detected] ≤  then f is at most ”  -dependent on I ” I-independence test I f(f( )= w f(f(  g independent of I, d(f,g)≤  variation f (I)

Claim: If Pr[ I is detected] ≤  then f is at most ”  -dependent on I ” I-independence test Proof: Let g(w  z 0 )=Maj z {f(w  z)} Define p(w)=Pr z [f(w  z)  ≠g(w  z)]  variation f (I) I f(f( )= w f(f(  g independent of I, d(f,g)≤  variation f (I)

1. Partition the coordinates into r subsets. The j-Junta Test I1I1 IrIr r=10j 2 A j-junta is independent of all but j subsets !

1. Partition the coordinates into r subsets. 2. Run the independence-test  r/  times on each subset. The j-Junta Test I1I1 IrIr If f has variation “  /r” on a subset, it is almost surely detected!

1. Partition the coordinates into r subsets. 2. Run the independence-test  r/  times on each subset. 3. Accept if ≤j of the subsets are detected. The j-Junta Test I1I1 IrIr Completeness: Soundness: If f is  far from being a junta, then the test rejects with probability ½. Soundness: If f is  far from being a junta, then the test rejects with probability ½.

Lemma: For every Boolean f, unless f is  –close to a j-junta, w.h.p., the test rejects. Soundness at least j+1 subsets have variation  /r  J  [n], | J |≤j, variation f ([n]\ J )<  over the partitions of [n],

Variations 1

I

I

We’ll prove that unless f is  –close to a j-junta, w.h.p. the test rejects. at least j+1 subsets have variation  /r | J |≤j, and variation f ([n]\ J )<  over the partitions of [n], For t   /r, let I1I1 IrIr If | J |>j, easy !!

We’ll prove that unless f is  –close to a j-junta, w.h.p. the test rejects. at least j+1 subsets have variation  /r | J |≤j, and variation f ([n]\ J )<  over the partitions of [n], Fix t   /r and let I1I1 IrIr Assume variation f ([n]\ J )> . Then !!! Claim: w.h.p.

o Recall: For each i in Claim: w.h.p.

I J o Recall: For each i in Claim: w.h.p.

I J The Unique-Variation

I J

I J

I J

I J

I J

I J

I J Q.E.D

Other Results o Shown number of queries: j 4 /  o Using adaptivity: j 3 /  o Using two-sidedness: j 2 /  o Allowing (2j)-juntas: j 2 /  o Variables in General probability spaces. o “f” is “g” test, where g is a j-junta. o Lower Bound: at least (j) 1/2 queries are needed

Open Problems o Improve lower bound to j 2 /  (perhaps via random-walk convergence on Z 2 ) o “f is g” for non-juntas? o Characterize efficiently testable properties via Fourier transform??