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

2. 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) ]

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

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

5. Juntas Boolean Functions: 11111 1 11 f(f( )=

6. Juntas 11111 1 11 f(f( )= j-junta: depends on at most j coordinates. 11111 1 11

7. Juntas 1111 1 1 1 1 f(f( )= 111 11 j-junta: depends on at most j coordinates.

8. 1 1111 111 11 f(f( )= Definition of j-Junta Test 11111 1 11 111 1 111 11111 11 1 1 1 1 11 1 1 111 1 1 1 11 11 1 1 f(f( )= f(f( f(f( f(f( 1 1 1

9. 1 1111 111 11 f(f( )= 11111 1 11 111 1 111 11111 11 1 1 1 1 11 1 1 111 1 1 1 11 11 1 1 f(f( )= f(f( f(f( f(f( 1 1 1 Accept? Reject? Definition of j-Junta Test

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

11. I-independence test

12. I 1 1 111 1 1 1 111 1

13. I-independence test I w f(f( )= f(f( 1 1 111 1 1 1 111 1 1 1 1 1 1 1

14. I-independence test I f(f( )= w f(f( 1 1 111 1 1 1 111 1 1 1 1 1 1 1 Claim: If Pr[ I is detected] ≤  then f is at most ”  -dependent on I ”  g independent of I, d(f,g)≤  variation f (I)

15. Claim: If Pr[ I is detected] ≤  then f is at most ”  -dependent on I ” I-independence test I f(f( )= w f(f( 1 1 111 1 1 1 111 1 1 1 1 1 1 1  g independent of I, d(f,g)≤  variation f (I)

16. 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( 1 1 111 1 1 1 111 1 1 1 1 1 1 1  g independent of I, d(f,g)≤  variation f (I)

17. 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 !

18. 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!

19. 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.

20. 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],

21. Variations 1

22. I

23. I

24. 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 !!

25. 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.

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

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

28. I J The Unique-Variation

29. I J

30. I J

31. I J

32. I J

33. I J

34. I J

35. I J Q.E.D

36. 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

37. 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??