Rajat Mittal IIT Kanpur Boolean functions or Central object of study in Computer Science AND OR Majority Parity With real range real vector space of dimension Parities for all ID: 1030736
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1. Exact degree of a symmetric Boolean functionRajat Mittal (IIT Kanpur)
2. Boolean functions or Central object of study in Computer ScienceAND, OR, Majority, ParityWith real range, real vector space of dimension Parities for all , , form a basis
3. Polynomial representationUsing interpolation, any can be written as, UniqueMultilinear (degree one in each variable) are Fourier coefficients if in domain
4. Degree of a representationDegree*: size of biggest , s.t., * Degree remains same even if
5. Symmetric functionImportant subclass of Boolean functionsInvariant under permutation of variablesDepends only on the weight of inputForms a real subspace of dimension Elementary symmetric polynomials, They form a basis,
6. What is the minimum possible degree of a Boolean symmetric function?non-constant
7.
8. Equivalence: multivariate and univariate Trivial bound: Minimum degree at least
9. OutlineProblem descriptionBest bound (GR97)Known resultsPossible approaches ConjectureRandom function (GR97)More than two values (CST10)Number theory approachesDual witness
10. What is the minimum possible degree of a non-constant Boolean symmetric function?
11. Finding the coefficientsWe pick the representation when
12. Matrix equation |x|=j
13. Taking inverse j
14. Lemma: Idea: Look at If then if and only if is constant
15. If then any symmetric Boolean f on variables has full degree (. Every non-constant symmetric Boolean f has degree at least Distance between primes
16. Random Lemma: for Idea: Fix At most one choice of , such that, Fraction of without full degree less than
17. More than two outputsOriginal Q: minimum possible degree of a univariate function ? New Q: minimum possible degree of a univariate function ?
18. CST10: bound on Upper boundLower boundTrivial case23001(Upper boundLower boundTrivial case23001
19. OutlineProblem descriptionBest bound (GR97)Known resultsPossible approaches ConjectureRandom function (GR97)More than two values (CST10)Number theory approachesDual witness
20. Results
21. ConjectureIf then Every non-constant symmetric Boolean f has degree at least
22. Experimental results (GR97) up to for Constructed families with gap two and three
23. OutlineProblem descriptionBest bound (GR97)Known resultsPossible approaches ConjectureRandom function (GR97)More than two values (CST10)Number theory approachesDual witness
24. Good sequencesDef: Sequence is good if Thm: Degree less than iff all sequences are good.
25. Good sequencesObs: Given a sequence , only one good extension is possible (in reals). Conjecture: Given a non-constant sequence , can’t good extend it to more than a constant times.
26. Conjecture for school studentsAt what level does this become a constant sequence?
27. OutlineProblem descriptionBest bound (GR97)Known resultsPossible approaches ConjectureRandom function (GR97)More than two values (CST10)Number theory approachesDual witness
28. Approximate degreeMinimum degree of , such that, PrimalDual
29. Dual WitnessA function has approximate degree more than iffVery useful lower bound technique Lower bounds for explicit functions [BT14, BKT17, BMTW19]Composition of approximate degree [She09, Lee09, BT17,BKT17, BKT18]
30. Witness: more formulationsA function has degree more than iff has degree less than Taking the univariate analogue has degree less than
31. Restrict to , need of small degree, s.t.,
32. Def: For , function if otherwise Another conjecture: For all , there is a small non-zero Fourier coefficient.
33. Almost Vandermonde identity ji Another conjecture: Arbitrary sum of first few rows can’t be zero up to first columns
34. ConclusionMinimum degree of a non-constant symmetric Boolean function?A very natural questionSeems it should be almost full degreeWe know We conjecture Many related problemsCan we generalize it to intervals [KLMMV09,ST10]?For what ?Many possible approaches in different domains
35. THANKS