On K-wise Independent Distributions and Boolean Functions

We pursue a systematic study of the following problem. Let f:{0,1}^n -> {0,1} be a (usually monotone) Boolean function whose behaviour is well understood when the input bits are identically independently distributed. What can be said about the behaviour of the function when the input bits are not completely independent, but only k-wise independent, i.e. every subset of k bits is independent? more precisely, how high should k be so that any k-wise independent distribution "fools" the function, i.e. causes it to behave nearly the same as when the bits are completely independent? We analyze several well known Boolean functions (including AND, Majority, Tribes and Percolation among others), some of which turn out to have surprising properties. In some of our results we use tools from the theory of the classical moment problem, seemingly for the first time in this subject, to shed light on these questions.

[1]  Avi Wigderson,et al.  Pairwise Independence and Derandomization , 2006, Found. Trends Theor. Comput. Sci..

[2]  Sergey Yekhanin,et al.  Long nonbinary codes exceeding the Gilbert-Varshamov bound for any fixed distance , 2004, IEEE Transactions on Information Theory.

[3]  I. Benjamini,et al.  Random walks with k-wise independent increments , 2004, math/0405601.

[4]  Ryan O'Donnell,et al.  New degree bounds for polynomial threshold functions , 2003, STOC '03.

[5]  Oded Goldreich,et al.  Simple Constructions of Almost k -wise Independent Random Variables , 2002 .

[6]  Johan Håstad,et al.  A Slight Sharpening of LMN , 2001, J. Comput. Syst. Sci..

[7]  Aravind Srinivasan,et al.  Improved Algorithms via Approximations of Probability Distributions , 2000, J. Comput. Syst. Sci..

[8]  Lih-Yuan Deng,et al.  Orthogonal Arrays: Theory and Applications , 1999, Technometrics.

[9]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998, math/9811157.

[10]  Olle Häggström,et al.  Infinite clusters in dependent automorphism invariant percolation on trees , 1997 .

[11]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[12]  Roy Meshulam,et al.  On Subsets of Finite Abelian Groups with No 3-Term Arithmetic Progressions , 1995, J. Comb. Theory A.

[13]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[14]  Y. Mansour,et al.  On construction of k-wise independent random variables , 1994, STOC '94.

[15]  Nimrod Megiddo,et al.  Constructing small sample spaces satisfying given constraints , 1993, SIAM J. Discret. Math..

[16]  Leonard J. Schulman.  Sample spaces uniform on neighborhoods , 1992, STOC '92.

[17]  Noam Nisan,et al.  Approximations of general independent distributions , 1992, STOC '92.

[18]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[19]  Bonnie Berger,et al.  Simulating (logcn)-wise independence in NC , 1991, JACM.

[20]  Noga Alon,et al.  Construction Of Asymptotically Good Low-rate Error-correcting Codes Through Pseudo-random Graphs , 1991, Proceedings. 1991 IEEE International Symposium on Information Theory.

[21]  James Aspnes,et al.  The expressive power of voting polynomials , 1991, STOC '91.

[22]  R. Durrett.  Probability: Theory and Examples , 1993 .

[23]  Noam Nisan,et al.  Approximate Inclusion-Exclusion , 1990, STOC '90.

[24]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[25]  Noam Nisan,et al.  Constant depth circuits, Fourier transform, and learnability , 1989, 30th Annual Symposium on Foundations of Computer Science.

[26]  Moni Naor,et al.  The probabilistic method yields deterministic parallel algorithms , 1989, 30th Annual Symposium on Foundations of Computer Science.

[27]  Endre Boros,et al.  Closed Form Two-Sided Bounds for Probabilities that At Least r and Exactly r Out of n Events Occur , 1989, Math. Oper. Res..

[28]  Oded Goldreich,et al.  On the power of two-point based sampling , 1989, J. Complex..

[29]  M. Luby.  Removing randomness in parallel computation without a processor penalty , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[30]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[31]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[32]  Oded Goldreich,et al.  The bit extraction problem or t-resilient functions , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[33]  A. Wigderson,et al.  A fast parallel algorithm for the maximal independent set problem , 1985, JACM.

[34]  Richard M. Karp,et al.  A fast parallel algorithm for the maximal independent set problem , 1984, STOC '84.

[35]  János Komlós,et al.  Storing a sparse table with O(1) worst case access time , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[36]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[37]  M. Kreĭn,et al.  The Markov moment problem and extremal problems : ideas and problems of P.L. Čebyšev and A.A. Markov and their further development , 1977 .

[38]  Neil J. A. Sloane,et al.  The theory of error-correcting codes (north-holland , 1977 .

[39]  A. Joffe.  On a Set of Almost Deterministic $k$-Independent Random Variables , 1974 .

[40]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .