Polynomial Time with Restricted Use of Randomness

We define a hierarchy of complexity classes that lie between P and RP, yielding a new way of quantifying partial progress towards the derandomization of RP. A standard approach in derandomization is to reduce the number of random bits an algorithm uses. We instead focus on a model of computation that allows us to quantify the extent to which random bits are being used. More specifically, we consider Stack Machines (SMs), which are log-space Turing Machines that have access to an unbounded stack, an input tape of length N , and a random tape of length N O(1) . We parameterize these machines by al

[1]  Zhi-Zhong Chen,et al.  Reducing randomness via irrational numbers , 1997, STOC '97.

[2]  H. Venkateswaran.  Properties that Characterize LOGCFL , 1991, J. Comput. Syst. Sci..

[3]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[4]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[5]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[6]  Daniel A. Spielman,et al.  Randomness efficient identity testing of multivariate polynomials , 2001, STOC '01.

[7]  Nicole Schweikardt,et al.  Randomized computations on large data sets: tight lower bounds , 2006, PODS.

[8]  Noga Alon,et al.  The Space Complexity of Approximating the Frequency Moments , 1999 .

[9]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[10]  Allan Borodin,et al.  Two Applications of Inductive Counting for Complementation Problems , 1989, SIAM J. Comput..

[11]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[12]  Salil P. Vadhan,et al.  Checking polynomial identities over any field: towards a derandomization? , 1998, STOC '98.

[13]  Richard S. Varga,et al.  Proof of Theorem 6 , 1983 .

[14]  E. Kushilevitz,et al.  Communication Complexity: Basics , 1996 .

[15]  David P. Woodruff,et al.  Optimal approximations of the frequency moments of data streams , 2005, STOC '05.

[16]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[17]  Nicole Schweikardt,et al.  Lower bounds for sorting with few random accesses to external memory , 2005, PODS.

[18]  H. Venkateswaran,et al.  Derandomization of probabilistic auxiliary pushdown automata classes , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[19]  Michael E. Saks,et al.  Space lower bounds for distance approximation in the data stream model , 2002, STOC '02.

[20]  Walter L. Ruzzo,et al.  Tree-size bounded alternation(Extended Abstract) , 1979, J. Comput. Syst. Sci..

[21]  Andrew Chi-Chih Yao,et al.  Theory and Applications of Trapdoor Functions (Extended Abstract) , 1982, FOCS.

[22]  Subhash Khot,et al.  Near-optimal lower bounds on the multi-party communication complexity of set disjointness , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[23]  A. Selman.  Structure in Complexity Theory , 1986, Lecture Notes in Computer Science.

[24]  Paul Beame,et al.  On the Value of Multiple Read/Write Streams for Approximating Frequency Moments , 2008, FOCS.

[25]  Nicole Schweikardt,et al.  Reversal complexity revisited , 2006, Theor. Comput. Sci..

[26]  Manindra Agrawal,et al.  Primality and identity testing via Chinese remaindering , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[27]  Walter L. Ruzzo.  On Uniform Circuit Complexity , 1981, J. Comput. Syst. Sci..

[28]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[29]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computations , 1990, STOC '90.

[30]  Peter Bro Miltersen,et al.  2 The Task of a Numerical Analyst , 2022 .

[31]  Avi Wigderson,et al.  P = BPP if E requires exponential circuits: derandomizing the XOR lemma , 1997, STOC '97.

[32]  Ravi Kumar,et al.  An information statistics approach to data stream and communication complexity , 2004, J. Comput. Syst. Sci..

[33]  Eric Allender,et al.  P-uniform circuit complexity , 1989, JACM.

[34]  Luca Trevisan,et al.  Pseudorandom generators without the XOR lemma , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[35]  Stephen A. Cook,et al.  Characterizations of Pushdown Machines in Terms of Time-Bounded Computers , 1971, J. ACM.

[36]  Sumit Ganguly,et al.  Simpler algorithm for estimating frequency moments of data streams , 2006, SODA '06.

[37]  Noam Nisan,et al.  On read-once vs. multiple access to randomness in logspace , 1990, Proceedings Fifth Annual Structure in Complexity Theory Conference.

[38]  Atri Rudra,et al.  Lower bounds for randomized read/write stream algorithms , 2007, STOC '07.

[39]  Christopher Umans,et al.  Pseudo-random generators for all hardnesses , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.