Abstract
The (block-)composition of two Boolean functions \(f : \{0, 1\}^{m} \rightarrow \{0, 1\}, g : \{0, 1\}^{n} \rightarrow \{0, 1\}\) is the function \(f \diamond g\) that takes as inputs m strings \(x_{1}, \ldots , x_{m} \in \{0, 1\}^{n}\) and computes
This operation has been used several times in the past for amplifying different hardness measures of f and g. This comes at a cost: the function \(f \diamond g\) has input length \(m \cdot n\) rather than m or n, which is a bottleneck for some applications.
In this paper, we propose to decrease this cost by “derandomizing” the composition: instead of feeding into \(f \diamond g\) independent inputs \(x_{1}, \ldots , x_{m},\) we generate \(x_{1}, \ldots , x_{m}\) using a shorter seed. We show that this idea can be realized in the particular setting of the composition of functions and universal relations (Gavinsky et al. in SIAM J Comput 46(1):114–131, 2017; Karchmer et al. in Computat Complex 5(3/4):191–204, 1995b). To this end, we provide two different techniques for achieving such a derandomization: a technique based on averaging samplers and a technique based on Reed–Solomon codes.
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References
Arkadev Chattopadhyay, Michal Koucký, Bruno Loff & Sagnik Mukhopadhyay (2017). Simulation Theorems via Pseudorandom Properties. CoRR. http://arxiv.org/abs/1704.06807.
Irit Dinur, Prahladh Harsha, Srikanth Srinivasan & Girish Varma (2015). Derandomized Graph Product Results Using the Low Degree Long Code. In 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, 275–287.
Irit Dinur & Or Meir (2016). Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, 3:1–3:51.
Edmonds, Jeff, Impagliazzo, Russell, Rudich, Steven, Sgall, Jiri: Communication complexity towards lower bounds on circuit depth. Computational Complexity 10(3), 210–246 (2001)
Peter Frankl & Norihide Tokushige: The Erdős-Ko-Rado Theorem for Integer Sequences. Combinatorica 19(1), 55–63 (1999)
Gavinsky, Dmitry, Meir, Or, Weinstein, Omri, Wigderson, Avi: Toward Better Formula Lower Bounds: The Composition of a Function and a Universal Relation. SIAM J. Comput. 46(1), 114–131 (2017)
Justin Gilmer, Michael E. Saks & Srikanth Srinivasan (2013). Composition Limits and Separating Examples for Some Boolean Function Complexity Measures. In Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5-7 June, 2013, 185–196.
Mika Göös (2015). Lower Bounds for Clique vs. Independent Set. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, 1066–1076.
Mika Göös, Pritish Kamath, Toniann Pitassi & Thomas Watson (2017). Query-to-Communication Lifting for PˆNP. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, 12:1–12:16.
Göös, Mika, Lovett, Shachar, Meka, Raghu, Watson, Thomas, Zuckerman, David: Rectangles Are Nonnegative Juntas. SIAM J. Comput. 45(5), 1835–1869 (2016)
Mika Göös, Toniann Pitassi & Thomas Watson (2015). Deterministic Communication vs. Partition Number. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, 1077–1088.
Michelangelo Grigni & Michael Sipser (1991). Monotone Separation of Logspace from NC. In Structure in Complexity Theory Conference, 294–298.
Johan Håstad & Avi Wigderson: Composition of the universal relation. In Advances in computational complexity theory, AMS-DIMACS (1993)
Russell Impagliazzo (1995). Hard-Core Distributions for Somewhat Hard Problems. In 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23-25 October 1995, 538–545.
Russell Impagliazzo & Avi Wigderson (1997). P = BPP if E Requires Exponential Circuits: Derandomizing the XOR Lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, 220–229.
Karchmer, Mauricio, Kushilevitz, Eyal, Nisan, Noam: Fractional Covers and Communication Complexity. SIAM J. Discrete Math. 8(1), 76–92 (1995a)
Karchmer, Mauricio, Raz, Ran, Wigderson, Avi: Super-Logarithmic Depth Lower Bounds Via the Direct Sum in Communication Complexity. Computational Complexity 5(3/4), 191–204 (1995b)
Mauricio Karchmer & Avi Wigderson: Monotone Circuits for Connectivity Require Super-Logarithmic Depth. SIAM J. Discrete Math. 3(2), 255–265 (1990)
Sajin Koroth & Or Meir (2018). Improved composition theorems for functions and relations. In RANDOM.
Pravesh K. Kothari, Raghu Meka & Prasad Raghavendra (2017). Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, 590–603.
Raghav Kulkarni & Avishay Tal: On Fractional Block Sensitivity, p. 2016. Chicago J. Theor. Comput, Sci (2016)
James R. Lee, Prasad Raghavendra & David Steurer (2015). Lower Bounds on the Size of Semidefinite Programming Relaxations. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015, 567–576.
Florence Jessie MacWilliams & Neil James Alexander Sloane (1978). The Theory of Error-Correcting Codes. North-holland Publishing Company, 2nd edition.
Mossel, Elchanan, O'Donnell, Ryan, Servedio, Rocco A.: Learning functions of k relevant variables. J. Comput. Syst. Sci. 69(3), 421–434 (2004)
Noam Nisan & Avi Wigderson: On Rank vs. Communication Complexity. Combinatorica 15(4), 557–565 (1995)
Noam Nisan & David Zuckerman: Randomness is Linear in Space. J. Comput. Syst. Sci. 52(1), 43–52 (1996)
Ran Raz & Pierre McKenzie (1997). Separation of the Monotone NC Hierarchy. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, 234–243.
Ran Raz, Omer Reingold & Salil P. Vadhan (2002). Extracting all the Randomness and Reducing the Error in Trevisan's Extractors. J. Comput. Syst. Sci. 65(1), 97–128. URL https://doi.org/10.1006/jcss.2002.1824.
Ran Raz & Avi Wigderson: Monotone Circuits for Matching Require Linear Depth. J. ACM 39(3), 736–744 (1992)
Razborov, Alexander A.: Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica 10(1), 81–93 (1990)
Alexander A. Razborov (2016). A New Kind of Tradeoffs in Propositional Proof Complexity. J. ACM 63(2), 16:1–16:14.
Ronen Shaltiel (2010). Derandomized Parallel Repetition Theorems for Free Games. In Proceedings of the 25th Annual IEEE Conference on Computational Complexity, CCC 2010, Cambridge, Massachusetts, June 9-12, 2010, 28–37.
Sherstov, Alexander A.: The Pattern Matrix Method. SIAM J. Comput. 40(6), 1969–2000 (2011)
Yaoyun Shi & Yufan Zhu: Quantum communication complexity of block-composed functions. Quantum Information & Computation 9(5), 444–460 (2009)
Avishay Tal (2013). Properties and applications of boolean function composition. In Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, 441–454.
Gábor Tardos & Uri Zwick (1997). The Communication Complexity of the Universal Relation. In Proceedings of the Twelfth Annual IEEE Conference on Computational Complexity, Ulm, Germany, June 24-27, 1997, 247–259.
Xiaodi, Wu, Yao, Penghui, Yuen, Henry S.: Raz-McKenzie simulation with the inner product gadget. Electronic Colloquium on Computational Complexity (ECCC) 24, 10 (2017)
Andrew Chi-Chih Yao (1979). Some Complexity Questions Related to Distributive Computing (Preliminary Report). In STOC, 209–213.
Zuckerman, David: Randomness-optimal oblivious sampling. Random Struct. Algorithms 11(4), 345–367 (1997)
Acknowledgements
We are grateful to Ronen Shaltiel for explaining to us his paper on derandomized parallel repetition (Shaltiel 2010) which served as an inspiration to this work, for pointers to the extractors’ literature, and for numerous valuable discussions. We would also like to thank anonymous referees for comments that improved the presentation of this work. This work was partially supported by the Israel Science Foundation (Grant No. 1445/16).
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Meir, O. On Derandomized Composition of Boolean Functions. comput. complex. 28, 661–708 (2019). https://doi.org/10.1007/s00037-019-00188-1
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DOI: https://doi.org/10.1007/s00037-019-00188-1
Keywords
- Circuit complexity
- circuit lower bounds
- formula complexity
- formula lower bounds
- derandomization
- communication complexity
- Karchmer–Wigderson relations
- KRW conjecture