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On the Power of Regular and Permutation Branching Programs

Authors Chin Ho Lee, Edward Pyne, Salil Vadhan

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Author Details

Chin Ho Lee
  • Harvard University, Cambridge, MA, USA
Edward Pyne
  • MIT, Cambridge, MA, USA
Salil Vadhan
  • Harvard University, Cambridge, MA, USA

Funding

  • Lee, Chin Ho: Supported by Madhu Sudan’s and Salil Vadhan’s Simons Investigator Awards.
  • Pyne, Edward: Supported by an Akamai Presidental Fellowship.
  • Vadhan, Salil: Supported by a Simons Investigator Award.

Acknowledgements

Chin Ho Lee thanks Sepehr Assadi for answering his questions on [Assadi and N, 2021]. We thank the anonymous reviewers for helpful comments.

Cite AsGet BibTex

Chin Ho Lee, Edward Pyne, and Salil Vadhan. On the Power of Regular and Permutation Branching Programs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 44:1-44:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.44

Abstract

We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation. - Regular SOBPs of length n and width ⌊w(n+1)/2⌋ can exactly simulate general SOBPs of length n and width w, and moreover an n/2-o(n) blow-up in width is necessary for such a simulation. Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width poly(n) or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs. - There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod 2 is average-case hard for arbitrary-order permutation ROBPs of exponential width. - There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs. - Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Pseudorandomness
  • Branching Programs

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References

  1. Sepehr Assadi and Vishvajeet N. Graph streaming lower bounds for parameter estimation and property testing via a streaming XOR lemma. In STOC '21 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 612-625. ACM, New York, 2021. URL: https://doi.org/10.1145/3406325.3451110.
  2. Sepehr Assadi and Vishvajeet N. Personal communication, 2022. Google Scholar
  3. Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, 2009. URL: https://doi.org/10.1137/070691954.
  4. C. H. Bennett. Logical reversibility of computation. IBM J. Res. Develop., 17:525-532, 1973. URL: https://doi.org/10.1147/rd.176.0525.
  5. Andrej Bogdanov, William M. Hoza, Gautam Prakriya, and Edward Pyne. Hitting sets for regular branching programs. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 3:1-3:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.3.
  6. Andrej Bogdanov, Periklis A. Papakonstantinou, and Andrew Wan. Pseudorandomness for read-once formulas. In 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science - FOCS 2011, pages 240-246. IEEE Computer Soc., Los Alamitos, CA, 2011. URL: https://doi.org/10.1109/FOCS.2011.57.
  7. Mark Braverman, Gil Cohen, and Sumegha Garg. Pseudorandom pseudo-distributions with near-optimal error for read-once branching programs. SIAM J. Comput., 49(5), 2020. URL: https://doi.org/10.1137/18M1197734.
  8. Mark Braverman, Anup Rao, Ran Raz, and Amir Yehudayoff. Pseudorandom generators for regular branching programs. SIAM J. Comput., 43(3):973-986, 2014. URL: https://doi.org/10.1137/120875673.
  9. Joshua Brody and Elad Verbin. The coin problem, and pseudorandomness for branching programs. In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science - FOCS 2010, pages 30-39. IEEE Computer Soc., Los Alamitos, CA, 2010. Google Scholar
  10. Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom generators from polarizing random walks. Theory Comput., 15:Paper No. 10, 26, 2019. URL: https://doi.org/10.4086/toc.2019.v015a010.
  11. Eshan Chattopadhyay and Jyun-Jie Liao. Optimal error pseudodistributions for read-once branching programs. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 25:1-25:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.25.
  12. Lijie Chen, Xin Lyu, Avishay Tal, and Hongxun Wu. New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs. In Kousha Etessami, Uriel Feige, and Gabriele Puppis, editors, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023), volume 261 of Leibniz International Proceedings in Informatics (LIPIcs), pages 39:1-39:20, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ICALP.2023.39.
  13. Sitan Chen, Thomas Steinke, and Salil P. Vadhan. Pseudorandomness for read-once, constant-depth circuits. CoRR, abs/1504.04675, 2015. URL: https://arxiv.org/abs/1504.04675.
  14. Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, and Amnon Ta-Shma. Error Reduction for Weighted PRGs Against Read Once Branching Programs. In Proceedings of the 36th Computational Complexity Conference (CCC), pages 22:1-22:17, 2021. Google Scholar
  15. Matei David, Periklis A. Papakonstantinou, and Anastasios Sidiropoulos. How strong is nisan’s pseudo-random generator? Inf. Process. Lett., 111(16):804-808, 2011. URL: https://doi.org/10.1016/j.ipl.2011.04.013.
  16. Anindya De. Pseudorandomness for permutation and regular branching programs. In 26th Annual IEEE Conference on Computational Complexity, pages 221-231. IEEE Computer Soc., Los Alamitos, CA, 2011. Google Scholar
  17. Dean Doron, Pooya Hatami, and William M. Hoza. Near-optimal pseudorandom generators for constant-depth read-once formulas. In 34th Computational Complexity Conference, volume 137 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 16, 34. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2019. Google Scholar
  18. Dean Doron, Pooya Hatami, and William M. Hoza. Log-seed pseudorandom generators via iterated restrictions. In 35th computational complexity conference, volume 169 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 6, 36, 2020. Google Scholar
  19. Dean Doron, Raghu Meka, Omer Reingold, Avishay Tal, and Salil Vadhan. Pseudorandom Generators for Read-Once Monotone Branching Programs. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), volume 207 of Leibniz International Proceedings in Informatics (LIPIcs), pages 58:1-58:21, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.58.
  20. P. Erdös and G. Szekeres. A combinatorial problem in geometry. Compositio Math., 2:463-470, 1935. URL: http://www.numdam.org/item?id=CM_1935__2__463_0.
  21. Michael A. Forbes and Zander Kelley. Pseudorandom generators for read-once branching programs, in any order. In 59th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2018, pages 946-955. IEEE Computer Soc., Los Alamitos, CA, 2018. URL: https://doi.org/10.1109/FOCS.2018.00093.
  22. Louis Golowich and Salil P. Vadhan. Pseudorandomness of expander random walks for symmetric functions and permutation branching programs. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 27:1-27:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.27.
  23. Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, and Salil P. Vadhan. Better pseudorandom generators from milder pseudorandom restrictions. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 120-129. IEEE Computer Society, 2012. URL: https://doi.org/10.1109/FOCS.2012.77.
  24. Rohit Gurjar and Ben Lee Volk. Pseudorandom bits for oblivious branching programs. ACM Trans. Comput. Theory, 12(2):Art. 8, 12, 2020. URL: https://doi.org/10.1145/3378663.
  25. Elad Haramaty, Chin Ho Lee, and Emanuele Viola. Bounded independence plus noise fools products. SIAM J. Comput., 47(2):493-523, 2018. URL: https://doi.org/10.1137/17M1129088.
  26. Pooya Hatami and William Hoza. Theory of unconditional pseudorandom generators. Electron. Colloquium Comput. Complex., TR23-019, 2023. URL: https://arxiv.org/abs/TR23-019.
  27. William M. Hoza. Better pseudodistributions and derandomization for space-bounded computation. In Proceedings of the 25th International Conference on Randomization and Computation (RANDOM), pages 28:1-28:23, 2021. Google Scholar
  28. William M. Hoza. Recent progress on derandomizing space-bounded computation. Electron. Colloquium Comput. Complex., TR22-121, 2022. URL: https://arxiv.org/abs/TR22-121.
  29. William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom generators for unbounded-width permutation branching programs. In 12th Innovations in Theoretical Computer Science Conference, volume 185 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 7, 20, 2021. Google Scholar
  30. Russell Impagliazzo, Raghu Meka, and David Zuckerman. Pseudorandomness from shrinkage. J. ACM, 66(2):Art. 11, 16, 2019. URL: https://doi.org/10.1145/3230630.
  31. Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, pages 356-364, Montréal, Québec, Canada, 23-25 May 1994. Google Scholar
  32. Michal Koucký, Prajakta Nimbhorkar, and Pavel Pudlák. Pseudorandom generators for group products: extended abstract. In STOC, pages 263-272, 2011. URL: https://doi.org/10.1145/1993636.1993672.
  33. Chin Ho Lee. Fourier bounds and pseudorandom generators for product tests. In 34th Computational Complexity Conference, volume 137, 2019. Google Scholar
  34. Chin Ho Lee, Edward Pyne, and Salil P. Vadhan. Fourier growth of regular branching programs. In Amit Chakrabarti and Chaitanya Swamy, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022, September 19-21, 2022, University of Illinois, Urbana-Champaign, USA (Virtual Conference), volume 245 of LIPIcs, pages 2:1-2:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.2.
  35. Chin Ho Lee and Emanuele Viola. More on bounded independence plus noise: pseudorandom generators for read-once polynomials. Theory Comput., 16:Paper No. 7, 50, 2020. URL: https://doi.org/10.4086/toc.2020.v016a007.
  36. Raghu Meka, Omer Reingold, and Avishay Tal. Pseudorandom generators for width-3 branching programs. In STOC'19 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 626-637. ACM, New York, 2019. URL: https://doi.org/10.1145/3313276.3316319.
  37. Raghu Meka and David Zuckerman. Small-bias spaces for group products. In Approximation, randomization, and combinatorial optimization, volume 5687 of Lecture Notes in Comput. Sci., pages 658-672. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_49.
  38. Noam Nisan. Pseudorandom generators for space-bounded computation. Combinatorica, 12(4):449-461, 1992. URL: https://doi.org/10.1007/BF01305237.
  39. Edward Pyne and Salil Vadhan. Pseudodistributions that beat all pseudorandom generators (extended abstract). In 36th Computational Complexity Conference, volume 200 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 33, 15, 2021. Google Scholar
  40. Oded Regev. Entropy-based bounds on dimension reduction in L₁. Israel Journal of Mathematics, 195(2):825-832, 2013. Google Scholar
  41. Omer Reingold, Thomas Steinke, and Salil Vadhan. Pseudorandomness for regular branching programs via Fourier analysis. In Approximation, randomization, and combinatorial optimization, volume 8096 of Lecture Notes in Comput. Sci., pages 655-670. Springer, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_45.
  42. Omer Reingold, Luca Trevisan, and Salil Vadhan. Pseudorandom walks on regular digraphs and the RL vs. L problem. In STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 457-466. ACM, New York, 2006. URL: https://doi.org/10.1145/1132516.1132583.
  43. Michael E. Saks and Shiyu Zhou. BP _hspace(s) subseteq dspace(s^3/2). J. Comput. Syst. Sci., 58(2):376-403, 1999. URL: https://doi.org/10.1006/jcss.1998.1616.
  44. Thomas Steinke. Pseudorandomness for permutation branching programs without the group theory. Electron. Colloquium Comput. Complex., 2012. URL: http://eccc.hpi-web.de/report/2012/083.
  45. Thomas Steinke, Salil Vadhan, and Andrew Wan. Pseudorandomness and Fourier-growth bounds for width-3 branching programs. Theory Comput., 13:Paper No. 12, 50, 2017. URL: https://doi.org/10.4086/toc.2017.v013a012.
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