Document Open Access Logo

Towards Identity Testing for Sums of Products of Read-Once and Multilinear Bounded-Read Formulae

Authors Pranav Bisht , Nikhil Gupta , Ilya Volkovich

PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2023.9.pdf
  • Filesize: 0.93 MB
  • 23 pages

Document Identifiers

Author Details

Pranav Bisht
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA
  • Department of Computer Science and Engineering, IIT(ISM) Dhanbad, India
Nikhil Gupta
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA
Ilya Volkovich
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA

Acknowledgements

The authors would like to thank the anonymous referees for their detailed comments and suggestions on the previous version of the paper.

Cite AsGet BibTex

Pranav Bisht, Nikhil Gupta, and Ilya Volkovich. Towards Identity Testing for Sums of Products of Read-Once and Multilinear Bounded-Read Formulae. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 9:1-9:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.FSTTCS.2023.9

Abstract

An arithmetic formula is an arithmetic circuit where each gate has fan-out one. An arithmetic read-once formula (ROF in short) is an arithmetic formula where each input variable labels at most one leaf. In this paper we present several efficient blackbox polynomial identity testing (PIT) algorithms for some classes of polynomials related to read-once formulas. Namely, for polynomial of the form: - f = Φ_1 ⋅ … ⋅ Φ_m + Ψ₁ ⋅ … ⋅ Ψ_r, where Φ_i,Ψ_j are ROFs for every i ∈ [m], j ∈ [r]. - f = Φ_1^{e₁} + Φ₂^{e₂} + Φ₃^{e₃}, where each Φ_i is an ROF and e_i-s are arbitrary positive integers. Earlier, only a whitebox polynomial-time algorithm was known for the former class by Mahajan, Rao and Sreenivasaiah (Algorithmica 2016). In the same paper, they also posed an open problem to come up with an efficient PIT algorithm for the class of polynomials of the form f = Φ_1^{e₁} + Φ_2^{e₂} + … + Φ_k^{e_k}, where each Φ_i is an ROF and k is some constant. Our second result answers this partially by giving a polynomial-time algorithm when k = 3. More generally, when each Φ₁,Φ₂,Φ₃ is a multilinear bounded-read formulae, we also give a quasi-polynomial-time blackbox PIT algorithm. Our main technique relies on the hardness of representation approach introduced in Shpilka and Volkovich (Computational Complexity 2015). Specifically, we show hardness of representation for the resultant polynomial of two ROFs in our first result. For our second result, we lift hardness of representation for a sum of three ROFs to sum of their powers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Identity Testing
  • Derandomization
  • Bounded-Read Formulae
  • Arithmetic Formulas

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. M. Agrawal. Proving lower bounds via pseudo-random generators. In Proceedings of the 25th FSTTCS, volume 3821 of LNCS, pages 92-105, 2005. Google Scholar
  2. M. Agrawal, N. Kayal, and N. Saxena. Primes is in P. Annals of Mathematics, 160(2):781-793, 2004. Google Scholar
  3. M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hits circuits: Hitting sets, lower bounds for depth-d occur-k formulas and depth-3 transcendence degree-k circuits. SIAM J. Comput., 45(4):1533-1562, 2016. Google Scholar
  4. M. Agrawal, C. Saha, and N. Saxena. Quasi-polynomial hitting-set for set-depth-delta formulas. In Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), pages 321-330, 2013. Google Scholar
  5. M. Anderson, D. van Melkebeek, and I. Volkovich. Derandomizing polynomial identity testing for multilinear constant-read formulae. Computational Complexity, 24(4):695-776, 2015. Google Scholar
  6. D. Angluin, L. Hellerstein, and M. Karpinski. Learning read-once formulas with queries. J. ACM, 40(1):185-210, January 1993. Google Scholar
  7. V. Bhargava, S. Saraf, and I. Volkovich. Linear independence, alternants and applications. In STOC '23: 55th Annual ACM SIGACT Symposium on Theory of Computing, Orlando, Florida, June 20-23, 2023. ACM, 2023. Google Scholar
  8. P. Bisht, N. Gupta, and I. Volkovich. Towards identity testing for sums of products of read-once and multilinear bounded-read formulae. In Electronic Colloquium on Computational Complexity, 2023. URL: https://eccc.weizmann.ac.il/report/2023/109/.
  9. P. Bisht and I. Volkovich. On solving sparse polynomial factorization related problems. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2022, December 18-20, 2022, IIT Madras, Chennai, India, volume 250 of LIPIcs, pages 10:1-10:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2022.10.
  10. D. Bshouty and N. H. Bshouty. On interpolating arithmetic read-once formulas with exponentiation. JCSS, 56(1):112-124, 1998. Google Scholar
  11. N. H. Bshouty and R. Cleve. Interpolating arithmetic read-once formulas in parallel. SIAM J. on Computing, 27(2):401-413, 1998. Google Scholar
  12. N. H. Bshouty, T. R. Hancock, and L. Hellerstein. Learning arithmetic read-once formulas. SIAM J. on Computing, 24(4):706-735, 1995. Google Scholar
  13. N. H. Bshouty, T. R. Hancock, and L. Hellerstein. Learning boolean read-once formulas with arbitrary symmetric and constant fan-in gates. JCSS, 50:521-542, 1995. Google Scholar
  14. N.H. Bshouty, T.R. Hancock, and L. Hellerstein. Learning boolean read-once formulas over generalized bases. J. Comput. Syst. Sci., 50(3):521-542, June 1995. Google Scholar
  15. D. A. Cox, J. Little, and D. O'Shea. Ideals, varieties, and algorithms - An introduction to computational algebraic geometry and commutative algebra (4. ed.). Undergraduate texts in mathematics. Springer, 2015. Google Scholar
  16. R. A. DeMillo and R. J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. Google Scholar
  17. P. Dutta, P. Dwivedi, and N. Saxena. Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 11:1-11:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  18. Z. Dvir and A. Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. SIAM J. on Computing, 36(5):1404-1434, 2007. Google Scholar
  19. M. A. Forbes. Deterministic divisibility testing via shifted partial derivatives. In FOCS, 2015. Google Scholar
  20. M. A. Forbes, R. Saptharishi, and A. Shpilka. Pseudorandomness for multilinear read-once algebraic branching programs, in any order. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC), pages 867-875, 2014. Full version at https://eccc.weizmann.ac.il/report/2013/132. URL: https://doi.org/10.1145/2591796.2591816.
  21. M. A. Forbes and A. Shpilka. Explicit noether normalization for simultaneous conjugation via polynomial identity testing. In APPROX-RANDOM, pages 527-542, 2013. Google Scholar
  22. J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999. Google Scholar
  23. K. O. Geddes, S. R. Czapor, and G. Labahn. Algorithms for computer algebra. Kluwer, 1992. Google Scholar
  24. N. Gupta, C. Saha, and B. Thankey. Equivalence test for read-once arithmetic formulas. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 4205-4272. SIAM, 2023. Google Scholar
  25. R. Gurjar, A. Korwar, and N. Saxena. Identity testing for constant-width, and commutative, read-once oblivious abps. In 31st Conference on Computational Complexity, CCC, pages 29:1-29:16, 2016. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.29.
  26. R. Gurjar, A. Korwar, N. Saxena, and N. Thierauf. Deterministic identity testing for sum of read-once oblivious arithmetic branching programs. In 30th Conference on Computational Complexity, CCC, pages 323-346, 2015. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.323.
  27. T. R. Hancock and L. Hellerstein. Learning read-once formulas over fields and extended bases. In Proceedings of the 4th Annual Workshop on Computational Learning Theory (COLT), pages 326-336, 1991. Google Scholar
  28. J. Heintz and C. P. Schnorr. Testing polynomials which are easy to compute (extended abstract). In Proceedings of the 12th Annual ACM Symposium on Theory of Computing (STOC), pages 262-272, 1980. Google Scholar
  29. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pages 355-364, 2003. Google Scholar
  30. M. Karchmer, N. Linial, I. Newman, M. Saks, and A. Wigderson. Combinatorial characterization of read-once formulae. Discrete Math., 114(1–3):275-282, April 1993. Google Scholar
  31. Z. S. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth 4 multilinear circuits with bounded top fan-in. SIAM J. on Computing, 42(6):2114-2131, 2013. Google Scholar
  32. Z. S. Karnin and A. Shpilka. Deterministic black box polynomial identity testing of depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (CCC), pages 280-291, 2008. Google Scholar
  33. N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 198-207, 2009. Full version at URL: https://eccc.weizmann.ac.il/report/2009/032.
  34. N. Kayal and N. Saxena. Polynomial identity testing for depth 3 circuits. Computational Complexity, 16(2):115-138, 2007. Google Scholar
  35. A. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 216-223, 2001. Google Scholar
  36. M. Kumar and R. Saptharishi. Hardness-randomness tradeoffs for algebraic computation. Bulletin of EATCS, 3(129), 2019. Google Scholar
  37. N. Limaye, S. Srinivasan, and S. Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 804-814. IEEE, 2021. Google Scholar
  38. L. Lovasz. On determinants, matchings, and random algorithms. In L. Budach, editor, Fundamentals of Computing Theory. Akademia-Verlag, 1979. Google Scholar
  39. M. Mahajan, B.V.R. Rao, and K. Sreenivasaiah. Building above read-once polynomials: Identity testing and hardness of representation. Algorithmica, 76:890-909, 2016. Google Scholar
  40. D. Medini and A. Shpilka. Hitting sets and reconstruction for dense orbits in vp_eand ΣΠΣ circuits. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 19:1-19:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  41. D. Minahan and I. Volkovich. Complete derandomization of identity testing and reconstruction of read-once formulas. TOCT, 10(3):10:1-10:11, 2018. URL: https://doi.org/10.1145/3196836.
  42. M. Neunhöffer, 2007. Lecture notes on finite fields - Module MT 5826, Chapter 4, Link - URL: http://www.math.rwth-aachen.de/homes/Max.Neunhoeffer/Teaching/ff/ffchap4.pdf.
  43. S. Peleg and A. Shpilka. Polynomial time deterministic identity testing algorithm for Σ^[3]ΠΣΠ^[2] circuits via edelstein-kelly type theorem for quadratic polynomials. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 259-271. ACM, 2021. Google Scholar
  44. C. Ramya and B.V.R. Rao. Lower bounds for sum and sum of products of read-once formulas. ACM Transactions on Computation Theory (TOCT), 11(2):1-27, 2019. Google Scholar
  45. C. Saha and B. Thankey. Hitting sets for orbits of circuit classes and polynomial families. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 50:1-50:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  46. S. Saraf and I. Volkovich. Blackbox identity testing for depth-4 multilinear circuits. Combinatorica, 38(5):1205-1238, 2018. Google Scholar
  47. N. Saxena and C. Seshadhri. An almost optimal rank bound for depth-3 identities. SIAM J. Comput., 40(1):200-224, 2011. Google Scholar
  48. N. Saxena and C. Seshadhri. Blackbox identity testing for bounded top-fanin depth-3 circuits: The field doesn't matter. SIAM J. Comput., 41(5):1285-1298, 2012. Google Scholar
  49. N. Saxena and C. Seshadhri. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits. J. ACM, 60(5):33, 2013. Google Scholar
  50. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. Google Scholar
  51. A. Shpilka and I. Volkovich. On the relation between polynomial identity testing and finding variable disjoint factors. In Automata, Languages and Programming, 37th International Colloquium (ICALP), pages 408-419, 2010. Full version at URL: https://eccc.weizmann.ac.il/report/2010/036.
  52. A. Shpilka and I. Volkovich. On reconstruction and testing of read-once formulas. Theory of Computing, 10:465-514, 2014. Google Scholar
  53. A. Shpilka and I. Volkovich. Read-once polynomial identity testing. Computational Complexity, 24(3):477-532, 2015. Google Scholar
  54. A. Sinhababu and T. Thierauf. Factorization of polynomials given by arithmetic branching programs. computational complexity, 30(2):1-47, 2021. Google Scholar
  55. I. Volkovich. Deterministically factoring sparse polynomials into multilinear factors and sums of univariate polynomials. In APPROX-RANDOM, pages 943-958, 2015. Google Scholar
  56. I. Volkovich. Characterizing arithmetic read-once formulae. ACM Transactions on Computation Theory (ToCT), 8(1):2, 2016. URL: https://doi.org/10.1145/2858783.
  57. R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation, pages 216-226, 1979. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact