research-article

Identity Testing for Radical Expressions

Authors:

Mahsa Shirmohammadi,

James WorrellAuthors Info & Claims

LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

Article No.: 8, Pages 1 - 11

https://doi.org/10.1145/3531130.3533331

Published: 04 August 2022 Publication History

Abstract

We study the Radical Identity Testing problem (RIT): Given an algebraic circuit representing a polynomial and nonnegative integers a1, …, ak and d1, …, dk, written in binary, test whether the polynomial vanishes at the real radicals, i.e., test whether . We place the problem in coNP assuming the Generalised Riemann Hypothesis (GRH), improving on the straightforward PSPACE upper bound obtained by reduction to the existential theory of reals. Next we consider a restricted version, called 2-RIT, where the radicals are square roots of prime numbers, written in binary. It was known since the work of Chen and Kao [16] that 2-RIT is at least as hard as the polynomial identity testing problem, however no better upper bound than PSPACE was known prior to our work. We show that 2-RIT is in coRP assuming GRH and in coNP unconditionally. Our proof relies on theorems from algebraic and analytic number theory, such as the Chebotarev density theorem and quadratic reciprocity.

References

[1]

M. Agrawal and S. Biswas. 2003. Primality and identity testing via chinese remaindering. Journal of the ACM (JACM) 50, 4 (2003), 429–443.

Digital Library

[2]

M. Agrawal, N. Kayal, and N. Saxena. 2004. PRIMES is in P. Annals of mathematics(2004), 781–793.

[3]

S. Akshay, N. Balaji, A. Murhekar, R. Varma, and N. Vyas. 2020. Near-Optimal Complexity Bounds for Fragments of the Skolem Problem. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik.

[4]

E. Allender, P. Bürgisser, J. Kjeldgaard-Pedersen, and P. B. Miltersen. 2006. On the complexity of numerical analysis. In CCC ’06. 331–339.

[5]

V. Arvind and P. P. Kurur. 2003. Upper Bounds on the Complexity of Some Galois Theory Problems. In Algorithms and Computation, 14th International Symposium, ISAAC 2003, Kyoto, Japan, December 15-17, 2003, Proceedings(Lecture Notes in Computer Science, Vol. 2906). Springer, 716–725.

[6]

E. Bach, J. Driscoll, and J. Shallit. 1993. Factor Refinement. Journal of Algorithms 15, 2 (1993), 199–222.

Digital Library

[7]

N. Balaji, K. Nosan, M. Shirmohammadi, and J. Worrell. 2022. Identity testing for radical expressions. https://arxiv.org/abs/2202.07961

[8]

N. Balaji, S. Perifel, M. Shirmohammadi, and J. Worrell. 2021. Cyclotomic Identity Testing and Applications. In ISSAC ’21: International Symposium on Symbolic and Algebraic Computation, Virtual Event, Russia, July 18-23, 2021. ACM, 35–42.

[9]

M. Benedikt, R. Lenhardt, and J. Worrell. 2013. LTL model checking of interval Markov chains. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Springer, 32–46.

[10]

P. Berman, M. Karpinski, L. Larmore, W. Plandowski, and W. Rytter. 2002. On the complexity of pattern matching for highly compressed two-dimensional texts. J. Comput. System Sci. 65, 2 (2002), 332–350.

[11]

J. Blömer. 1992. How to Denest Ramanujan’s Nested Radicals. In 33rd Annual Symposium on Foundations of Computer Science. IEEE Computer Society, 447–456.

[12]

J. Blömer. 1997. Denesting by Bounded Degree Radicals. In Algorithms - ESA ’97, 5th Annual European Symposium, Proceedings(Lecture Notes in Computer Science, Vol. 1284), R. E. Burkard and G. J. Woeginger (Eds.). Springer, 53–63.

[13]

J. Blömer. 1998. A Probabilistic Zero-Test for Expressions Involving Roots of Rational Numbers. In Algorithms — ESA’ 98. 151–162.

[14]

J. Canny. 1988. Some algebraic and geometric computations in PSPACE. In Proceedings of the twentieth annual ACM symposium on Theory of computing. 460–467.

Digital Library

[15]

K. Chatterjee and R. Ibsen-Jensen. 2014. The complexity of ergodic mean-payoff games. In International Colloquium on Automata, Languages, and Programming. Springer, 122–133.

[16]

Z. Chen and M. Kao. 2000. Reducing randomness via irrational numbers. SIAM J. Comput. 29, 4 (2000), 1247–1256.

Digital Library

[17]

V. Chonev, J. Ouaknine, and J. Worrell. 2016. On the complexity of the orbit problem. Journal of the ACM (JACM) 63, 3 (2016), 1–18.

Digital Library

[18]

H. Cohen. 2013. A Course in Computational Algebraic Number Theory. Springer Berlin Heidelberg.

Digital Library

[19]

Max Dehn. 1912. Transformation der Kurven auf zweiseitigen Flächen. Math. Ann. 72, 3 (1912), 413–421.

[20]

E. D. Demaine, J. S. B. Mitchell, and J. O’Rourke. 2006. The open problems project: Problem 33.

[21]

R. A. Demillo and R. J. Lipton. 1978. A probabilistic remark on algebraic program testing. Inform. Process. Lett. 7, 4 (1978), 193–195.

[22]

J. Esparza, S. Kiefer, and M. Luttenberger. 2008. Solving monotone polynomial equations. In Fifth Ifip International Conference On Theoretical Computer Science–Tcs 2008. Springer, 285–298.

[23]

K. Etessami and M. Yannakakis. 2010. On the Complexity of Nash Equilibria and Other Fixed Points. SIAM J. Comput. 39, 6 (2010), 2531–2597.

Digital Library

[24]

M. R. Garey, R. L. Graham, and D. S. Johnson. 1976. Some NP-complete geometric problems. In Proceedings of the eighth annual ACM symposium on Theory of computing. ACM, 10–22.

Digital Library

[25]

C. Haase and S. Kiefer. 2015. The odds of staying on budget. In International Colloquium on Automata, Languages, and Programming. Springer, 234–246.

[26]

C. Haase, S. Kiefer, and M. Lohrey. 2017. Counting problems for Parikh images. In Leibniz International Proceedings in Informatics, LIPIcs, Vol. 83. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 12.

[27]

H. Iwaniec and E. Kowalski. 2004. Analytic number theory. Vol. 53. AMS.

[28]

E. Kaltofen. 1989. Factorization of Polynomials Given by Straight-Line Programs.Adv. Comput. Res. 5(1989), 375–412.

[29]

N. Kayal and C. Saha. 2012. On the sum of square roots of polynomials and related problems. ACM Transactions on Computation Theory (TOCT) 4, 4 (2012), 1–15.

Digital Library

[30]

S. Kiefer, I. Marusic, and J. Worrell. 2015. Minimisation of Multiplicity Tree Automata. Foundations of Software Science and Computation Structures LNCS 9034 (2015), 297.

[31]

S. Kiefer, A. Murawski, J. Ouaknine, B. Wachter, and J. Worrell. 2013. On the Complexity of Equivalence and Minimisation for Q-weighted Automata. Logical Methods in Computer Science 9 (2013).

[32]

S. Kiefer and D. Wojtczak. 2011. On probabilistic parallel programs with process creation and synchronisation. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Springer, 296–310.

[33]

P. Koiran. 1996. Hilbert’s Nullstellensatz Is in the Polynomial Hierarchy. Journal of Complexity 12, 4 (1996), 273–286.

Digital Library

[34]

D. König and M. Lohrey. 2018. Parallel identity testing for skew circuits with big powers and applications. Int. J. Algebra Comput. 28, 6 (2018), 979–1004.

[35]

H. Davenport H. L. and Montgomery. 2013. Multiplicative Number Theory. Springer New York.

[36]

J. C. Lagarias and A. M. Odlyzko. 1977. Effective versions of the Chebotarev density theorem. In Algebraic Number Fields. Academic Press, 409–464.

[37]

S. Landau. 1984. Polynomial Time Algorithms for Galois Groups. In EUROSAM 84, International Symposium on Symbolic and Algebraic Computation, Cambridge, England, UK, July 9-11, 1984, Proceedings(Lecture Notes in Computer Science, Vol. 174), J. P. Fitch (Ed.). Springer, 225–236.

[38]

A. Lechner, J. Ouaknine, and J. Worrell. 2015. On the complexity of linear arithmetic with divisibility. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, 667–676.

[39]

Markus Lohrey. 2014. The compressed word problem for groups. Springer.

[40]

L. Lovász. 1979. On determinants, matchings, and random algorithms. In FCT, Vol. 79. 565–574.

[41]

D. Richardson. 1992. The elementary constant problem. In Papers from the international symposium on Symbolic and algebraic computation. 108–116.

[42]

D. Richardson. 1997. How to recognize zero. Journal of Symbolic Computation 24, 6 (1997), 627–645.

Digital Library

[43]

S. and G. L. Miller. 1983. Solvability by Radicals is in Polynomial Time. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA. ACM, 140–151.

[44]

Jean-Pierre Serre. 1981. Quelques applications du théoreme de densité de Chebotarev. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 54, 1 (1981), 123–201.

[45]

P. Stevenhagen, H. W. Lenstra, and Jr.1995. Chebotarev and his density theorem.

[46]

I. Stewart and D. Tall. 2001. Algebraic Number Theory and Fermat’s Last Theorem: Third Edition. Taylor & Francis.

[47]

I. N. Stewart. 1973. Galois theory. Chapman and Hall.

[48]

M. Ummels and D. Wojtczak. 2011. The complexity of Nash equilibria in limit-average games. In International Conference on Concurrency Theory. Springer, 482–496.

Cited By

Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704862
Jindal GGaillard L(2023)On the Order of Power Series and the Sum of Square Roots ProblemProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597079(354-362)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597079

Recommendations

Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ABP). In this work, we give an exponential lower bound of exp (n/k^O(k)) on the width of any read-k ...
Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits
CCC '21: Proceedings of the 36th Computational Complexity Conference

Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates ...
Derandomizing polynomial identity tests means proving circuit lower bounds

We show that derandomizing Polynomial Identity Testing is essentially equivalent to proving arithmetic circuit lower bounds for NEXP. More precisely, we prove that if one can test in polynomial time (or even nondeterministic subexponential time, ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

August 2022

817 pages

ISBN:9781450393515

DOI:10.1145/3531130

Editor:
Christel Baier

Copyright © 2022 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

Sponsors

SIGLOG: ACM Special Interest Group on Logic and Computation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 August 2022

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed limited

Conference

LICS '22

Sponsor:

SIGLOG

LICS '22: 37th Annual ACM/IEEE Symposium on Logic in Computer Science

August 2 - 5, 2022

Haifa, Israel

Acceptance Rates

Overall Acceptance Rate 215 of 622 submissions, 35%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
88
Total Downloads

Downloads (Last 12 months)29
Downloads (Last 6 weeks)18

Reflects downloads up to 13 Jan 2025

Other Metrics

View Author Metrics

Citations

Cited By

Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704862
Jindal GGaillard L(2023)On the Order of Power Series and the Sum of Square Roots ProblemProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597079(354-362)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597079

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

HTML Format

View this article in HTML Format.

Media

Figures

Other

Tables

View Table of Contents