Skip to main content
SpringerLink
Log in

Evaluation of Circuits Over Nilpotent and Polycyclic Groups

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study the circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups. The best upper bound for this problem is coRP (the complements of problems in randomized polynomial time), which is shown by a reduction to polynomial identity testing for arithmetic circuits. Conversely, the compressed word problem for the linear group \({\mathsf {SL}}_3({\mathbb {Z}})\) is equivalent to polynomial identity testing. In the paper, we show that the compressed word problem for every finitely generated nilpotent group is in \({\mathsf {DET}} \subseteq {\mathsf {NC}}^2\). Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits. It is a major open problem, whether polynomial identity testing for skew arithmetic circuits can be solved in polynomial time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Explicitly, the result is stated in [18, Corollary 6.5], where the authors note that Eberly’s reduction [17] from iterated polynomial multiplication to iterated integer multiplication is actually an \({\mathsf {AC}}^0\)-reduction, which yields a \({\mathsf {DLOGTIME}}\)-uniform \({\mathsf {TC}}^0\) bound with the main result from [18].

  2. Every finitely generated extension field of a perfect field has a separating transcendence base and every prime field is perfect.

  3. It is probably known to experts that G is polycyclic. Since we could not find an explicit proof, we present the arguments for completeness.

References

  1. Agrawal, M., Biswas, S.: Primality and identity testing via chinese remaindering. J. Assoc. Comput. Mach. 50(4), 429–443 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allender, E., Beals, R., Ogihara, M.: The complexity of matrix rank and feasible systems of linear equations. Comput. Complex. 8(2), 99–126 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allender, E., Bürgisser, P., Kjeldgaard-Pedersen, J., Miltersen, P.B.: On the complexity of numerical analysis. SIAM J. Comput. 38(5), 1987–2006 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Allender, E., Jiao, J., Mahajan, M., Vinay, V.: Non-commutative arithmetic circuits: depth reduction and size lower bounds. Theor. Comput. Sci. 209(1–2), 47–86 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Àlvarez, C., Jenner, B.: A very hard log-space counting class. Theor. Comput. Sci. 107(1), 3–30 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arora, S., Barak, B.: Computational Complexity—A Modern Approach. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  7. Arvind, V., Joglekar, P.S.: Arithmetic circuit size, identity testing, and finite automata. Electron. Colloq. Comput. Complex. (ECCC) 16, 26 (2009)

    Google Scholar 

  8. Auslander, L.: On a problem of Philip Hall. Ann. Math. 86(2), 112–116 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  9. Barrington, D.A.M.: Bounded-width polynomial-size branching programs recognize exactly those languages in \(\text{ NC }^1\). J. Comput. Syst. Sci. 38, 150–164 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Barrington, D.A.M., Thérien, D.: Finite monoids and the fine structure of \(\text{ NC }^{1}\). J. Assoc. Comput. Mach. 35(4), 941–952 (1988)

    Article  MathSciNet  Google Scholar 

  11. Beaudry, M., McKenzie, P., Péladeau, P., Thérien, D.: Finite monoids: from word to circuit evaluation. SIAM J. Comput. 26(1), 138–152 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ben-Or, M., Cleve, R.: Computing algebraic formulas using a constant number of registers. SIAM J. Comput. 21(1), 54–58 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Biss, D.K., Dasgupta, S.: A presentation for the unipotent group over rings with identity. J. Algebra 237(2), 691–707 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cook, S.A.: A taxonomy of problems with fast parallel algorithms. Inf. Control 64, 2–22 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cook, S.A., Fontes, L.: Formal theories for linear algebra. Log. Methods Comput. Sci. 8(1), (2012). doi:10.2168/LMCS-8(1:25)2012

  16. Diekert, V., Myasnikov, A.G., Weiß, A.: Conjugacy in Baumslag’s group, generic case complexity, and division in power circuits. In: Proceedings of the 11th Symposium on Latin American Theoretical Informatics, LATIN 2014. Volume 8392 of Lecture Notes in Computer Science, pp. 1–12. Springer (2014)

  17. Eberly, W.: Very fast parallel polynomial arithmetic. SIAM J. Comput. 18(5), 955–976 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constant-depth threshold circuits for division and iterated multiplication. J. Comput. Syst. Sci. 65, 695–716 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ibarra, O.H., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. Assoc. Comput. Mach. 30(1), 217–228 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, STOC 1997, pp. 220–229. ACM Press (1997)

  21. Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1–2), 1–46 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kargapolov, M.I., Merzljakov, J.I.: Fundamentals of the Theory of Groups, Volume 62 of Graduate Texts in Mathematics. Springer-Verlag, New York (1979)

    MATH  Google Scholar 

  23. Kharlampovič, O.G.: A finitely presented solvable group with unsolvable word problem (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 45(4), 852–873 (1981)

    MathSciNet  Google Scholar 

  24. Knuth, D .E.: The Art of Computer Programming, Volume 2: Seminumerical Algorithmus, 3rd edn. Addison-Wesley, Boston (1998)

    MATH  Google Scholar 

  25. König, D., Lohrey, M.: Parallel identity testing for algebraic branching programs with big powers and applications. In: Proceedings of Mathematical Foundations of Computer Science, MFCS 2015. Lecture Notes in Computer Science 9235, pp. 445–458. Springer (2015)

  26. Lipton, R.J., Zalcstein, Y.: Word problems solvable in logspace. J. Assoc. Comput. Mach. 24(3), 522–526 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lohrey, M.: Word problems and membership problems on compressed words. SIAM J. Comput. 35(5), 1210–1240 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lohrey, M.: Algorithmics on SLP-compressed strings: a survey. Groups Complex. Cryptol. 4(2), 241–299 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lohrey, M.: The Compressed Word Problem for Groups. Springer Briefs in Mathematics. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

  30. Lohrey, M.: Rational subsets of unitriangluar groups. Int. J. Algebra Comput. 25(1–2), 113–121 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mal’cev, A.I.: On certain classes of infinite soluble groups. Am. Math. Soc. Transl. Ser. 2(2), 1–21 (1956)

    MathSciNet  Google Scholar 

  32. Miller, G.: The commutator subgroup of a group generated by two operators. Proc. Natl. Acad. Sci. USA 18, 665–668 (1932)

    Article  MATH  Google Scholar 

  33. Moore, C.: Predicting nonlinear cellular automata quickly by decomposing them into linear ones. Phys. D Nonlinear Phenom. 111, 27–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Myasnikov, A.G., Weiß, A.: TC\(\hat{\,\,}\)0 circuits for algorithmic problems in nilpotent groups. CoRR, abs/1702.06616 (2017)

  35. Nikolaev, A., Ushakov, A.: Subset sum problem in polycyclic groups. CoRR, abs/1703.07406 (2017)

  36. Robinson, D.: Parallel algorithms for group word problems. Ph.D. thesis, University of California, San Diego (1993)

  37. Rotman, J .J.: An Introduction to the Theory of Groups, 4th edn. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  38. Ruzzo, W.L.: On uniform circuit complexity. J. Comput. Syst. Sci. 22, 365–383 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  39. Simon, H.U.: Word problems for groups and contextfree recognition. In: Proceedings of Fundamentals of Computation Theory, FCT 1979, pp. 417–422. Akademie-Verlag (1979)

  40. Swan, R.: Representations of polycyclic groups. Proc. Am. Math. Soc. 18, 573–574 (1967)

    MathSciNet  MATH  Google Scholar 

  41. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  42. Toda, S.: Counting problems computationally equivalent to computing the determinant. Technical Report CSIM 91-07, University of Electro-Communications, Tokyo (1991)

  43. Travers, S.D.: The complexity of membership problems for circuits over sets of integers. Theor. Comput. Sci. 369(1–3), 211–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Proceedings of the Sixth Annual Structure in Complexity Theory Conference, pp. 270–284. IEEE Computer Society (1991)

  45. Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  46. Waack, S.: On the parallel complexity of linear groups. R.A.I.R.O. Inf. Théor. Appl. 25(4), 265–281 (1991)

    MathSciNet  MATH  Google Scholar 

  47. Wehrfritz, B .A .F.: Infinite Linear Groups. Springer, Berlin (1977)

    MATH  Google Scholar 

  48. Weiß, A.: On the complexity of conjugacy in amalgamated products and HNN extensions. Ph.D. thesis, Universität Stuttgart (2015)

  49. Zariski, O., Samuel, P.: Commutative Algebra, Volume I, Volume 28 of Graduate Texts in Mathematics. Springer, Berlin (1958)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Siegen, Siegen, Germany

    Daniel König & Markus Lohrey

Authors
  1. Daniel König
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Lohrey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Lohrey.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

König, D., Lohrey, M. Evaluation of Circuits Over Nilpotent and Polycyclic Groups. Algorithmica 80, 1459–1492 (2018). https://doi.org/10.1007/s00453-017-0343-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-017-0343-z

Search

Navigation