Symmetric Computation (Invited Talk)

Author Anuj Dawar

Thumbnail PDF


  • Filesize: 441 kB
  • 12 pages

Document Identifiers

Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, U.K.

Cite AsGet BibTex

Anuj Dawar. Symmetric Computation (Invited Talk). In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 2:1-2:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We discuss a recent convergence of notions of symmetric computation arising in the theory of linear programming, in logic and in circuit complexity. This leads us to a coherent and robust definition of problems that are efficiently and symmetrically solvable. This is at once a rich class of problems and one for which we have methods for proving lower bounds. In this paper, we take a tour through results which show applications of these methods in a number of areas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity classes
  • Descriptive Complexity
  • Fixed-point Logic with Counting
  • Circuit Complexity
  • Linear Programming
  • Hardness of Approximation
  • Arithmetic Circuits


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


  1. A. V. Aho and J. D. Ullman. Foundations of Computer Science, C Edition. Computer Science Press / W. H. Freeman, 1992. Google Scholar
  2. M. Anderson and A. Dawar. On Symmetric Circuits and Fixed-Point Logics. Theory Comput. Syst., 60(3):521-551, 2017. URL:
  3. M. Anderson, A. Dawar, and B. Holm. Solving Linear Programs without Breaking Abstractions. J. ACM, 62, 2015. Google Scholar
  4. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof Verification and the Hardness of Approximation Problems. J. ACM, 45(3):501-555, 1998. Google Scholar
  5. A. Atserias, A. Bulatov, and A. Dawar. Affine Systems of Equations and Counting Infinitary Logic. Theoretical Computer Science, 410(18):1666-1683, 2009. Google Scholar
  6. A. Atserias, A. Dawar, and J. Ochremiak. On the Power of Symmetric Linear Programs. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, pages 1-13, 2019. URL:
  7. A. Atserias and E. N. Maneva. Sherali-Adams relaxations and indistinguishability in counting logics. SIAM J. Comput., 42:112-137, 2013. Google Scholar
  8. A. Atserias and J. Ochremiak. Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Isomorphism Problem. In 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, 2018. Google Scholar
  9. Albert Atserias and Anuj Dawar. Definable Inapproximability: New Challenges for Duplicator. In 27th EACSL Annual Conference on Computer Science Logic, CSL, pages 7:1-7:21, 2018. URL:
  10. L. Barto and M. Kozik. Constraint Satisfaction Problems Solvable by Local Consistency Methods. J. ACM, 61:3:1-3:19, 2014. Google Scholar
  11. A. Blass, Y. Gurevich, and S. Shelah. Choiceless Polynomial Time. Annals of Pure and Applied Logic, 100:141-187, 1999. Google Scholar
  12. J-Y. Cai, M. Fürer, and N. Immerman. An Optimal Lower Bound on the Number of Variables for Graph Identification. Combinatorica, 12(4):389-410, 1992. Google Scholar
  13. A. Chandra and D. Harel. Structure and Complexity of Relational Queries. Journal of Computer and System Sciences, 25:99-128, 1982. Google Scholar
  14. A. Dawar. A Restricted Second Order Logic for Finite Structures. Information and Computation, 143:154-174, 1998. Google Scholar
  15. A. Dawar. The Nature and Power of Fixed-Point Logic with Counting. ACM SIGLOG News, 2(1):8-21, 2015. Google Scholar
  16. A. Dawar and P. Wang. A Definability Dichotomy for Finite Valued CSPs. In 24th EACSL Annual Conference on Computer Science Logic, CSL 2015, pages 60-77, 2015. Google Scholar
  17. A. Dawar and P. Wang. Definability of semidefinite programming and Lasserre lower bounds for CSPs. In 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS, 2017. URL:
  18. A. Dawar and G. Wilsenach. Symmetric Arithmetic Circuits. forthcoming. Google Scholar
  19. Anuj Dawar. On Symmetric and Choiceless Computation. In Topics in Theoretical Computer Science - The First IFIP WG 1.8 International Conference, TTCS, pages 23-29, 2015. URL:
  20. Anuj Dawar and Gregory Wilsenach. Symmetric Circuits for Rank Logic. In 27th EACSL Annual Conference on Computer Science Logic, CSL, pages 20:1-20:16, 2018. URL:
  21. L. Denenberg, Y. Gurevich, and S. Shelah. Definability by Constant-depth Polynomial-size Circuits. Information and Control, 70:216-240, 1986. Google Scholar
  22. R. Fagin. Generalized First-Order Spectra and Polynomial-Time Recognizable Sets. In R. M. Karp, editor, Complexity of Computation, SIAM-AMS Proceedings, Vol 7, pages 43-73, 1974. Google Scholar
  23. T. Feder and M.Y. Vardi. Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study Through Datalog and Group Theory. SIAM Journal of Computing, 28:57-104, 1998. Google Scholar
  24. M. Grohe. The Quest for a Logic Capturing PTIME. In Proc. 22nd IEEE Symp. on Logic in Computer Science, pages 267-271, 2008. Google Scholar
  25. M. Grohe. Descriptive Complexity, Canonisation, and Definable Graph Structure Theory. Lecture Notes in Logic. Cambridge University Press, 2017. Google Scholar
  26. M. Grohe and M. Otto. Pebble Games and linear equations. J. Symb. Log., 80:797-844, 2015. Google Scholar
  27. J. Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. Google Scholar
  28. L. Hella. Logical Hierarchies in PTIME. Information and Computation, 129:1-19, 1996. Google Scholar
  29. L. G. Khachiyan. Polynomial Algorithms in Linear Programming. USSR Computational Mathematics and Mathematical Physics, 20(1):53-72, 1980. Google Scholar
  30. S. Khot, D. Minzer, and M. Safra. Pseudorandom Sets in Grassmann Graph Have Near-Perfect Expansion. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 592-601, 2018. URL:
  31. M. Laurent. A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0-1 Programming. Math. Oper. Res., 28:470-496, 2003. URL:
  32. P. N. Malkin. Sherali-Adams relaxations of graph isomorphism polytopes. Discrete Optimization, 12:73-97, 2014. Google Scholar
  33. M. Otto. The Logic of Explicitly Presentation-Invariant Circuits. In Computer Science Logic, 10th International Workshop, CSL '96, Annual Conference of the EACSL, pages 369-384, 1996. Google Scholar
  34. M. Otto. Bounded Variable Logics and Counting - A Study in Finite Models, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997. Google Scholar
  35. T. Rothvoß. The Matching Polytope has Exponential Extension Complexity. In Symp. Theory of Computing, STOC 2014, pages 263-272, 2014. Google Scholar
  36. L. G. Valiant. Completeness Classes in Algebra. In Proceedings of the 11h Annual ACM Symposium on Theory of Computing STOC, pages 249-261, 1979. URL:
  37. J. von zur Gathen. Algebraic Complexity Theory. Ann. Rev. Comput. Sci., 3:317-347, 1988. URL:
  38. M. Yannakakis. Expressing Combinatorial Optimization Problems by Linear Programs. J. Comput. Syst. Sci., 43(3):441-466, 1991. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail