Complexity of Fault Tolerant Query Complexity

Authors Ramita Maharjan, Thomas Watson

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Author Details

Ramita Maharjan
  • University of Memphis, TN, USA
Thomas Watson
  • University of Memphis, TN, USA

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Ramita Maharjan and Thomas Watson. Complexity of Fault Tolerant Query Complexity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In the model of fault tolerant decision trees introduced by Kenyon and Yao, there is a known upper bound E on the total number of queries that may be faulty (i.e., get the wrong bit). We consider this computational problem: Given as input the truth table of a function f: {0,1}ⁿ → {0,1} and a value of E, find the minimum possible height (worst-case number of queries) of any decision tree that computes f while tolerating up to E many faults. We design an algorithm for this problem that runs in time Õ(binom(n+E,E)⋅(2E+3)ⁿ), which is polynomial in the size of the truth table when E is a constant. This generalizes a standard algorithm for the non-fault tolerant setting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Fault
  • Tolerant
  • Query
  • Complexity


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  1. Scott Aaronson. Algorithms for boolean function query properties. SIAM Journal on Computing, 32(5):1140-1157, 2003. URL:
  2. Eric Allender. The new complexity landscape around circuit minimization. In Proceedings of the 14th Conference on Language and Automata Theory and Applications (LATA), pages 3-16. Springer, 2020. URL:
  3. Eric Allender and Bireswar Das. Zero knowledge and circuit minimization. Information and Computation, 256:2-8, 2017. URL:
  4. Eric Allender, Joshua Grochow, Dieter van Melkebeek, Cristopher Moore, and Andrew Morgan. Minimum circuit size, graph isomorphism, and related problems. SIAM Journal on Computing, 47(4):1339-1372, 2018. URL:
  5. Eric Allender, Lisa Hellerstein, Paul McCabe, Toniann Pitassi, and Michael Saks. Minimizing DNF formulas and AC^0_d circuits given a truth table. In Proceedings of the 21st Conference on Computational Complexity (CCC), pages 237-251. IEEE, 2006. URL:
  6. Eric Allender, Michal Koucký, Detlef Ronneburger, and Sambuddha Roy. Derandomization and distinguishing complexity. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 209-220. IEEE, 2003. URL:
  7. Guy Blanc, Jane Lange, Mingda Qiao, and Li-Yang Tan. Properly learning decision trees in almost polynomial time. In Proceedings of the 62nd Symposium on Foundations of Computer Science (FOCS), pages 920-929. IEEE, 2021. URL:
  8. Nader Bshouty and Catherine Haddad-Zaknoon. Adaptive exact learning of decision trees from membership queries. In Proceedings of the 30th International Conference on Algorithmic Learning Theory (ALT), pages 207-234. PMLR, 2019. Google Scholar
  9. Uriel Feige, Prabhakar Raghavan, David Peleg, and Eli Upfal. Computing with noisy information. SIAM Journal on Computing, 23(5):1001-1018, 1994. URL:
  10. Steven Friedman and Kenneth Supowit. Finding the optimal variable ordering for binary decision diagrams. IEEE Transactions on Computers, 39(5):710-713, 1990. URL:
  11. David Guijarro, Víctor Lavín, and Vijay Raghavan. Exact learning when irrelevant variables abound. Information Processing Letters, 70(5):233-239, 1999. URL:
  12. Shuichi Hirahara, Rahul Ilango, and Bruno Loff. Hardness of constant-round communication complexity. In Proceedings of the 36th Computational Complexity Conference (CCC), pages 31:1-31:30. Schloss Dagstuhl, 2021. URL:
  13. Rahul Ilango. Constant depth formula and partial function versions of MCSP are hard. In Proceedings of the 61st Symposium on Foundations of Computer Science (FOCS), pages 424-433. IEEE, 2020. URL:
  14. Rahul Ilango. The minimum formula size problem is (ETH) hard. In Proceedings of the 62nd Symposium on Foundations of Computer Science (FOCS), pages 427-432. IEEE, 2021. URL:
  15. Rahul Ilango, Bruno Loff, and Igor Carboni Oliveira. NP-hardness of circuit minimization for multi-output functions. In Proceedings of the 35th Computational Complexity Conference (CCC), pages 22:1-22:36. Schloss Dagstuhl, 2020. URL:
  16. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proceedings of the 32nd Symposium on Theory of Computing (STOC), pages 73-79, 2000. URL:
  17. Claire Kenyon and Andrew Yao. On evaluating boolean functions with unreliable tests. International Journal of Foundations of Computer Science, 1(1):1-10, 1990. URL:
  18. Eyal Kushilevitz and Enav Weinreb. On the complexity of communication complexity. In Proceedings of the 41st Symposium on Theory of Computing (STOC), pages 465-474. ACM, 2009. URL:
  19. Netanel Raviv. Truth table minimization of computational models. Technical report, arXiv, 2013. URL:
  20. Rüdiger Reischuk and Bernd Schmeltz. Reliable computation with noisy circuits and decision trees - A general nlog n lower bound. In Proceedings of the 32nd Symposium on Foundations of Computer Science (FOCS), pages 602-611. IEEE, 1991. URL:
  21. Mario Szegedy and Xiaomin Chen. Computing boolean functions from multiple faulty copies of input bits. Theoretical Computer Science, 321(1):149-170, 2004. URL:
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