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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In this paper, we introduce a measure of Boolean functions we call diameter, that captures the relationship between certificate complexity and several other measures of Boolean functions. Our measure can be viewed as a variation on alternating number, but while alternating number can be exponentially larger than certificate complexity, we show that diameter is always upper bounded by certificate complexity. We argue that estimating diameter may help to get improved bounds on certificate complexity in terms of sensitivity, and other measures.
Previous results due to Lin and Zhang [Krishnamoorthy Dinesh and Jayalal Sarma, 2018] imply that s(f) ≥ Ω(n^{1/3}) for transitive functions with constant alternating number. We improve and extend this bound and prove that s(f) ≥ √n for transitive functions with constant alternating number, as well as for transitive functions with constant diameter. {We also show that bs(f) ≥ Ω(n^{3/7}) for transitive functions under the weaker condition that the "minimum" diameter is constant.}
Furthermore, we prove that the log-rank conjecture holds for functions of the form f(x ⊕ y) for functions f with diameter bounded above by a polynomial of the logarithm of the Fourier sparsity of the function f.

Siddhesh Chaubal and Anna Gál. Diameter Versus Certificate Complexity of Boolean Functions. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 31:1-31:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{chaubal_et_al:LIPIcs.MFCS.2021.31, author = {Chaubal, Siddhesh and G\'{a}l, Anna}, title = {{Diameter Versus Certificate Complexity of Boolean Functions}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {31:1--31:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.31}, URN = {urn:nbn:de:0030-drops-144713}, doi = {10.4230/LIPIcs.MFCS.2021.31}, annote = {Keywords: Sensitivity Conjecture, Boolean Functions, Certificate Complexity, Block Sensitivity, Log-rank Conjecture, Alternating Number} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Nisan and Szegedy [Nisan and Szegedy, 1994] conjectured that block sensitivity is at most polynomial in sensitivity for any Boolean function. There is a huge gap between the best known upper bound on block sensitivity in terms of sensitivity - which is exponential, and the best known separating examples - which give only a quadratic separation between block sensitivity and sensitivity.
In this paper we give various new constructions of families of Boolean functions that exhibit quadratic separation between sensitivity and block sensitivity. Our constructions have several novel aspects. For example, we give the first direct constructions of families of Boolean functions that have both 0-block sensitivity and 1-block sensitivity quadratically larger than sensitivity.

Siddhesh Chaubal and Anna Gál. New Constructions with Quadratic Separation between Sensitivity and Block Sensitivity. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chaubal_et_al:LIPIcs.FSTTCS.2018.13, author = {Chaubal, Siddhesh and G\'{a}l, Anna}, title = {{New Constructions with Quadratic Separation between Sensitivity and Block Sensitivity}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.13}, URN = {urn:nbn:de:0030-drops-99129}, doi = {10.4230/LIPIcs.FSTTCS.2018.13}, annote = {Keywords: Sensitivity Conjecture, Boolean Functions, Complexity Measures} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

One central issue in the formal design and analysis of reactive systems is the notion of refinement that asks whether all behaviors of the implementation is allowed by the specification. The local interpretation of behavior leads to the notion of simulation. Alternating transition systems (ATSs) provide a general model for composite reactive systems, and the simulation relation for ATSs is known as alternating simulation. The simulation relation for fair transition systems is called fair simulation. In this work our main contributions are as follows: (1) We present an improved algorithm for fair simulation with Büchi fairness constraints; our algorithm requires O(n^3 * m) time as compared to the previous known O(n^6)-time algorithm, where n is the number of states and m is the number of transitions. (2) We present a game based algorithm for alternating simulation that requires O(m^2)-time as compared to the previous known O((n*m)^2)-time algorithm, where n is the number of states and m is the size of transition relation. (3) We present an iterative algorithm for alternating simulation that matches the time complexity of the game based algorithm, but is more space efficient than the game based algorithm.

Krishnendu Chatterjee, Siddhesh Chaubal, and Pritish Kamath. Faster Algorithms for Alternating Refinement Relations. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 167-182, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chatterjee_et_al:LIPIcs.CSL.2012.167, author = {Chatterjee, Krishnendu and Chaubal, Siddhesh and Kamath, Pritish}, title = {{Faster Algorithms for Alternating Refinement Relations}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {167--182}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.167}, URN = {urn:nbn:de:0030-drops-36713}, doi = {10.4230/LIPIcs.CSL.2012.167}, annote = {Keywords: Simulation and fair simulation, Alternating simulation, Graph games} }

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