Document Open Access Logo

On the Complexity of Hazard-Free Formulas

Authors Leah London Arazi , Amir Shpilka

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.115.pdf
  • Filesize: 1.04 MB
  • 19 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Communication complexity
Keywords
  • Hazard-free computation
  • Boolean formulas
  • monotone formulas
  • Karchmer-Wigderson games
  • communication complexity
  • lower bounds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

This paper studies the hazard-free formula complexity of Boolean functions.
Our first result shows that unate functions are the only Boolean functions for which the monotone formula complexity of the hazard-derivative equals the hazard-free formula complexity of the function itself. Consequently, they are the only functions for which the hazard-derivative approach of Ikenmeyer et al. (J. ACM, 2019) yields optimal bounds. 
Our second result proves that the hazard-free formula complexity of a uniformly random Boolean function is at most 2^{(1+o(1))n}. Prior to this, no better upper bound than O(3ⁿ) was known. Notably, unlike in the general case of Boolean circuits and formulas, where the typical complexity is derived from that of the multiplexer function with n-bit selector, the hazard-free formula complexity of a random function is smaller than the optimal hazard-free formula for the multiplexer by an exponential factor in n.
We provide two proofs of this fact. The first is direct, bounding the number of prime implicants of a random Boolean function and using this bound to construct a DNF of the claimed size. The second introduces a new and independently interesting result: a weak converse to the hazard-derivative lower bound method, which gives an upper bound on the hazard-free complexity of a function in terms of the monotone complexity of a subset of its hazard-derivatives. 
Additionally, we explore the hazard-free formula complexity of block composition of Boolean functions and obtain a result in the hazard-free setting that is analogous to a result of Karchmer, Raz, and Wigderson (Computational Complexity, 1995) in the monotone setting. We show that our result implies a stronger lower bound on the hazard-free formula depth of the block composition of the set covering function with the multiplexer function than the bound obtained via the hazard-derivative method.

Cite As Get BibTex

Leah London Arazi and Amir Shpilka. On the Complexity of Hazard-Free Formulas. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 115:1-115:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.115

Author Details

Leah London Arazi
  • School of Computer Science, Tel Aviv University, Israel
Amir Shpilka
  • School of Computer Science, Tel Aviv University, Israel

Funding

This research was co-funded by the European Union (ERC, EACTP, 101142020), the Israel Science Foundation (grant number 514/20) and the Len Blavatnik and the Blavatnik Family Foundation. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

The first author thanks Maya Baruch for invaluable discussions and feedback. She also thanks Ido Weinstein and Rotem Zluf for discussing early ideas, which helped identify and correct initial mistakes.

References

  1. Noga Alon and Ravi B. Boppana. The monotone circuit complexity of boolean functions. Combinatorica, 7:1-22, 1987. URL: https://doi.org/10.1007/BF02579196.
  2. Leah London Arazi and Amir Shpilka. On the complexity of hazard-free formulas, 2025. URL: https://arxiv.org/abs/2411.09026.
  3. Johannes Bund, Christoph Lenzen, and Moti Medina. Optimal metastability-containing sorting via parallel prefix computation. IEEE Transactions on Computers, 69(2):198-211, 2020. URL: https://doi.org/10.1109/TC.2019.2939818.
  4. Irit Dinur and Or Meir. Toward the krw composition conjecture: Cubic formula lower bounds via communication complexity. Computational Complexity, 27:375-462, September 2018. URL: https://doi.org/10.1007/s00037-017-0159-x.
  5. Jeff Edmonds, Russell Impagliazzo, Steven Rudich, and Jiří Sgall. Communication complexity towards lower bounds on circuit depth. Comput. Complexity, 10(3):210-246, 2001. URL: https://doi.org/10.1007/s00037-001-8195-x.
  6. Edward B. Eichelberger. Hazard detection in combinational and sequential switching circuits. IBM Journal of Research and Development, 9(2):90-99, 1965. URL: https://doi.org/10.1147/rd.92.0090.
  7. Stephan Friedrichs, Matthias Függer, and Christoph Lenzen. Metastability-containing circuits. IEEE Transactions on Computers, 67(8):1167-1183, 2018. URL: https://doi.org/10.1109/TC.2018.2808185.
  8. Dmitry Gavinsky, Or Meir, Omri Weinstein, and Avi Wigderson. Toward better formula lower bounds: The composition of a function and a universal relation. SIAM Journal on Computing, 46(1):114-131, 2017. URL: https://doi.org/10.1137/15M1018319.
  9. Johan Håstad and Avi Wigderson. Composition of the universal relation. In Advances in computational complexity theory (New Brunswick, NJ, 1990), volume 13 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 119-134. Amer. Math. Soc., Providence, RI, 1993. URL: https://doi.org/10.1090/dimacs/013/07.
  10. David A. Huffman. The design and use of hazard-free switching networks. J. ACM, 4(1):47-62, January 1957. URL: https://doi.org/10.1145/320856.320866.
  11. Christian Ikenmeyer, Balagopal Komarath, Christoph Lenzen, Vladimir Lysikov, Andrey Mokhov, and Karteek Sreenivasaiah. On the complexity of hazard-free circuits. J. ACM, 66(4), August 2019. URL: https://doi.org/10.1145/3320123.
  12. Christian Ikenmeyer, Balagopal Komarath, and Nitin Saurabh. Karchmer-Wigderson Games for Hazard-Free Computation. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251 of Leibniz International Proceedings in Informatics (LIPIcs), pages 74:1-74:25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.ITCS.2023.74.
  13. Stasys Jukna. Notes on hazard-free circuits. SIAM Journal on Discrete Mathematics, 35(2):770-787, 2021. URL: https://doi.org/10.1137/20M1355240.
  14. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5:191-204, 1995. URL: https://doi.org/10.1007/BF01206317.
  15. Mauricio Karchmer and Avi Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3(2):255-265, 1990. URL: https://doi.org/10.1137/0403021.
  16. Stephen Cole Kleene. Introduction to metamathematics. D. Van Nostrand Co., Inc., New York, 1952. Google Scholar
  17. Balagopal Komarath and Nitin Saurabh. On the complexity of detecting hazards. Inf. Process. Lett., 162:105980, 2020. URL: https://doi.org/10.1016/J.IPL.2020.105980.
  18. Sajin Koroth and Or Meir. Improved composition theorems for functions and relations. In Eric Blais, Klaus Jansen, José D. P. Rolim, and David Steurer, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, volume 116 of LIPIcs, pages 48:1-48:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.APPROX-RANDOM.2018.48.
  19. Edward J. McCluskey. Logic design principles - with emphasis on testable semicustom circuits. Prentice Hall series in computer engineering. Prentice Hall, 1986. Google Scholar
  20. EJ McCluskey. Transients in combinational logic circuits. Redundancy Technique for Computing System, 9, 1962. Google Scholar
  21. Or Meir. Toward better depth lower bounds: Two results on the multiplexor relation. Computational Complexity, 29(1):4, 2020. URL: https://doi.org/10.1007/s00037-020-00194-8.
  22. Masao Mukaidono. On the B-ternary logical function - A ternary logic considering ambiguity. Syst. Comput. Controls, 3(3):27-36, 1972. Google Scholar
  23. David E. Muller. Complexity in electronic switching circuits. IRE Transactions on Electronic Computers, EC-5(1):15-19, 1956. URL: https://doi.org/10.1109/TEC.1956.5219786.
  24. Steven M. Nowick and David L. Dill. Exact two-level minimization of hazard-free logic with multiple-input changes. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 14(8):986-997, 1995. URL: https://doi.org/10.1109/43.402498.
  25. Alexander A. Razborov. Lower bounds on the monotone complexity of some Boolean function. Soviet Math. Dokl., 31:354-357, 1985. Google Scholar
  26. Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, and Robert Robere. Krw composition theorems via lifting. Computational Complexity, 33(1):1-97, 2024. URL: https://doi.org/10.1007/s00037-024-00250-7.
  27. John Riordan and Claude E. Shannon. The number of two-terminal series-parallel networks. Journal of Mathematics and Physics, 21(1-4):83-93, 1942. URL: https://doi.org/10.1002/sapm194221183.
  28. Claude E. Shannon. The synthesis of two-terminal switching circuits. The Bell System Technical Journal, 28(1):59-98, 1949. URL: https://doi.org/10.1002/j.1538-7305.1949.tb03624.x.
  29. Éva Tardos. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica, 8:141-142, 1988. URL: https://doi.org/10.1007/BF02122563.
  30. S. Unger. Hazards and delays in asynchronous sequential switching circuits. IRE Transactions on Circuit Theory, 6(1):12-25, 1959. URL: https://doi.org/10.1109/TCT.1959.1086527.
  31. Michael Yoeli and Shlomo Rinon. Application of ternary algebra to the study of static hazards. J. ACM, 11(1):84-97, January 1964. URL: https://doi.org/10.1145/321203.321214.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact