Open Problems Column
Abstract
Juris Hartmanis, one of the founders of modern complexity theory, passed away on July 29, 2022 at the age of 94. This column is a tribute to him. It is Open Problems by or Inspired by Juris Hartmanis
References
[1]
[AAW10] Eric Allender, Vikraman Arvind, and Fengming Wang. Uniform derandomization from pathetic lower bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th International Workshop, RANDOM 2010, pages 380--393, 2010. https://link.springer.com/content/pdf/10.1007/ 978--3--642--15369--3_29.pdf.
[2]
[ABDPar] Eric Allender, Nikhil Balaji, Samir Datta, and Rameshwar Pratap. On the complexity of algebraic numbers, and the bitcomplexity of straight-line programs. Computability (The Journal of the Association Computability in Europe), 2022 (to appear). https://eccc.weizmann.ac.il/report/2022/053.
[3]
[ABI97] Eric Allender, Jos´e L. Balc´azar, and Neil Immerman. A firstorder isomorphism theorem. SIAM Journal of Computing, 26(2):557--567, 1997. https://doi.org/10.1137/S0097539794270236.
[4]
[ACG20] Boris Adamczewski, Julien Cassigne, and Marion Le Gonidec. On the computational complexity of algebraic numbers: TheHartmanis-Stearns problem revisited. Transactions of the the American Mathematics Society, 373:3085--3115, 2020. https://hal.archives-ouvertes.fr/hal-01254293/ document.
[5]
[Agr01] Manindra Agrawal. The first-order isomorphism theorem. In Ramesh Hariharan, Madhavan Mukund, and V. Vinay, editors, FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science, 21st Conference, Bangalore, India, December 13--15, 2001, Proceedings, volume 2245 of Lecture Notes in Computer Science, pages 70--82. Springer, 2001. https://doi.org/10.1007/3--540--45294-X\_7.
[6]
[AK10] Eric Allender and Michal Kouck´y. Amplifying lower bounds by means of self-reducibility. Journal of the ACM, 57(3):14:1-- 14:36, 2010. https://iuuk.mff.cuni.cz/~koucky/papers/gap.pdf.
[7]
[All91] Eric Allender. Limitations of the upward separation technique. Math. Syst. Theory, 24(1):53--67, 1991. 10.1007/BF02090390.
[8]
[AW09a] Scott Aaronson and Avi Wigderson. Algebraization: A new barrier to complexity theory. ACM Transactions on Computing Theory, 1, 2009. https://www.scottaaronson.com/papers/alg.pdf.
[9]
[AW09b] Manindra Agrawal and Osamu Watanabe. One-way functions and the Berman-Hartmanis conjecture. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity, CCC 2009, Paris, France, 15--18 July 2009, pages 194--202. IEEE Computer Society, 2009. https://doi.org/10.1109/CCC.2009.17.
[10]
[BB87] Jonathan Borwein and Peter Borwein. Pi and the AGM. John Wiley and Sons, 1987. https://doi.org/10.1016/S0022-0000(70)80012--5.
[11]
[BBP97] David Bailey, Peter Borwein, and Simon Plouffe. On the rapid computation of various polylogarithmic constants. Mathematics of Computation, pages 903--913, 1997. https://www.davidhbailey.com/dhbpapers/digits.pdf.
[12]
[Ber97] Daniel Bertrand. Theta functions and transcendence. Ramanjuan Journal, pages 339--350, 1997. https://link.springer.com/content/pdf/10.1023/A: 1009749608672.pdf.
[13]
[BG16] Richard Beigel and William I. Gasarch. On the sizes of DPDAs, PDAs, LBAs. Theor. Comput. Sci., 638:63--75, 2016. https://doi.org/10.1016/j.tcs.2015.08.028.
[14]
[BH77] Leonard Berman and Juris Hartmanis. On isomorphism and density of NP and other complete sets. SIAM Journal on Computing, 6:305--322, 1977. https://dl.acm.org/doi/10.1145/800113.803628.
[15]
[CGW16] Jin-Yi Cai, Heng Guo, and Tyson Williams. The complexity of counting edge colorings and a dichotomy for some higher domain holant problems. Reseach Math Sciences, 3(18), 2016. https://link.springer.com/article/10.1186/ s40687-016-0067--8.
[16]
[CHO+22] Lijie Chen, Shuichi Hirahara, Igor Carboni Oliveira, J´an Pich, Ninad Rajgopal, and Rahul Santhanam. Beyond natural proofs: Hardness magnification and locality. Journal of ACM, 69(4):25:1--25:49, 2022. https://doi.org/10.1145/3538391.
[17]
[CILM18] Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin. Hardness amplification for non-commutative arithmetic circuits. In 33rd Computational Complexity Conference, CCC 2018, June 22--24, 2018, San Diego, CA, USA, pages 12:1--12:16, 2018. https://drops.dagstuhl.de/opus/volltexte/2018/8877/ pdf/LIPIcs-CCC-2018--12.pdf.
[18]
[CJW19] Lijie Chen, Ce Jin, and R. Ryan Williams. Hardness magnification for all sparse NP languages. In IEEE Annual Symposium on Foundations of Computer Science, (FOCS), pages 1240--1255,2019. https://ieeexplore.ieee.org/document/8948681.
[19]
[CJW20] Lijie Chen, Ce Jin, and R. Ryan Williams. Sharp threshold results for computational complexity. In ACM Symposium on Theory of Computing, (STOC), pages 1335--1348, 2020. https://dl.acm.org/doi/pdf/10.1145/3357713.3384283.
[20]
[CLY22] Lijie Chen, Jiatu Li, and Tianqi Yang. Extremely efficient constructions of hash functions, with applications to hardness magnification and PRFs. In Computational Complexity Conference, (CCC), pages 23:1--23:37, 2022. https://drops.dagstuhl.de/opus/volltexte/2022/16585/ pdf/LIPIcs-CCC-2022--23.pdf.
[21]
[CMMW19] Lijie Chen, Dylan M. McKay, Cody D. Murray, and R. Ryan Williams. Relations and equivalences between circuit lower bounds and Karp-Lipton theorems. In Computational Complexity Conference (CCC), pages 30:1--30:21, 2019. https://drops.dagstuhl.de/opus/volltexte/2019/10852/ pdf/LIPIcs-CCC-2019--30.pdf.
[22]
[Coo71] Stephen Cook. The complexity of theorem proving procedures. In Proceedings of the Third Annual ACM Symposium on the Theory of Computing, Shaker Heights OH, pages 151--158, 1971. https://www.inf.unibz.it/~calvanese/teaching/ 11--12-tc/material/cook-1971-NP-completeness-of-SAT. pdf.
[23]
[CT19] Lijie Chen and Roei Tell. Bootstrapping results for threshold circuits "just beyond" known lower bounds. In ACM Symposium on Theory of Computing, pages 34--41, 2019. https://dl.acm.org/doi/10.1145/3313276.3316333.
[24]
[DNNS96] Daniel Duverney, Keij Nishiika, Kumiko Nishioka, and Iekata Shiokawa. Transcendence of Jacobi's theta series. Proceedings of the Japanese Academy of Mathematical Science, pages 202-- 203, 1996. file:///C:/Users/BillLoc/Downloads/pjaa.72.202.pdf.
[25]
[Fil11] Yuval Filmus. Lower bounds for context-free grammars. Information Processing Letters, 111(18):895--898, 2011. https://www.sciencedirect.com/science/article/pii/ S0020019011001712.
[26]
[FLvMV00] Lance Fortnow, Richard J. Lipton, Dieter van Melkebeek, and Anastasios Viglas. Time-space lower bounds for satisfiability. Journal of the ACM, 52(6):835--865, 2005. Merger of FOCS'99 and CCC'00. https://people.cs.uchicago.edu/~fortnow/papers/tst. pdf.
[27]
[FMR70] Patrick C. Fischer, Albert R. Meyer, and Arnold L. Rosenberg. Time-restricted sequence generation. J. Comput. Syst. Sci., 4(1):50--73, 1970. https://doi.org/10.1016/S0022-0000(70)80012--5.
[28]
[For00] Lance Fortnow. Time-space tradeoffs for satisfiability. J. Comput. Syst. Sci., 60(2):337--353, 2000. https://lance.fortnow.com/papers/files/npvsnl.pdf.
[29]
[Fre12] Rusins Freivalds. Hartmanis-Stearns conjecture on real time and transcendence. In Michael J. Dinneen, Bakhadyr Khoussainov, and Andr´e Nies, editors, Computation, Physics and Beyond - International Workshop on Theoretical Computer Science, WTCS 2012, Dedicated to Cristian S. Calude on the Occasion of His 60th Birthday, Auckland, New Zealand, February 21--24, 2012, Revised Selected and Invited Papers, volume 7160 of Lecture Notes in Computer Science, pages 105--119. Springer, 2012. https://doi.org/10.1007/978--3--642--27654--5\_9.
[30]
[Fu20] Bin Fu. Hardness of sparse sets and minimal circuit size problem. In Computing and Combinatorics - 26th International Conference (COCOON), pages 484--495, 2020. https://arxiv.org/abs/2003.00669.
[31]
[Gol11] Oded Goldreich. A candidate counterexample to the easy cylinders conjecture. In Oded Goldreich, editor, Studies in Complexity and Cryptography. Miscellanea on the Interplay between Randomness and Computation - In Collaboration with Lidor Avigad, Mihir Bellare, Zvika Brakerski, Shafi Goldwasser, Shai Halevi, Tali Kaufman, Leonid Levin, Noam Nisan, Dana Ron, Madhu Sudan, Luca Trevisan, Salil Vadhan, Avi Wigderson, David Zuckerman, volume 6650 of Lecture Notes in Computer Science, pages 136--140. Springer, 2011. https://doi.org/10.1007/978--3--642--22670-0\_16.
[32]
[Har80] Juris Hartmanis. On the succinctness of different representations of languages. SIAM Journal on Computing, 9(1):114--120, 1980. https://epubs.siam.org/doi/abs/10.1137/0209010.
[33]
[HIM78] Juris Hartmanis, Neil Immerman, and Stephen R. Mahaney. One-way log-tape reductions. In 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16--18 October 1978, pages 65--72. IEEE Computer Society, 1978. https://doi.org/10.1109/SFCS.1978.31.
[34]
[Hir20] Shuichi Hirahara. Non-disjoint promise problems from metacomputational view of pseudorandom generator constructions. In Computational Complexity Conference (CCC), pages 20:1-- 20:47, 2020. https://drops.dagstuhl.de/opus/volltexte/2020/ 12572/.
[35]
[HIS85] Juris Hartmanis, Neil Immerman, and Vivian Sewelson. Sparse sets in NP?P: EXPTIME versus NEXPTIME. Information and Computation, 65:158--181, 1985. https://dl.acm.org/doi/pdf/10.1145/800061.808769.
[36]
[HS65] Juris Hartmanis and Richard E. Stearns. On the computational complexity of algorithms. Transactions of the American Math Society, 117:285--306, 1965. https://www.ams.org/journals/tran/ 1965--117-00/S0002--9947--1965-0170805--7/ S0002--9947--1965-0170805--7.pdf.
[37]
[Imm99] Neil Immerman. Descriptive Complexity Theory. Springer Verlag, New York, Heidelberg, Berlin, 1999.
[38]
[Jon75] Neil D. Jones. Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci., 11(1):68--85, 1975. https://doi.org/10.1016/S0022-0000(75)80050-X.
[39]
[JY85] Deborah Joseph and Paul Young. Some remarks on witness functions for nonpolynomial and noncomplete sets in NP. Theor. Comput. Sci., 39:225--237, 1985. https://doi.org/10.1016/0304--3975(85)90140--9.
[40]
[Kar72] Richard Karp. Reducibilities among combinatorial problems. In Ray Miller and J.W. Thatcher, editors, Complexity of computer computations, pages 85--103. Plenum Press, 1972. https://www.cs.umd.edu/~gasarch/BLOGPAPERS/Karp.pdf.
[41]
[KMR95] Stuart A. Kurtz, Stephen R. Mahaney, and James S. Royer. The isomorphism conjecture fails relative to a random oracle. J. ACM, 42(2):401--420, 1995. https://doi.org/10.1145/201019.201030.
[42]
[LR10] Richard Lipton and Ken Regan. Making an algorithm an algorithm-BBP, 2010. https://rjlipton.wpcomstaging.com/2010/07/14/ making-an-algorithm-an-algorithm-bbp/.
[43]
[LR13] Richard Lipton and Kenneth Regan. People, problems, and proof: essays from Godel's last letter: 2010. Springer, New York, Heidelberg, Berlin, 2013.
[44]
[LW10] Richard J. Lipton and Ryan Williams. Amplifying circuit lower bounds against polynomial time, with applications. Computational Complexity, 22(2):311--343, 2013. Conference version in CCC'10. https://people.csail.mit.edu/rrw/qbf-nc-journal-rev. pdf.
[45]
[Mah82] Steven Mahaney. Sparse complete sets for NP: Solution to a conjecture of Berman and Hartmanis. Journal of Computer and System Sciences, 25:130--143, 1982. https://ieeexplore.ieee.org/document/4567805.
[46]
[MMW19] Dylan M. McKay, Cody D. Murray, and R. Ryan Williams. Weak lower bounds on resource-bounded compression imply strong separations of complexity classes. In ACM Symposium on Theory of Computing (STOC), pages 1215--1225, 2019. https://dl.acm.org/doi/10.1145/3313276.3316396.
[47]
[MP20] Moritz M¨uller and J´an Pich. Feasibly constructive proofs of succinct weak circuit lower bounds. Ann. Pure Appl. Log., 171(2), 2020. https://doi.org/10.1016/j.apal.2019.102735.
[48]
[Myh55] John Myhill. Creative sets. Zeitschrift f¨ur mathematische Logik und Grundlagen der Mathematik, 1:97--108, 1955.
[49]
[OPS19] Igor Carboni Oliveira, J´an Pich, and Rahul Santhanam. Hardness magnification near state-of-the-art lower bounds. In Computational Complexity Conference (CCC), pages 27:1-- 27:29, 2019. https://theoryofcomputing.org/articles/v017a011/ v017a011.pdf.
[50]
[OS18] Igor Carboni Oliveira and Rahul Santhanam. Hardness magnification for natural problems. In IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 65--76, 2018. https://ieeexplore.ieee.org/document/8555094.
[51]
[Sri00] Aravind Srinivasan. The value of strong inapproximability results for clique. In ACM Symposium on Theory of Computing (STOC), pages 144--152, 2000. https://dl.acm.org/doi/10.1145/335305.335322.
[52]
[Val76] Leslie G. Valiant. A note on the succinctness of descriptions of deterministic languages. Inf. Control., 32(2):139--145, 1976. https://doi.org/10.1016/S0019--9958(76)90173-X.
[53]
[Val82] Leslie Valiant. Reduciblity by algebraic projections. L'Enseignement Mathematique, T. XXVIII:253--268, 1982.
[54]
[Wil08] Ryan Williams. Time space tradeoffs for counting NP solutions modulo integers. In Computational Complexity, pages 179--219, New York, 2008. IEEE. http://www.stanford.edu/~rrwill/projects.html.
[55]
[Yap10] Chee Yap. is in log space, 2010. https://cs.nyu.edu/exact/doc/pi-log.pdf.
Recommendations
Open Problems Column
There have been two polls asking theorists (and others) what they thought of P vs NP (and other questions) [4, 5]. Both were written by William Gasarch and appeared in the Complexity Column of SIGACT News, edited by Lane A. Hemaspaandra. They were in ...
Open Problems Column
There have been two polls asking theorists (and others) what they thought of P vs NP and related questions [5, 6]. Both were written by me (William Gasarch) and appeared in the Complexity Column of SIGACT News, edited by Lane A. Hemaspaandra.
I have ...
Open Problems Column
This issue's Open Problems Column is by Dana Moshkovitz. It is titled: Sliding Scale Conjec- tures in PCP. The PCP theorem has been around since 1992, but some of the most fundamental questions about it are still open. This column explains the questions,...
Comments
Information & Contributors
Information
Published In
December 2022
42 pages
Copyright © 2022 Copyright is held by the owner/author(s).
Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 19 December 2022
Published in SIGACT Volume 53, Issue 4
Check for updates
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 58Total Downloads
- Downloads (Last 12 months)13
- Downloads (Last 6 weeks)2
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in