Abstract
This paper characterizes alternation trading-based proofs that the Boolean satisfiability problem is not in the time- and space-bounded class DTISP\({(n^c,n^\epsilon)}\), for various values c < 2 and \({\epsilon < 1}\). We characterize exactly what can be proved in the \({\epsilon = 0}\) case with currently known methods and prove the conjecture of Williams that \({c=2\cos(\pi/7)}\) is optimal for this. For time–space trade-offs and lower bounds on satisfiability, we give a theoretical and computational analysis of the alternation trading proofs for \({0 < \epsilon < 1}\).
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References
Scott Aaronson & Avi Wigderson (2009). Algebrization: A New Barrier in Complexity Theory. ACM Transactions on Computation Theory 1(1).
Theodore Baker, John Gill & Robert Solovay (1975). Relativizations of the P=?NP Question. SIAM Journal on Computing 4, 431–442.
James Bennett (1962). On Spectra. Ph.D. thesis, Princeton University.
Scott Diehl & Dieter van Melkebeek (2006). Time-Space Lower Bounds for the Polynomial-Time Hierarchy on Randomized Machines. SIAM Journal on Computing 36, 563–594.
Lance Fortnow (1997). Nondeterministic Polynomial Time Versus Nondeterministic Logarithmic Space: Time-Space Tradeoffs for Satisfiability. In Proc. IEEE Conference on Computational Complexity (CCC), 52–60.
Lance Fortnow, Richard Lipton, Dieter van Melkebeek & Anastasios Viglas (2005). Time-Space Lower Bounds for Satisfiability. Journal of the ACM 52(6), 835–865.
Lance Fortnow & Dieter van Melkebeek (2000). Time-Space Tradeoffs for Nondeterministic Computation. In Proc. IEEE Conference on Computational Complexity (CCC), 2–13.
Kannan Ravi: Towards Separating Nondeterminism from Determinism. Mathematical Systems Theory 17, 29–45 (1984)
Richard Lipton & Anastasios Viglas (1999). On the Complexity of SAT. In Proc. 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 459–464.
Dieter van Melkebeek (2004). Time-Space Lower Bounds for NPComplete Problems. In Current Trends in Theoretical Computer Science, 265–291. World Scientific.
Dieter van Melkebeek (2007). A Survey of Lower Bounds for Satisfiability and Related Problems. Foundations and Trends in Theoretical Computer Science 2(3), 197–303.
Dieter van Melkebeek & Ran Raz (2005). A Time Lower Bound for Satisfiability. Theoretical Computer Science 348, 311–320.
V. A. Nepomnjaščiĭ (1970). Rudimentary Predicates and Turing Computations. Dokl. Akad. Nauk SSSR 195, 282–284. English translation in Soviet Math. Dokl. 11 (1970) 1462–1465.
Alexander A. Razborov & Steven Rudich (1997). Natural Proofs. Journal of Computer and System Sciences 55(1), 24–35.
Iannis Tourlakis: Time-Space Tradeoffs for SAT and Related Problems. Journal of Computer and System Sciences 63(2), 268–287 (2001)
Ryan Williams: Inductive Time-Space Lower Bounds for SAT and Related Problems. Computational Complexity 15(4), 433–470 (2006)
Ryan Williams: Time-Space Tradeoffs for Counting NP Solutions Modulo Integers. Computational Complexity 17(2), 179–219 (2008)
Ryan Williams (2010). Alternation-Trading Proofs, Linear Programming, and Lower Bounds. In Proc. 27th Intl. Symp. on Theory of Computings (STACS 2010), 669–680.
Ryan Williams (2013). Alternation-Trading Proofs, Linear Programming, and Lower Bounds. ACM Transactions of Computation Theory 5(2), article 6.
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Buss, S.R., Williams, R. Limits on Alternation Trading Proofs for Time–Space Lower Bounds. comput. complex. 24, 533–600 (2015). https://doi.org/10.1007/s00037-015-0104-9
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DOI: https://doi.org/10.1007/s00037-015-0104-9