computational complexity

, Volume 15, Issue 4, pp 433–470

Inductive Time-Space Lower Bounds for Sat and Related Problems


    • Computer Science DepartmentCarnegie Mellon University
Open AccessArticle

DOI: 10.1007/s00037-007-0221-1

Cite this article as:
Williams, R. comput. complex. (2006) 15: 433. doi:10.1007/s00037-007-0221-1


We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows.
  1. 1.

    We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism and alternating computation, on both subpolynomial (n o(1)) space RAMs and sequential one-tape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NP-complete problems that have efficient reductions from SAT, and ∑ k -SAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using \(n^{\sqrt{3}}\) time and subpolynomial space.

  2. 2.

    We show how indirect diagonalization leads to time-space lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant c k > 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic \(n^{{c}_{k}}\) time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in \({n^{{k \cdot {c}}_{k}}}\) time and subpolynomial space.



Time-space tradeoffs lower bounds polynomial-time hierarchy satisfiability diagonalization bounded nondeterminism

Subject classification.


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© Birkhäuser Verlag, Basel 2007