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Space complexity

Space complexity. [AB 4]. Input/Work/Output TM. Configurations. Brain Hurts. Time and space complexity. Def: The running time of T on input x is the number of δ transitions from the initial state to an accept/reject state.

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Space complexity

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  1. Space complexity [AB 4]

  2. Input/Work/Output TM

  3. Configurations

  4. Brain Hurts

  5. Time and space complexity Def: The running time of T on input x is the number of δ transitions from the initial state to an accept/reject state. Def: The space complexity of T on input x is the maximal number of tape cells used throughout the computation.

  6. Time-Complexity

  7. Space-Complexity

  8. Space vs. Time

  9. Configurations The recorded state of a Turing machine at a specific time • How many distinct configurations may a Turing machine that uses s cells have? ||s  s  |Q|

  10. The Configuration graph Vertices – All possible configurations

  11. PSPACE  EXP Proof: A deterministic run that halts must avoid repeating a configuration  its running time is bounded from above by the number of configurations the machine has, which, for a PSPACE machine, is at most exponential

  12. Name the Class

  13. The Strong Church-Turing thesis "A probabilistic Turing machine can efficiently simulate any realistic model of computation.“

  14. New Evidence that Quantum Mechanics is Hard to Simulate on Classical Computers I'll discuss new types of evidence that quantum mechanics is hard to simulate classically -- evidence that's more complexity-theoretic in character than (say) Shor's factoring algorithm, and that also corresponds to experiments that seem easier than building a universal quantum computer. Specifically:(1) I'll show that, by using linear optics (that is, systems of non-interacting bosonic particles), one can generate probability distributions that can't be efficiently sampled by a classical computer, unless P^#P = BPP^NP and hence the polynomial hierarchy collapses. The proof exploits an old observation: that computing the amplitude for n bosons to evolve to a given configuration involves taking the Permanent of an n-by-n matrix. I'll also discuss an extension of this result to samplers that only approximate the boson distribution. (Based on recent joint work with Alex Arkhipov) (2) Time permitting, I'll also discuss new oracle evidence that BQP has capabilities outside the entire polynomial hierarchy. (arXiv:0910.4698)

  15. “Can machines Think?” Turing (1950): I PROPOSE to consider the question, 'Can machi The question of whether it is possible for machines to think has a long history, which is firmly entrenched in the distinction between dualist and materialist views of the mind. From the perspective of dualism, the mind is non-physical (or, at the very least, has non-physical properties[6]) and, therefore, cannot be explained in purely physical terms. The materialist perspective argues that the mind can be explained physically, and thus leaves open the possibility of minds that are artificially produced.[7] Are there imaginable digital computers which would do as well as human beings?

  16. What are we? • Was Alan Turing a computer mistreated by other computers? • Will there ever be a computer passing Turing’s test? • Can everything in our universe be captured as computation? • Is there free choice?

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