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Space complexity [AB 4]

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2 Input/Work/Output TM Output

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Input tape Output tape Work tape State Input head Work head Output head Configurations (First try) The recorded state of a Turing machine at a specific time How many distinct configurations may a Turing machine that uses s cells have?

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Input tape Output tape Work tape State Input head Work head Output head Configurations The input stays fixed (Read only tape) The output (or the output tape head) does not affect the next transitions (Right only, write only tape).

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Work tape State Input head Work head Configurations For a given input string x {0,1} n

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Space complexity Def: The space complexity of a Turing Machine T on input x is the maximal number of tape cells used throughout the computation.

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7 Space-Complexity

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8 Space vs. Time

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The Configuration graph Vertices – All possible configurations

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PSPACE EXP

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Name the Class 11

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

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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)

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“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]dualistmaterialistmindnon-physicalnon-physical properties [6] [7] Are there imaginable digital computers which would do as well as human beings?

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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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Recall last lecture and Nondeterministic TMs Ola Svensson.

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