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Space complexity. [AB 4]. Input/Work/Output TM. 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?. | | T. | | n. . | | s. | Q |. n. . s. . T. . . . - PowerPoint PPT Presentation
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Space complexity
[AB 4]
•Read only!
Input Tape
•Only tape counted
Work Tape
•Write only! No going back
Output Tape
2
Input/Work/Output TM
_ _ _ _ _ - -
_ _ _ _ _ - -
a a b a b - -
Input
Work
Output
Input tape
Output tape
Work tape
State Input head
Work head
Output head
Configurations (First try)
||n |Q| s
The recorded state of a Turing machine at a specific time
How many distinct configurations may a Turing machine that uses s cells have?
||T ||s n T
Input tape
Output tape
Work tape
State Input head
Work head
Output head
Configurations
||n |Q| s
•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).
||T ||s n T
Work tape State Input head Work head
Configurations
|Q| s||s n
For a given input string x {0,1}n
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.
• Let t:NN be a complexity function
Definition:
• Deterministic space:
•
Det. Log space:
•
Det polynomial space:
7
Space-Complexity
nSPACEL log
TM space-by decided | ticdeterminisntOLLntSPACE
k
knSPACEPSPACE
PSPACE
L
• PPSPACE
Claim:
•a TM that runs t(n) stepsuses at most t(n) space
Proof:
• PSPACEEXPTIME
Claim:
• Next
Proof:
8
Space vs. Time
EXPTIME
PSPACEP
The Configuration graph
Vertices – All possible configurations
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
anbncn
Minimum Spanning Tree
Seating:
Hamiltonian Cycle
Tour: Hamiltonian
Path
Halting Problem
Name the Class
11
EXPPSPACE
NP
P
NL
L
The Strong Church-Turing thesis
"A probabilistic Turing machine can efficiently simulate any realistic model of computation.“
New Evidence that Quantum Mechanics is Hard to Simulate on Classical
ComputersI'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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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?
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?