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

Content: input tape

||N

Head position:

Input

N

Content: work tape

||S

Head position:

Work

S

the machine’s

state

|Q|

How many distinct configurations may a TM with input-size N and work-tape of size S have?

3

ConfigurationsWhat

about output

?

anbn

anbnan

anb2nan

Palindrome

anb2na4nb8n…

Brain Hurts

4

EXPPSPACE

NP

P

NL

L

Find

A problem in NL

Not known to

be in L

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 .

• Let t:NN be a complexity function

Definition:

Deterministic time:

•

Det. Polynomial time:

•

Det Exponential time:

6

Time-Complexity

k

knTIMEP

TM time-by decided | ticdeterminisntOLLntTIME

k

nkeTIMEEXP

EXPP

• 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 content of the tape

the position

of the head

the machine’s state

Configurations

||s |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?

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

12

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)

“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?

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?