What More Than Turing-Universality Do You Want?

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What More Than Turing-Universality Do You Want?
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• Scott Aaronson (MIT)
• Papers and slides at www.scottaaronson.com

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The Pervasiveness of Universality
Almost any programming language or cellular
automaton you can think to invent, provided its
sufficiently complicated, will be able to
simulate a Turing machine For n large enough,
almost any n-bit logic gate will be capable of
expressing all Boolean functions Almost any
2-qubit unitary transformation can be used to
approximate any unitary transformation on any
number of qubits, to any desired precision
Yet precisely because universality is common as
dirt, its not useful for distinguishing among
candidate physical theories
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versus
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What We Could Ask of Physical LawsBeyond just
Turing-universality
Simplicity Symmetry Relativity (at least
Galilean) Quantum Mechanics (but why?) Robustness
(i.e., fault-tolerant universality) Physical
Universality (cf. constructor theory) Interesting
Structure Formation in Generic Cases
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Symmetry
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Classical Reversible Gates
Flip second bit iff first bit is 1 Not universal
(affine)
CNOT
Flip third bit iff first two bits are both
1 Universal can generate all permutations of
n-bit strings
Toffoli
Swap second and third bits iff first bit is
1 Computationally universal, but has a symmetry
(preserves Hamming weight)
Fredkin
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A.-Grier-Schaefer 2015 Classified all sets of
reversible gates in terms of which n-bit
reversible transformations they generate
(assuming swaps and ancilla bits are free)
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Physical Universality
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Schaeffer 2014 The first known
physically-universal cellular automaton (able
to implement any transformation in any bounded
region, by suitably initializing the complement
of that region) Solved open problem of Janzing
2010
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One of My Favorite Open Questions
For every n-qubit unitary transformation U, is
there a Boolean function f such that U can be
realized by a polynomial-time quantum algorithm
with an oracle for f?
(Im giving you any computational capability f
you could possibly wantbut its still far from
obvious how to get the physical capability U!)
Can show For every n-qubit state ??, theres a
Boolean function f such that ?? can be prepared
by a polynomial-time quantum algorithm with an
oracle for f
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Interesting Structure Formation
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How to Measure Interesting Structure?
Many people have studied this Jim Crutchfield
will tell you about how to define structure in
terms of predictability
One simpleminded measure the Kolmogorov-Chaitin
complexity of a coarse-grained description of our
cellular automaton or other system
Sean Carrolls example
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The Coffee Automaton
A., Carroll, Mohan, Ouellette, Werness 2015 A
probabilistic n?n reversible system that starts
half coffee and half cream. At each time
step, we randomly shear half the coffee cup
horizontally or vertically (assuming a toroidal
cup)
We prove that the apparent complexity of this
image has a rising-falling pattern, with a
maximum of at least n1/6
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Interesting Computations Should Be Not Merely
Expressible, But Succinctly Expressible?
BB(n) the maximum number of steps that a
1-tape, 2-symbol, n-state Turing machine can take
on an initially blank tape before halting
BB(1)1 BB(2)6 BB(3)21 BB(4)107 BB(5)?47,176,8
70 BB(6)?7.4?1036534
(Famous uncomputably-rapidly growing function)
Gödel ? beyond some finite point, the values of
BB(n) are not even provable in ZF set theory!
(assuming ZF is consistent)
Yedidia 2015 (building on Harvey Friedman) This
happens at n?533,482Also, 10,000 states suffice
to test Goldbach