Explorations in Universality

Scott Aaronson (MIT)Ada Lovelace Bicentenary Lecture on

Computability, Hebrew University, Dec. 31. 2015

Two Meanings of “Universality”Turing’s “Existence of the Software Industry Lemma”:There exists a machine U such that U(P,x)=P(x) for all P

The Church-Turing “Ceiling of Expressive Power”

My vision, when I first learned what programming was as an 11-year-old:

ScottLang

PascalQBasic

GW-BASIC

Nope! GW-BASIC can already do everything, in principle

Expressiveness Ceilings Everywhere!Almost any programming language or cellular automaton you can think to invent, provided it’s “sufficiently complicated,” can simulate Turing machines

For n 3 or 4, almost any n-bit logic gate will be able to express 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

What could there possibly be to say about such phenomena that’s new?

versus

If Turing-universality is

“common as dirt,” then it gives no

basis to distinguish “our” laws from,

say, Conway’s Game of Life

This TalkRecent explorations, by me and my students, into more “fine-grained” notions of universality

Luke Schaeffer, ICALP 2014: Physically universal cellular automaton

Adam Yedidia, work in progress: Small Turing machines that do interesting things

A.-Grier-Schaeffer 2015: Classification of reversible Boolean logic gates

Some favorite open problems

Physical UniversalityAbility to do universal computation Ability to manipulate physical world in any desired way

Suppose I gave you an oracle for NP-complete problems. Could you use it to- Take over the world?- Cure cancer?- Make someone fall in love with you?

The Toaster-Enhanced Turing Machine (TETM)

Janzing 2010: Call a cellular automaton physically universal if it can implement any desired transformation on any finite region R, provided we appropriately initialize the complement of R

Does there exist a physically universal CA?

Conway’s Life: Not physically universal, because not reversible. No detectors outside R can tell whether the center of R contained an isolated particle that died

Many Reversible CAs: Still not physically universal, because of the possibility of constructing “impenetrable walls”

Luke Schaeffer’s Automaton (2014)The first known example of a physically universal CA

My Favorite Open Problem About Physical Universality

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?

(I’m giving you any computational capability f you could possibly want—but it’s still far from obvious how to get the physical capability U!)

Can show: For every n-qubit state |, there’s a Boolean function f such that | can be prepared by a polynomial-time quantum algorithm with an oracle for f

How Weird / Improbable / Complex / Artificial Are Universal Computers?Clearly related to the origin of life debate!

Even CAs / programming languages that are equivalent in the Church-Turing sense, could differ dramatically in “how much encoding we need to get over the threshold of interesting behavior”

Pr[Life arising on a given planet]:“A constant factor that matters!”

Stephen Wolfram 2007: $25,000 prize for proving that a particular 2-state, 3-symbol TM was universal

One reason to doubt his claim: In this game, one gets “tiny universal machines” only at the cost of fantastically complicated input and output encodings!

Minsky, McCarthy, et al. 1950s: What’s the “simplest” universal TM? (Found one with 7 states and 4 symbols)

Won later that year by Alex Smith (who had “nothing better to do that weekend”)

Wolfram claims this sort of thing vindicates his “Principle

of Computational Universality”

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

Famously 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)

At How Many States Do Turing Machines “Cross The Threshold”?

Question: Where does this happen? Could BB(6) already be independent of ZF? What about BB(60)? BB(600)?

BB(1)=1

BB(2)=6

BB(3)=21

BB(4)=107

BB(5)47,176,870

BB(6)7.41036534

Adam Yedidia, 2015: Building on work of Harvey Friedman, constructed a 380,000-state Turing machine whose halting or non-halting is independent of ZF set theory

Also an 8,000-state machine that tests Goldbach’s Conjecture, and a 30,000-state machine that tests the Riemann Hypothesis

Constructions use a special-purpose programming language, compiled to multi-tape TM and then one-tape TM

Ongoing work: Inline interpreter run by the TM itself improvement to ~40,000 states for ZF set theory, 12,000 for Riemann Hypothesis

Conclusion: For humanity to know BB(40,000) would entail knowing Con(ZF). What about BB(400)? BB(6)?

Post’s Lattice (1920s)All possible sets of Boolean functions that are closed under composing them into circuits

If you assume the constant functions (0 and 1) are available for free, then it looks like this (otherwise it’s way more complicated)

Meaning: from any non-affine gate you can extract AND or OR; from any non-monotone gate you can extract NOT, etc.

What’s the analogue of Post’s lattice for classical reversible gates?

Flip y iff x=1

Not universal (affine)

CNOT ToffoliFlip z iff x=y=1

Universal; generates all permutations of n-

bit strings

FredkinSwap y,z iff x=1

Like Toffoli but conservative (preserves

Hamming weight)

I.e. permutations of {0,1}k

Ground rules: Swaps are free, as are ancilla bits, as long as they’re returned to their initial states by the end

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)

Why Toffoli Generates All Reversible Transformations

xf

xxfgarxf

xxfgarxfx

xfx

xfxfxgarx

xfxgarxx

,,

,,,

,

,,,

,,

3 Combinatorial Results that Shape the Classification

(Proofs left as exercises for the audience)

1. If a reversible G satisfies |G(x)||x|+j (mod k) for all x, then either k=2 or j=0

2. If a nontrivial affine reversible transformation G satisfies |G(x)||x| (mod k) for all x, then either k=2 or k=4

3. If a non-conservative reversible transformation G:{0,1}n{0,1}n satisfies G(x)G(y)xy (mod k) for all x,y, then k=2

My Favorite Open Problem HereClassification of all possible sets of quantum gates acting on qubits, in terms of which unitaries they generate

A complicated problem in representation theory and Lie algebras, but seems fundamental for QC

Grier-Schaeffer 2015: Classification of all possible sets of “stabilizer” quantum gates (discrete subgroup used in quantum error-correction)

Aaronson-Bouland 2015: Classification of all 2-mode beamsplitters

Concrete questions: Are there interesting discrete subgroups other than the stabilizer group and its conjugates? Does every nontrivial continuous subgroup yield universal QC?

Discussion QuestionsWhat would a universe look like where the Physical Church-Turing Thesis was false?

Our quantum world? Falsifies the polynomial-time Physical C-T thesis, but not the original computability one

“Hypercomputing” universes: would allow ways to solve the halting problem that, in our world, appear precluded by quantum gravity (e.g. Bekenstein’s bound)

David Deutsch says that, with different laws of physics, the halting problem would be easy and the AND function would be hard. What are those laws?

What would an interesting model of computation look like that could solve its own halting problem?

What would an interesting universe look like that couldn’t do universal computation?

The Digi-Comp II(In the lobby of the MIT Stata Center)

“It can multiply! It can divide! It can

sort!”

Me: “Real question is, what can’t it do?”

Is this silly device with metal balls a universal computer? If so, wow! If not,

what prevents it from being one?

CC (Comparator Circuit): Complexity class defined by Subramanian in 1990, between NL and P

A CC-complete problem is Stable Marriage

Piles

Theorem (A.): A natural idealization of the Digi-Comp II solves exactly the problems in CC

DigiComp Comparators

x/2 x/2

SORT

1 0

0 1

ConclusionThe subject of universality is almost 100 years old—almost 200 if we go back to Lady Ada—yet there are still basic questions that remain to be answered.