The Church-Turing Thesis and Physics

1. The Church-Turing Thesis and Physics Scott Aaronson (University of Texas at Austin) Bernays Lecture, ETH Zurich, Sep. 2, 2019

2. The Church-Turing Thesis “Computable” = Turing-computable Fundamental principle linking computer science to the real world

3. Discrete state of unbounded size, reading, writing, looping, conditionals—and that’s about it! Turing machines = Lisp = cellular automata = -calculus = … What makes a “Turing computer”? “Once you know one way to program, you basically know them all” (defun depth-first-search (start goal been-list moves) (cond ((equal start goal) (reverse (cons start been-list))) (t (try-moves start goal been-list moves moves)))) (defun try-moves (start goal been-list moves-to-try moves) (cond ((null moves-to-try) nil) ((member start been-list :test #'equal) nil) (t (let ((child (funcall (car moves-to-try) start))) (if child (or (depth-first-search (funcall (car moves-to-try) start) goal (cons start been-list) moves) (try-moves start goal been-list (cdr moves-to-try) moves)) (try-moves start goal been-list (cdr moves-to-try) moves)))))) (defun run-depth (start goal moves) (depth-first-search start goal () moves))

4. COMPUTABLE UNCOMPUTABLE The halting problem Multiplication 3  5 = 15 Primality Diophantine equations Knot Isomorphism Tiling Almost all functions f:N{0,1} Presburger arithmetic Matrix mortality “Whether God exists” (!)

5. If we want a mathematics of computation, we need a precise definition of “computable”—much like the (,) definition of continuity What does the CT Thesis mean?A 1930s View… Where could we get such a definition? Turing’s approach: Psychologically analyze what a human “computer” does with pen, paper, and no creativity Pragmatic approach: Invent a bunch of definitions and see if they’re all equivalent

6. Take it as an empirical claim about physics—say, Any bounded physical system can be simulated by a Turing machine, to any desired precision More Modern View of CT Thesis This would be falsified if we found a magic box on the beach to solve the halting problem!(To whatever extent we were convinced that the box worked) The more we can simulate, the better-supported the CT Thesis becomes—but it can never be “proved”

7. Challenges to the Church-Turing Thesis

8. Goldin and Wegner: The Church-Turing Thesis is false because it ignores interactivity, distributed systems, the Internet, … I parodied this with the…Toaster-Enhanced Turing Machine!

9. Some programming language semanticists: The Church-Turing Thesis is false because Turing machines can’t take functions as input, or do other “higher-order” things Response: Confuses the internal organization of a software system with a computer’s actual physical inputs and outputs More deeply, the CT Thesis is why higher-order formalisms are possible at all—because they can be compiled down to machine code!

10. “The Church-Turing Thesis is false because it doesn’t account for human consciousness” Response: Regardless of your beliefs about the hard problem of consciousness—here we’re only talking about simulation

11. “The Church-Turing Thesis is false because you can never know the initial conditions well enough to simulate all of Nature” Response: Computers could be defined as those physical objects for which we do think we know the initial conditions and laws well enough; we just can’t calculate all their consequences

12. “The Church-Turing Thesis is false because quantum mechanics says measurement outcomes are random, and infinite random strings are uncomputable” Response: Even in (finite-dimensional) QM, a Turing machine can calculate the probability distributions over outcomes—leaving only a tiny sampling step Enough with theft over honest toil! What would a REAL counterexample to the CT Thesis look like?

13. Relativity Computer(cf. Malament and Hogarth) DONE

14. Zeno’s Computer STEP 1 STEP 2 Time (seconds) STEP 3 STEP 4 STEP 5

15. Noise-Free Analog Computers More than Turing machines! Can multiply  and e in a single time step But also less than Turing machines! Can’t, e.g., extract one decimal digit of  In any case, probably can’t realize in our universe, again for quantum gravity reasons

16. Roger Penrose speculates, unconvincingly in my view, that this is where uncomputability could show up in physics. But… Quantum Gravity Computing? AdS/CFT has shown that, in toy models, even gravity can be subsumed into standard QM, which suggests (but doesn’t prove) that one could simulate it on a computer

17. “The Church-Turing Thesis is vacuous because the whole observable universe is finite—only ~47 billion light years in radius, and containing at most ~10122 bits. A lookup table suffices!” Response: C’mon! The CT Thesis seeks to compress a huge amount of data about computations in Nature—much like any other scientific hypothesis

18. Interpreted as a statement about the physical world, the Church-Turing Thesis remains on remarkably solid ground after 83 years Concluding Thoughts Should we be surprised that Church, Turing, et al. managed to “guess the right definition of computation from their armchairs”? Could we build sensible laws of physics around a notion of discrete computation totally different from Turing’s—say, where the halting problem was computable but the AND function wasn’t?

19. What’s efficiently computable in the physical world? Can we—very soon—build quantum computers that are good enough to bear on that question? Next Two Lectures