302 NOTES
27. For an interesting trust- based solution to the prisoner’s dilemma and other games, see
Joshua Letchford, Vincent Conitzer, and Kamal Jain, “An ‘ethical’ game- theoretic
solution concept for two- player perfect- information games,” in Proceedings of the 4th
International Workshop on Web and Internet Economics, ed. Christos Papadimitriou and
Shuzhong Zhang (Springer, 2008).
28. Origin of the tragedy of the commons: William Forster Lloyd, Two Lectures on the
Checks to Population (Oxford University, 1833).
29. Modern revival of the topic in the context of global ecology: Garrett Hardin, “The
tragedy of the commons,” Science 162 (1968): 1243– 48.
30. It’s quite possible that even if we had tried to build intelligent machines from chemical
reactions or biological cells, those assemblages would have turned out to be implemen-
tations of Turing machines in nontraditional materials. Whether an object is a general-
purpose computer has nothing to do with what it’s made of.
31. Turing’s breakthrough paper defined what is now known as the Turing machine, the
basis for modern computer science. The Entscheidungsproblem, or decision problem, in
the title is the problem of deciding entailment in first- order logic: Alan Turing, “On
computable numbers, with an application to the Entscheidungsproblem,” Proceedings of
the London Mathematical Society, 2nd ser., 42 (1936): 230– 65.
32. A good survey of research on negative capacitance by one of its inventors: Sayeef Sala-
huddin, “Review of negative capacitance transistors,” in International Symposium on
VLSI Technology, Systems and Application (IEEE Press, 2016).
33. For a much better explanation of quantum computation, see Scott Aaronson, Quan-
tum Computing since Democritus (Cambridge University Press, 2013).
34. The paper that established a clear complexity- theoretic distinction between classical
and quantum computation: Ethan Bernstein and Umesh Vazirani, “Quantum com-
plexity theory,” SIAM Journal on Computing 26 (1997): 1411– 73.
35. The following article by a renowned physicist provides a good introduction to the
current state of understanding and technology: John Preskill, “Quantum computing in
the NISQ era and beyond,” arXiv:1801.00862 (2018).
36. On the maximum computational ability of a one- kilogram object: Seth Lloyd, “Ulti-
mate physical limits to computation,” Nature 406 (2000): 1047– 54.
37. For an example of the suggestion that humans may be the pinnacle of physically
achievable intelligence, see Kevin Kelly, “The myth of a superhuman AI,” Wired, April
25, 2017: “We tend to believe that the limit is way beyond us, way ‘above’ us, as we are
‘above’ an ant.... What evidence do we have that the limit is not us?”
38. In case you are wondering about a simple trick to solve the halting problem: the obvi-
ous method of just running the program to see if it finishes doesn’t work, because that
method doesn’t necessarily finish. You might wait a million years and still not know if
the program is really stuck in an infinite loop or just taking its time.
39. The proof that the halting problem is undecidable is an elegant piece of trickery. The
question: Is there a LoopChecker(P,X) program that, for any program P and any input
X, decides correctly, in finite time, whether P applied to input
X will halt and produce
a result or keep chugging away forever? Suppose that LoopChecker exists. Now write
a program Q that calls LoopChecker as a subroutine, with Q itself and X as inputs, and
then does the opposite of what LoopChecker(Q,X) predicts. So, if LoopChecker says
that Q halts, Q doesn’t halt, and vice versa. Thus, the assumption that LoopChecker
exists leads to a contradiction, so LoopChecker cannot exist.
40. I say “appear” because, as yet, the claim that the class of NP- complete problems re-
quires superpolynomial time (usually referred to as P ≠ NP) is still an unproven con-
jecture. After almost fifty years of research, however, nearly all mathematicians and
computer scientists are convinced the claim is true.
41. Lovelace’s writings on computation appear mainly in her notes attached to her trans-
lation of an Italian engineer’s commentary on Babbage’s engine: L. F. Menabrea,
“Sketch of the Analytical Engine invented by Charles Babbage,” trans. Ada, Countess
of Lovelace, in Scientific Memoirs, vol. III, ed. R. Taylor (R. and J. E. Taylor, 1843).
Menabrea’s original article, written in French and based on lectures given by Babbage
in 1840, appears in Bibliothèque Universelle de Genève 82 (1842).
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