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No predictable technical breakthroughs in computers will help solve the traveling salesman problem, since even a computer a billion times faster will still be stumped by the addition of a few more cities.

The brain is a kind of computer, and thought is just a complex computation. Life is a complex chemical reaction. It takes nothing away from the wonder or value of human thought. Life and thought are both made all the more wonderful by the realization that they emerge from simple, understandable parts. I do not feel diminished by my kinship to Turing’s machine.

the physics of a neuron depends on quantum mechanics, just as the physics of a transistor does, but there is no evidence that neural processing takes place at the quantum mechanical level as opposed to the classical level; that is, there is no evidence that quantum
mechanics is necessary to explain human thought.

The bits in a quantum computer must remain entangled in order for the computation to work, but the smallest of disturbances—a passing cosmic ray, say, or possibly even the inherent noisiness of the vacuum itself—can destroy the entanglement. This loss of
entanglement, called decoherence, could turn out to be the Achilles heel of quantum mechanical computers.

A quantum computer would take advantage of entanglement. This is analagous to an atom being in many places at once: a bit that it is in many states (1 or 0) at once.

Atoms seem able to compute certain problems easily, such as how they stick together—problems that are very difficult to compute on a conventional computer. For instance, when two hydrogen atoms bind to an oxygen atom to form a water molecule, these atoms somehow “compute” that the angle between the two bonds should be 107 degrees. How can a single molecule be so much faster than a digital
computer?

A single subatomic particle exists everywhere at once, and we are
just more likely to observe such a particle at one place than at another. For most purposes, we can think of a particle as being where we observe it to be, but to explain all observed effects we have to acknowledge that the particle is in more than one place.

The only way we know how to achieve genuinely unpredictable effects is to rely on quantum mechanics. Unlike the classical physics of the roulette wheel, in which effects are determined by causes, quantum mechanics produces effects that are purely probabilistic. There is no way of predicting, for example, when a given uranium atom will decay into lead.

Statistically speaking, most mathematical functions are noncomputable. This is because any program can be specified in a finite number of bits, whereas specifying a function usually requires an infinite number of bits.

A roulette wheel is an example of what physicists call a chaotic system—a system in which a small change in the initial conditions (the throw, the mass of the ball, the diameter of the wheel, and so forth) can produce a large change in the state to which the system evolves (the resulting number). This notion of a chaotic system helps explain how a deterministic set of interactions can produce unpredictable results.

*pseudorandom sequence* only appears random to an observer who does not know how it was computed.

Can a deterministic system like a computer produce a truly random sequence of numbers? In a formal sense, the answer is No, since everything a digital computer does is determined by its design and its inputs. But the same could be said of a roulette wheel—after all, the ball’s final landing place is determined by the physics of the ball (its mass, its velocity) and the spinning wheel.

A true continuum is unrealizable in the physical world. The problem with analog computers is that their signals can achieve only a limited degree of accuracy. For example, an electrical signal can be disturbed by the random motion of molecules inside a wire, or by the magnetic field created when a light is turned on in the next room.

No device built in the physical universe can have any more computational power than a Turing machine. To put it more precisely, any computation that can be performed by any physical computing device can be performed by any universal computer, as long as the latter has sufficient time and memory.

Today, we call the combination of a finite-state machine with an infinitely long tape a Turing machine. A universal computer with the proper programming should be able to simulate the function of a human brain.

The discovery that our planet was just one of a number of planets in orbit around the Sun—was deeply disturbing to many people at the time, and the philosophical implications of astronomy became a topic of heated debate. A similar controversy arose over evolutionary theory. I am convinced that most of the current philosophical discussions about the limits of computers are based on a similar misjudgment.

Random-access memory; Central processing unit; Input-output processor

If every dot on the screen displays data stored in a different memory
location, the computer can draw any pattern on the screen just by writing the appropriate pattern into memory.

the array-based stack implementation is not necessarily ideal