Макс Тегмарк - Life 3.0 - Being Human in the Age of Artificial Intelligence

What Is Computation?

We’ve now seen how a physical object can remember information. But how can it compute?

A computation is a transformation of one memory state into another. In other words, a computation takes information and transforms it, implementing what mathematicians call a function . I think of a function as a meat grinder for information, as illustrated in figure 2.5: you put information in at the top, turn the crank and get processed information out at the bottom—and you can repeat this as many times as you want with different inputs. This information processing is deterministic in the sense that if you repeat it with the same input, you get the same output every time.

Figure 2.5: A computation takes information and transforms it, implementing what mathematicians call a function . The function f (left) takes bits representing a number and computes its square. The function g (middle) takes bits representing a chess position and computes the best move for White. The function h (right) takes bits representing an image and computes a text label describing it.

Although it sounds deceptively simple, this idea of a function is incredibly general. Some functions are rather trivial, such as the one called NOT that inputs a single bit and outputs the reverse, thus turning zero into one and vice versa. The functions we learn about in school typically correspond to buttons on a pocket calculator, inputting one or more numbers and outputting a single number—for example, the function x 2simply inputs a number and outputs it multiplied by itself. Other functions can be extremely complicated. For instance, if you’re in possession of a function that would input bits representing an arbitrary chess position and output bits representing the best possible next move, you can use it to win the World Computer Chess Championship. If you’re in possession of a function that inputs all the world’s financial data and outputs the best stocks to buy, you’ll soon be extremely rich. Many AI researchers dedicate their careers to figuring out how to implement certain functions. For example, the goal of machine-translation research is to implement a function inputting bits representing text in one language and outputting bits representing that same text in another language, and the goal of automatic-captioning research is inputting bits representing an image and outputting bits representing text describing it (figure 2.5).

Figure 2.6: A so-called NAND gate takes two bits, A and B, as inputs and computes one bit C as output, according to the rule that C = 0 if A = B = 1 and C = 1 otherwise. Many physical systems can be used as NAND gates. In the middle example, switches are interpreted as bits where 0 = open, 1= closed, and when switches A and B are both closed, an electromagnet opens the switch C. In the rightmost example, voltages (electrical potentials) are interpreted as bits where 1 = five volts, 0 = zero volts, and when wires A and B are both at five volts, the two transistors conduct electricity and the wire C drops to approximately zero volts.

In other words, if you can implement highly complex functions, then you can build an intelligent machine that’s able to accomplish highly complex goals. This brings our question of how matter can be intelligent into sharper focus: in particular, how can a clump of seemingly dumb matter compute a complicated function?

Rather than just remain immobile as a gold ring or other static memory device, it must exhibit complex dynamics so that its future state depends in some complicated (and hopefully controllable/programmable) way on the present state. Its atom arrangement must be less ordered than a rigid solid where nothing interesting changes, but more ordered than a liquid or gas. Specifically, we want the system to have the property that if we put it in a state that encodes the input information, let it evolve according to the laws of physics for some amount of time, and then interpret the resulting final state as the output information, then the output is the desired function of the input. If this is the case, then we can say that our system computes our function.

As a first example of this idea, let’s explore how we can build a very simple (but also very important) function called a NAND gate *3out of plain old dumb matter. This function inputs two bits and outputs one bit: it outputs 0 if both inputs are 1; in all other cases, it outputs 1. If we connect two switches in series with a battery and an electromagnet, then the electromagnet will only be on if the first switch and the second switch are closed (“on”). Let’s place a third switch under the electromagnet, as illustrated in figure 2.6, such that the magnet will pull it open whenever it’s powered on. If we interpret the first two switches as the input bits and the third one as the output bit (with 0 = switch open, and 1 = switch closed), then we have ourselves a NAND gate: the third switch is open only if the first two are closed. There are many other ways of building NAND gates that are more practical—for example, using transistors as illustrated in figure 2.6. In today’s computers, NAND gates are typically built from microscopic transistors and other components that can be automatically etched onto silicon wafers.

There’s a remarkable theorem in computer science that says that NAND gates are universal , meaning that you can implement any well-defined function simply by connecting together NAND gates. *4So if you can build enough NAND gates, you can build a device computing anything! In case you’d like a taste of how this works, I’ve illustrated in figure 2.7 how to multiply numbers using nothing but NAND gates.

MIT researchers Norman Margolus and Tommaso Toffoli coined the name computronium for any substance that can perform arbitrary computations. We’ve just seen that making computronium doesn’t have to be particularly hard: the substance just needs to be able to implement NAND gates connected together in any desired way. Indeed, there are myriad other kinds of computronium as well. A simple variant that also works involves replacing the NAND gates by NOR gates that output 1 only when both inputs are 0. In the next section, we’ll explore neural networks, which can also implement arbitrary computations, i.e., act as computronium. Scientist and entrepreneur Stephen Wolfram has shown that the same goes for simple devices called cellular automata, which repeatedly update bits based on what neighboring bits are doing. Already back in 1936, computer pioneer Alan Turing proved in a landmark paper that a simple machine (now known as a “universal Turing machine”) that could manipulate symbols on a strip of tape could also implement arbitrary computations. In summary, not only is it possible for matter to implement any well-defined computation, but it’s possible in a plethora of different ways.

As mentioned earlier, Turing also proved something even more profound in that 1936 paper of his: that if a type of computer can perform a certain bare minimum set of operations, then it’s universal in the sense that given enough resources, it can do anything that any other computer can do. He showed that his Turing machine was universal, and connecting back more closely to physics, we’ve just seen that this family of universal computers also includes objects as diverse as a network of NAND gates and a network of interconnected neurons. Indeed, Stephen Wolfram has argued that most non-trivial physical systems, from weather systems to brains, would be universal computers if they could be made arbitrarily large and long-lasting.