Moore designed his pinball machine to complete the analogy with the Turing machine. The starting position of the pinball represents the data that is fed to the tape of the Turing machine. In a crucial (unrealistic) way, the player must be able to adjust the starting position of the ball with infinite precision, meaning that the specified position of the ball requires a number, with endless numbers being made after the decimal point. Only in such numbers can Moore encode data of infinitely long Turing tape.
The bumper arrangement then takes the ball to a new position to correspond to reading and writing on the tape of the Turing machine. Certain curved bumpers move tape in a way that makes data stored in distant decimal places more important to reminisce about a chaotic system, while relatively curved bumpers are the opposite. The ball exits from the bottom of the box to mark the end of the calculation, and the result is the final position.
Moore sets up the flexibility of the computer for the pinball machine – one arrangement of the bumper might calculate the first thousand digits of the PI, while the other might calculate the best next step in the chess game. But in doing so, he also injects the attribute we usually don’t associate with computers: unpredictability.
Some algorithms stop and output the result. But others run forever. (Consider a program that tasked prints the final number of PI.) Turing asked, is there a program that can check any program and determine if it will stop? This problem is called a stop problem.
Turing shows that such procedures do not exist if it is taken into account its meaning. If one computer can predict the behavior of another machine, it is easy to modify the first machine (the machine that predicts the behavior) to run forever when the other machine stops. And vice versa: when another machine is running forever, it stops. Then – this is the shocking part – the imaginary imagination describes the description of one’s own adjustment predictor. If the machine stops, it will run forever. If it runs forever, it will also stop. Turing concluded that since neither of these options is possible, the predicted body cannot exist.
(His discovery is closely related to groundbreaking results since 1931, when logician Kurt Gödel developed a similar approach Feeding self-referential paradox Enter a strict mathematical framework. Goldell proves that mathematical statements exist and that their truth cannot be established. )
In short, Turing proves that solving the stop problem is impossible. The only general way to know if the algorithm stops is to run it as long as possible. If stopped, you have the answer. But if it isn’t, you’ll never know if it actually runs, or if it stops if you wait longer.
“We know there are some initial states that we can’t predict in advance what it’s going to do,” Walpert said.
since Moore designed his box To mimic any Turing machine, it may also act in unpredictable ways. The exit of the ball marks the end of the calculation, so whether there is any specific bumper arrangement that will capture the ball or turn the ball to the exit must also be undecided. “In fact, any question about the long-term dynamics of these more refined maps is undeterminable,” Moore said.
