Moore designed the pinball machine to complete the similarity of the Turing machine. The pinball start position represents the data on the tape supplied to the Turing machine. Critical (and unrealistically), the player must be able to adjust the starting position of the ball with infinite accuracy. In other words, to specify the position of the ball, the numbers are required in an infinite matrix of numbers after the decimal point. Only with such a number Moore can encode infinitely long Turing tape data.
The bumper positioning then leads the ball to a new position in a way that corresponds to reading and writing the tape on the Turing machine. Certain curved bumpers shift the tape one-way, making data stored in distant decimals more important in a way reminiscent of a chaotic system, while curved bumpers are reversed. The exit of the ball from the bottom of the ball marks the edge of the calculation and, as a result, indicates the final position.
Moore had computer flexibility in setting up a pinball machine. One bumper placement could calculate the first 1000 digits of the Pi, while another could calculate the best next move in a game of chess. But in doing so, he also injected an attribute that we might not normally be associated with a computer: unpredictability.
Some algorithms stop and output the results. But others run forever. (Consider the program responsible for printing the last digits of the PI.) Is there a step where Turing can look up the program and decide whether to stop it? This question has become known as a halt issue.
Turing showed that no such procedure existed, taking into account what it means. If one machine can predict the behavior of another, it can easily change the first machine (the one that predicts behavior) and run forever when the other machine stops. And the opposite: stops when other machines run forever. Next, here is the part that bends the mind. I imagine feeding this tweaked predictor explanation to itself. If the machine goes down, it will run forever too. And if it runs forever, it will stop too. As neither option may be, Turing concluded, the predictor itself should not exist.
(His findings were closely related to the groundbreaking results of 1931 when logician Kurt Godel developed a similar method. Supply to the self-reference paradox In a strict mathematical framework. Gödel has proven that there are mathematical statements that cannot establish the truth.
In short, Turing has proven that it is impossible to solve the stopping problem. The only common way to know if the algorithm has stopped is to do it as much as possible. If it stops, you have your answer. But if not, I don’t know if it really runs forever or if it stopped if I waited a little longer.
“We know there are these types of initial states.
Since Moore was designing his box To mimic the Turing machine, it could also work in an unpredictable way. The exit of the ball marks the end of the calculation, so the question of whether a particular arrangement of the bumper traps the ball or leads it to the exit must also be unmanageable. “Really, questions about the long-term dynamics of these more elaborate maps are promising,” Moore said.
