What is the halting problem? What does it have to do with self referential paradoxes?
How do I make sense of this?
"So suppose I have a black box which can decide the halting problem, that is I can give it a program P and the box tells me if P will ever stop.
Using this I can make a program P' that works as follows:
Given a program P ask the box if P will stop.
If P will halt run forever.
If P will run forever, halt.
Last step, give P' as input to itself, if P' will run forever it must halt. "
I don't understand why there is a P' here? And why we have regular P and P'. I'm so confused.
The halting problem is a fundamental concept in computer science and mathematics. It refers to the task of determining whether a particular program will ever halt (i.e., stop running) or if it will run forever.
Self-referential paradoxes play a role in the halting problem because they introduce a form of contradiction or paradox. In the context of programming, self-reference refers to a program's ability to refer to itself or manipulate its own code. Paradoxes arise when a program references itself in a way that leads to contradictions or logical inconsistencies.
To understand the provided explanation, let's break it down step-by-step:
1. Suppose we have a black box that can solve the halting problem. This means it can take any program P as input and determine if P will halt or run forever.
2. Now, we want to construct a program P' that uses our black box. P' takes a program P as its input.
3. P' asks the black box if program P will halt. If the black box says P will halt, then P' runs forever (i.e., it goes into an infinite loop).
4. On the other hand, if the black box says P will run forever, then P' halts (i.e., it stops running).
5. Finally, P' takes itself as input and asks the black box if P' will run forever. The intent here is to create a paradox. If the black box says P' will run forever, then it contradicts the earlier behavior where P' would halt in that case. If the black box says P' will halt, it contradicts the earlier behavior where P' would run forever.
The purpose of introducing P' is to create a paradoxical situation by making P' ask the black box about its own behavior. This is done to demonstrate that there are certain problems (like the halting problem) that are undecidable, meaning there is no algorithm that can always provide a correct answer for all possible inputs.
I hope this step-by-step explanation helps clarify the concept. Let me know if you have any further questions!
The halting problem is a fundamental question in computer science that asks whether it is possible to write a program that can determine whether any other program will halt or run forever. It states that there is no general algorithm that can solve this problem for all possible programs.
Now let's break down the given example to help you make sense of it. The example adopts a self-referential paradox to argue against the existence of a program that can decide the halting problem. Here is how it works:
1. Suppose we have a hypothetical black box that can decide the halting problem. This means that if we give it a program P, the box will always provide the correct answer, either "P will halt" or "P will run forever."
2. Based on this assumption, we construct a new program called P' as follows:
- Given a program P, we ask the black box if P will halt.
- If the black box says "P will halt," then P' enters an infinite loop and continues running forever.
- If the black box says "P will run forever," then P' halts and stops running.
3. Finally, we give P' as input to itself. If P' will run forever, according to our construction, it should halt. But if it will halt, according to our construction, it should run forever. This creates a paradox or contradiction.
The purpose of introducing P' is to demonstrate the paradoxical situation that arises from assuming the existence of the black box. By constructing P' in this way, we reach a contradiction where P' behaves in a way that contradicts both of its possible outcomes (halt and run forever). This implies that the black box, which claims to solve the halting problem, cannot exist.
The introduction of regular P and P' helps in distinguishing between the original program and the new program constructed for the self-referential paradox. While P represents any given program for which we want to determine if it halts or not, P' is the self-referential program that exhibits contradictory behavior based on the output of the black box.
Understanding these concepts can be challenging, so don't hesitate to ask more questions if you still have doubts!