Question at position 21 True or False: There exists an algorithm that takes as input a program P and any legal input to P, and always correctly determines whether P will halt or not.TrueFalse题目解析

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This question asks whether there exists an algorithm that, given any program P and any legal input to P, can always determine whether P halts. The core concept here is the Halting Problem from computability theory.
Option 1: True. Saying that there exists such an algorithm would imply we can decide for every possible program and input whether it halts. However, Alan Turing proved that the Halting Problem is undecidable; no general algorithm can solve this for all possible programs. This option contradicts that foundational result and is therefore incorrect.
Option 2: False. This statement aligns with the Halting Problem’s undecidability. The standard argument is a diagonalization-style proof: assuming a halting decider exists leads to a contradiction when analyzing its own behavior on its own code, which shows no universal halting algorithm can exist. This option accurately captures the theoretical limitation.
Additional nuance: The claim could tempt one to think about restricted cases—such as halting on specific languages, restricted input domains, or restricted program families—where halting is decidable. Yet the statement as written refers to “any program P and any legal input,” which encompasses the general, unrestricted case; that broad scope is precisely where undecidability arises.
In short, the assertion that a universal halting-detector exists for all programs and inputs would conflict with the established Halting Problem result, and that mismatch is why this option is considered incorrect. Understanding the distinction between general undecidability and decidability in restricted contexts helps clarify why the alternate view is invalid.

Question at position 21 True or False: There exists an algorithm that takes as input a program P and any legal input to P, and always correctly determines whether P will halt or not.TrueFalse题目解析

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