如何显示{w | M_w接受0x且接受1x}语言不是递归语言？

Question

我需要证明L = {w | M_w如果接受0x，则接受1x}不是递归的

我认为这应该是赖斯定理的简单应用，该定理指出对于递归可枚举语言的任何非平凡性质P，{w | M_w是图灵机，并且L（M_w）在P}中不是递归的。我对此不太确定，因为我不太确定什么是财产。而且，我也不知道该如何证明它是递归可枚举语言的特质。

Answer 1

I am a bit unsure of this because I am not too sure what constitutes as a property;

For the purposes of Rice's theorem, a property is any logical statement (proposition, predicate, etc.) about some language which is either true or false for that language. A property is nontrivial if the statement is neither tautological nor contradictory: that is, it is nontrivial if it is true of some languages but not others. The property "|L| < 0" is contradictory and is therefore a trivial property; "|L| >= 0" is tautological and therefore a trivial property; "|L| = 0" is a nontrivial property.

moreover, I do not know how to show that it is specifically a property of recursively enumerable languages, either.

Recursively enumerable languages are also "just languages" and their properties are also the properties of "just languages". They are sets of strings and their properties all involve what strings are in them. Do not get hung up on whether a particular property of languages is a property of recursively-enumerable languages: it is. Even if it were not, a proof using Rice's Theorem isn't affected: if the property is known not to apply to recursively-enumerable languages and to be nontrivial, then it is known the language is not recursive anyway - Rice's Theorem isn't even necessary.