Xrfgtjtk

Let $a_0, a_1 in mathbb{N} setminus {0}$ and $L={w in {0,1}^*|a_0·#_0(w)+a_1·#_1(w)- a_1a_0=0}$ . Let's assume problem $P$ that, language of Turing machine accepts at least one word from language $L$. Using membership problem, is $P$ undecidable?

Membership problem of $w in L land L in unrestricted$, is undecidable but is semi-decidable. Prove of semi-decitability is quite easy by simulating of Turing machine $T$, where $L(T)=L$. But how can we show that $P$ is undecidable?

To show that a language over Turing Machine encodings is undecidable, we can use Rice's Theorem, which states that any non-trivial semantic property of Turing Machines is undecidable. Here, our property is "Turing Machine $M$ accepts at least one word from language $L$." We first note that, for every $a_0,a_1 in mathbb{N} setminus {0}$, $L$ is nonempty and does not contain every string. For the former, $1^{a_0} in L$ for any $a_0,a_1$; for the latter, observe that $varepsilon notin L$ for any such $L$.

From here, we just need to show that this property is nontrivial. This requires showing that there is a Turing machine that accepts a language satisfying this property, and one that accepts a language which does not satisfy this property. Turing machines accepting the languages $Sigma^*$ and $emptyset$ suffice, respectively.