GO Classes DPP | TOC|DECIDABILITY

Answer 1

Understanding $\textbf{L}$:

We have to decide if a given Turing machine $\mathrm{M}$ halts on at least one input $w$ of length $\leq 100$.

Finite number of inputs:

The number of strings $w$ with $|w| \leq 100$ is finite. However, the halting problem for a fixed $w$ is undecidable.

Recognizability of $\textbf{L}$:

We can recognize $\mathrm{L}$ as follows:

• Run $\mathrm{M}$ on all $w$ with $|w| \leq 100$ in parallel (dovetailing).
   
• If any of these computations halts, we accept $\mathrm{\langle M\rangle}$.
  
  This procedure will halt if $\mathrm{M} \in \mathrm{L}$. If $\mathrm{M} \notin \mathrm{L}$, it runs forever.
  
  So $\mathrm{L}$ is recognizable (recursively enumerable).
   

Undecidability of $\textbf{L}$:

If $\text{L}$ were decidable, we could decide the halting problem for a fixed pair $(\mathrm{M}, w_0)$:

• Construct $\mathrm{M}^{\prime}$ that ignores its input and runs $\mathrm{M}$ on $w_0$.
   
• Then $\mathrm{M}^{\prime} \in \mathrm{L}$ iff $\mathrm{M}$ halts on $w_0$.
  
  This would solve the halting problem, which is impossible.
  
  So $\text{L}$ is undecidable.
   

Conclusion:

$\text{L}$ is recognizable but not decidable → Option A is correct.