Which of the following is recursive

Consider below two Languages

$$L1=\{|$$M takes atleast 2000 steps on some input $$\}$$

$$L2=\{|$$M takes atleast 2000 steps on all inputs$$\}$$

Where for each Turing machine M, $$$$ denotes specific encoding of M.

Here I think both are recursive, because let's say I bound the number of steps the Turing machine runs to 2000, then Turing machine will only be able to process inputs of length atmost 2000.

Now, there are finite number of inputs of length 2000, I can check whether for all such inputs, my Turing machine is making 2000 steps or not.This answer L2.

And when L2 is answered, I have an answer to L1, making L1 and L2 both recursive.

Am I correct in my reasoning?

• A Turing machine running in 2000 steps could accept, for example, all strings starting with 10 - of any length. Commented Dec 28, 2018 at 11:48

If $$x$$ and $$y$$ are words which are identical on their first 2000 symbols, then they are both accepted/rejected/neither in 2000 steps.