For example, finding out whether a time-bounded Turing machine outputs 0 all inputs is still undecidable. (coRE-complete, in fact. That's better in some sense than the Pi_2-completeness of the same problem for Turing machines without the time bound, but since neither admits a solution in the form of a computer program, a lot of people would consider the difference an academic curiosity only.)
No it's not; call the time bound N; the TM can only inspect N symbols, so there are 2^N possible inputs; a exhaustive check takes O(N*2^N) operations.
Call the time bound c|x|^k, where x is the input and c and k are constants.
I see what point you're making, and I agree that you get yourself a nice, merely coNP-complete problem if you're willing to cut off the space of inputs at a point by enforcing a constant time bound or whatever. More than anything, this kind of argument makes me shudder at just how hard the very worst stuff in NP must be.
But yes, it should reduce to NP territory. If you actually had a nondeterministic CPU, then for most use cases you could fully evaluate these machines in seconds.
This is backwards; I meant that if you have any time bound whatsoever, the number of inputs you can inspect is limited to (a architecture-specific finite multiple of) the number of operations you can execute (because inspecting a input is such a operation).
That is literally the problem statement you gave. I was pointing out that it is very much not undecidable.
It might be (almost certainly is, in this case) computationally intractable both in the general case and in practice, but it's still decidable.
Is that a problem? I mean, if you're installing a timeout then haven't you already decided that the answer is "no"?
I singled out "outputting 0" on every input because it's among the simplest possible specs you could try try to check with the kind of "very powerful semantic analysis" tool the top-level post was hoping for. In practice you probably want to check a more complex spec -- you want your code to solve a specific problem on whatever input it receives -- but if the "always output 0" spec is impossible to automatically check, then what hope do more relevant specs have?
So you can have a practical, useful, non-Turing-complete language by forcing every function to have a limit on the number of steps it can take. It can be implicit if you want. In practice, most embedded scripting systems use this route, which is why you don't really need a separate language just for non-Turing-completeness.