there are decision problems that are computable and are not in P

Question

I got this doubt, it is clear that there are non-computable problems that are not in P, for example Busy Beaver, but is this another true?

Answer 1

Yes -- the time hierarchy theorem guarantees that there are computable problems outside $O(f(n))$ for practically every function $f$ you can "reasonably" compute -- including any elementary function.

So, for example, there's a problem that cannot be decided in $O(2^n)$ time, and in particular not in any polynomially bounded time.

---

The problems you get out of the hierarchy theorem tend to be quite artificial, but there are also more "natural" problems that are known to be non-polynomial.

One example is type-checking implicitly-typed functional languages with a Hindley-Milner polymorphic type systems, such as ML or Haskell. The usual type-checking algorithm works very well for real programs because they tend not to have many levels of potentially polymorphic function definitions nested inside each other, but one can construct families of programs with ever more deeply nested definitions that prove the type-checking problem is in fact EXPTIME-complete.