Take a semi-decidable problem and an algorithm that finds the positive answer in finite time. The run-time of the algorithm, restricted to inputs with a positive answer, cannot be bounded by a computable function. (Otherwise we’d know how long to wait for a positive answer. If the algorithm runs longer than that we know that the answer is no and the problem would be solvable.)
My question is now: Can such an algorithm still have a, say, a run-time bound linear (polynomial, constant,...) in the input size, but with an uncomputable constant? Or would that still allow me to decide the problem? Are there example?