Decidable Program Equivalence
Given a Turing machine $M$, we can construct another Turing machine $T$, which on input $1^n$ simulates $M$ for $\log n$ steps, and returns whether $M$ halted within these steps. The new Turing machine runs in linear time, and is equivalent to the constant time Turing machine which always returns "No" iff $M$ doesn't halt. Therefore program equivalence is undecidable even for linear time Turing machines.