If there exists a Cook reduction of a decision problem $\mathcal{P}_1$ into another decision problem $\mathcal{P}_2$ and also a Cook reduction of $\mathcal{P}_2$ into $\mathcal{P}_1$, then is there also a Karp reduction (a polynomial transformation) between $\mathcal{P}_1$ and $\mathcal{P}_2$ (in both directions)?These are the definitions I use:Cook reduction  $\mathcal{P}_1$ polynomially reduces to $\mathcal{P}_2$ if there is a polynomial-time oracle algorithm for $\mathcal{P}_1$ using an oracle for $\mathcal{P}_2$.Karp reduction  $\mathcal{P}_1=(X_1,Y_1)$ polynomially transforms to $\mathcal{P}_2=(X_2,Y_2)$ if there is a function $f:X_1\rightarrow X_2$ computable in polynomial time such that for all $x\in X_1$, $x\in Y_1$ if and only if $f(x)\in X_2$.
If you have a Cook reduction in both directions, do you also have a Karp reduction?
complexity theory;reductions