A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of A to the category of copies of B.

• See this paper of Harrison-Trainor/Melnikov/Miller/Montalban for a precise definition. Roughly speaking, let $\mathcal{A}$ and $\mathcal{B}$ be respectively the categories whose objects are copies of $A$ and $B$ and whose morphisms are isomorphisms (in the usual sense); then a computable functor reducing $\mathcal{A}$ to $\mathcal{B}$ is a pair of Turing functionals, one sending objects in $\mathcal{A}$ to objects in $\mathcal{B}$ and the other sending morphisms in $\mathcal{A}$ to morphisms in $\mathcal{B}$. The claim that this yields a strict strengthening of strong (= Medvedev) reducibility is made without proof on page $5$ of the linked paper.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of B to the category of copies of A.

• See this paper of Harrison-Trainor/Melnikov/Miller/Montalban for a precise definition (and note that this isn't the starting definition; that definition is given on page $3$, and its characterization by functors is Theorem 5). Roughly speaking, let $\mathcal{A}$ and $\mathcal{B}$ be respectively the categories whose objects are copies of $A$ and $B$ and whose morphisms are isomorphisms (in the usual sense); then a computable functor reducing $\mathcal{A}$ to $\mathcal{B}$ is a functor from $\mathcal{B}$ to $\mathcal{A}$ given by a pair of Turing functionals, one sending objects in $\mathcal{B}$ to objects in $\mathcal{A}$ and the other sending morphisms in $\mathcal{B}$ to morphisms in $\mathcal{A}$. The claim that this yields a strict strengthening of strong (= Medvedev) reducibility is made without proof on page $5$ of the linked paper.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of A to the category of copies of B.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?

A countable structure A is strongly reducible to a structure B if there is a uniform turing functional which, given a copy of the atomic diagram of B, computes a copy of the atomic diagram of A.

A is said to be effectively interpretable in B if A is strongly reducible to B via a computable functor from the category of copies of A to the category of copies of B.

• See this paper of Harrison-Trainor/Melnikov/Miller/Montalban for a precise definition. Roughly speaking, let $\mathcal{A}$ and $\mathcal{B}$ be respectively the categories whose objects are copies of $A$ and $B$ and whose morphisms are isomorphisms (in the usual sense); then a computable functor reducing $\mathcal{A}$ to $\mathcal{B}$ is a pair of Turing functionals, one sending objects in $\mathcal{A}$ to objects in $\mathcal{B}$ and the other sending morphisms in $\mathcal{A}$ to morphisms in $\mathcal{B}$. The claim that this yields a strict strengthening of strong (= Medvedev) reducibility is made without proof on page $5$ of the linked paper.

What is an example of two countable structures A,B such that A is strongly reducible to B but A is not effectively interpretable in B?