How to describe Deterministic Transitive Closure in FOL?
Hence, taking into account that FOL captures LOGSPACE, it should be possible to:

Given a functional relation R(X,Y) (e.g. &quot;motherOf(X,Y)&quot;)
It should be possible to define, in FOL, a predicate T(X,Y) that captures its transitive closure (e.g. &quot;femaleAncestor(X, Y)&quot;)


Actually this is not correct: first-order logic does not capture logspace. As you said earlier in your question, FOL plus deterministic transitive closure captures logspace, that is
$$
\text{FO(DTC)} = \text{L}
$$
but
$$
\text{FO} \ne \text{FO(DTC)},
$$
i.e. deterministic transitive closure is not definable in first-order logic.
More generally, FOL is quite limited. FOL captures a class called uniform $\text{AC0}$ (see the complexity zoo and Immerman's diagram); it is the class of queries computable in &quot;constant parallel time&quot;. And transitive closure, even deterministic transitive closure, cannot be done in $\text{AC0}$ (with a circuit of constant depth). Even something very simple, calculating the parity of the number of input bits, can't be done in $\text{AC0}$.

How to describe Deterministic Transitive Closure in FOL?

To reiterate: it is impossible.