Can Circuit Value Problem or HORN-SAT be reduced to PATH problem?
Hint: Reduce from the monotone circuit value problem. Let $X$ be the set of all inputs and internal gates in the circuit. Choose $T$ and $R$ so that an element is admissible if the corresponding input or gate is true (each AND gate will require one triple in $R$, each OR gate will require two). Choose $S$ to be the indicator of the output gate to complete the proof.