We prove that TAUT has a $p$-optimal proof system if and only if a logic related to least fixed-point logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no {\em effective} $p$-optimal proof system if $\textup{NTIME}(h^{O(1)}) \not\subseteq \textup{DTIME}(h^{O(\log h)})$ for every time constructible and increasing function $h$.

Minor corrections.

We prove that TAUT has a $p$-optimal proof system if and only if $L_\le$, a logic introduced in [Gurevich, 88], is a P-bounded logic for P. Furthermore, using the method developed in [Chen and Flum, 10], we show that TAUT has no \emph{effective} $p$-optimal proof system under some reasonable complexity-theoretic assumption.