A test for completeness with respect to implicit reducibility in the chain super-intutionistic logics

We examine chain logics $C_2,C_3,\dots$, which are intermediary between classical and intuitionistic logics. They are also the logics of pseudo-Boolean algebras of type $\langle E_m,\&,\vee,\supset,\neg\rangle$, where $E_m$ is the chain $0<\tau_ 1<\tau_2<\dots<\tau_{m-2}<1$ ($m=2,3,\dots$). The formula $F$ is called to be implicitly expressible in logic $L$ by the system $\Sigma$ of formulas if the relation
$$ L\vdash(F\sim q)\sim((G_1\sim H_1)\&\dots\&(C_k\sim H_k)) $$
is true, where $q$ do not appear in $F$, and formula$G_i$ and $H_i$, for $i=1,\dots, k$, are explicitly expressible in $L$ via $\Sigma$ The formula $F$ is said to be implicitly reducible in logic $L$ to formulas of $\Sigma$ if there exists a finite sequence of formulas $G_1,G_2,\dots,G_l$ where $G_l$ coincides with $F$ and for $j = 1,\dots,l$ the formula $G_j$ is implicitly expressible in $L$ by $\Sigma\cup\{G_1,\dots,G_{j-1}\}$. The system $\Sigma$ is called complete relative to implicit reducibility in logic $L$ if any formula is implicitly reducible in $L$ to $\Sigma$.
The paper contains the criterion for recognition of completeness with respect to implicit reducibility in the logic $C_m$, for any $m=2,3,\dots$ . The criterion is based on 13 closed pre-complete classes of formulas.