Unification with Simple Variable Restrictions and Admissibility of ${\mathrm{\Pi }}_2$-rules

We develop a method to recognize admissibility of ${\mathrm{\Pi }}_2$-rules, relating this problem to a specific instance of the unification problem with linear constants restriction, called here “unification with simple variable restriction”. It is shown that for logical systems enjoying an appropriate algebraic semantics and a finite approximation of left uniform interpolation, this unification with simple variable restriction can be reduced to standard unification. As a corollary, we obtain the decidability of admissibility of ${\mathrm{\Pi }}_2$-rules for many logical systems.