The Decidability of Simultaneous Rigid E-Unification with One Variable

We show that simultaneous rigid E-unification, or SREU for short, is decidable and in fact EXPTIME-complete in the case of one variable. This result implies that the A*EA* fragment of intuitionistic logic with equality is decidable. Together with a previous result regarding the undecidability of the 99-fragment, we obtain a complete classification of decidability of the prenex fragment of intuitionistic logic with equality, in terms of the quantifier prefix. It is also proved that SREU with one variable and a constant bound on the number of rigid equations is Pcomplete.

The title problem is proved decidable and in fact EXPTime complete. Furthermore, the problem becomes PTime complete if the number of equations is bounded by any (positive) constant. It follows that the A*EA* fragment of intuitionistic logic with equality is decidable, which contrasts with the undecidability of the EE fragment [126]. Notice that simultaneous rigid E-unification with two variables and only three rigid equations is undecidable [126].