Undecidability of QLTL and QCTL with two variables and one monadic predicate letter

We study the algorithmic properties of quantified linear-time temporal logic QLTL in languages with restrictions on number individual variables as well and arity predicate letters. prove that satisfiability problem for two one monadic letter Σ 11 -hard. Thus, is Π -hard, so not recursively enumerable, such languages. The resultholds both increasing domain constant semantics obtained by reduction from a -hard N×N recurrent tiling problem. It follows proof similar results hold branching-time QCTL, hence alternating-time QATL. result presented this paper strengthens I. Hodkinson, F. Wolter, M. Zakharyaschev, who have shown variablesand an unlimited supply