First order modal temporal logics with generalized intervals

Following the work of G. Ligozat (1991) who described the interval algebras A(S) whose elements are relations on generalized intervals, the author proposes a class of first order modal temporal logics where the possible worlds are points, standard intervals or unions of convex intervals and where the accessibility relations are elements of A(S). These logics have standard syntax and semantics with the unique exception that, whereas predicates are generally interpreted on intervals, the terms are always interpreted on points which are considered as elements of the intervals. He also addresses the problem of automated reasoning in such logics and defines for that sake a satisfiability and validity preserving translation function into a standard two-sorted first order logic