Completeness and Decidability in Sequence Logic

Sequence logic is a parameterized logic where the formulas are sequences of formulas of some arbitrary underlying logic. The sequence formulas are interpreted in certain linearly ordered sets of models of the underlying logic. This interpretation induces an entailment relation between sequence formulas which strongly depends on which orderings one wishes to consider. Some important classes are: all linear orderings, all dense linear orderings and all (or some specific) wellorderings. For all these classes one can ask for a sound and complete proof system for the entailment relation, as well as for its decidability. For the class of dense linear orderings and all linear orderings we give sound and complete proof systems which also yield decidability (assuming that the underlying logic is sound, complete and decidable). The entailment relation can be expressed in the first-order theory of the ordering. Consequently, if the latter theory is decidable, then so is the corresponding entailment relation. Even if the first-order theory of every ordinal is known to be decidable, we did not find a natural proof system. We formulate some open problems on the entailment relation in the case of well-orderings.