Pure future local temporal logics are expressively complete for Mazurkiewicz traces

The paper settles a long standing problem for Mazurkiewicz traces: the pure future local temporal logic defined with the basic modalities exists-next and until is expressively complete. This means every first-order definable language of Mazurkiewicz traces can be defined in a pure future local temporal logic. The analogous result with a global interpretation has been known, but the treatment of a local interpretation turned out to be much more involved. Local logics are interesting because both the satisfiability problem and the model checking problem are solvable in Pspace for these logics whereas they are non-elementary for global logics. Both, the (previously known) global and the (new) local results generalize Kamp’s Theorem for words, because for sequences local and global viewpoints coincide.