On Expressive Powers of Timed Logics: Comparing Boundedness, Non-punctuality, and Deterministic Freezing

Timed temporal logics exhibit a bewildering diversity of operators and the resulting decidability and expressiveness properties also vary considerably. We study the expressive powers of timed logics TPTL[U,S] and MTL[UI ,SI ] as well as their several fragments. Extending the LTL EF games of Etessami and Wilke, we define MT L Ehrenfeucht-Fraisse games on a pair of timed words. Using the associated EF theorem we show that, expressively, the timed logics BoundedMTL[UI ,SI ], MTL[FI ,PI ] and MITL[UI ,SI ] (respectively incorporating the restrictions of boundedness, unary modalities and non-punctuality), are all pairwise incomparable. Going to more expressive logics, we show that MTL[UI ,SI ] is expressively incomparable with the unary freeze logic TPTL[F,P] extending the result of Bouyer et al. Finally, we consider the deterministic freeze logic TTL[Xθ,Yθ], which is expressively equivalent to partially ordered 2-way deterministic timed automata (po2DTA). As our second main result, we show by explicit reductions that TTL[Xθ,Yθ] lies strictly between the unary, nonpunctual logic MITL[FI ,PI ] and its bounded fragment BoundedMITL[FI ,PI ]. This also shows that deterministic punctuality is expressible by the non-punctual MITL[FI ,PI ].