Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the recent years. Even though most interval logics turn out to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing severe semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the negation depth of modalities. The decidability of the satisfiability problem for the resulting fragment, called CDTBS, over the class of all linear orders, is proved by embedding it into a well-known decidable quantifier prefix class of first-order logic, namely, Bernays-Schönfinkel class. In addition, we show that CDTBS is in fact NP-complete (Bernays-Schönfinkel class is NEXPTIME-complete), and we prove its expressive completeness with respect to a suitable fragment of Bernays-Schönfinkel class. Finally, we show that any increase in the negation depth of CDTBS modalities immediately yields undecidability. © 2013 Springer Science+Business Media Dordrecht.
The Light Side of Interval Temporal Logic: the Bernays-Schonfinkel Fragment of CDT
SCIAVICCO, Guido
2014
Abstract
Decidability and complexity of the satisfiability problem for the logics of time intervals have been extensively studied in the recent years. Even though most interval logics turn out to be undecidable, meaningful exceptions exist, such as the logics of temporal neighborhood and (some of) the logics of the subinterval relation. In this paper, we explore a different path to decidability: instead of restricting the set of modalities or imposing severe semantic restrictions, we take the most expressive interval temporal logic studied so far, namely, Venema's CDT, and we suitably limit the negation depth of modalities. The decidability of the satisfiability problem for the resulting fragment, called CDTBS, over the class of all linear orders, is proved by embedding it into a well-known decidable quantifier prefix class of first-order logic, namely, Bernays-Schönfinkel class. In addition, we show that CDTBS is in fact NP-complete (Bernays-Schönfinkel class is NEXPTIME-complete), and we prove its expressive completeness with respect to a suitable fragment of Bernays-Schönfinkel class. Finally, we show that any increase in the negation depth of CDTBS modalities immediately yields undecidability. © 2013 Springer Science+Business Media Dordrecht.I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
