In this paper, we study the finite satisfiability problem for the logic BE under the homogeneity assumption. BE is the cornerstone of Halpern and Shoham's interval temporal logic, and features modal operators corresponding to the prefix (a.k.a. "Begins") and suffix (a.k.a. "Ends") relations on intervals. In terms of complexity, BE lies in between the "Chop"logic C, whose satisfiability problem is known to be non-elementary, and the PSpace-complete interval logic D of the sub-interval (a.k.a. "During") relation. BE was shown to be ExpSpace-hard, and the only known satisfiability procedure is primitive recursive, but not elementary. Our contribution consists of tightening the complexity bounds of the satisfiability problem for BE, by proving it to be ExpSpace-complete. We do so by devising an equi-satisfiable normal form with boundedly many nested modalities. The normalization technique resembles Scott's quantifier elimination, but it turns out to be much more involved due to the limitations enforced by the homogeneity assumption.

The Logic of Prefixes and Suffixes is Elementary under Homogeneity*

Della Monica D.;Montanari A.;Puppis G.
;
2023-01-01

Abstract

In this paper, we study the finite satisfiability problem for the logic BE under the homogeneity assumption. BE is the cornerstone of Halpern and Shoham's interval temporal logic, and features modal operators corresponding to the prefix (a.k.a. "Begins") and suffix (a.k.a. "Ends") relations on intervals. In terms of complexity, BE lies in between the "Chop"logic C, whose satisfiability problem is known to be non-elementary, and the PSpace-complete interval logic D of the sub-interval (a.k.a. "During") relation. BE was shown to be ExpSpace-hard, and the only known satisfiability procedure is primitive recursive, but not elementary. Our contribution consists of tightening the complexity bounds of the satisfiability problem for BE, by proving it to be ExpSpace-complete. We do so by devising an equi-satisfiable normal form with boundedly many nested modalities. The normalization technique resembles Scott's quantifier elimination, but it turns out to be much more involved due to the limitations enforced by the homogeneity assumption.
2023
979-8-3503-3587-3
File in questo prodotto:
File Dimensione Formato  
LICS 2023.pdf

non disponibili

Tipologia: Documento in Pre-print
Licenza: Non pubblico
Dimensione 504.59 kB
Formato Adobe PDF
504.59 kB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/1260565
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact