Data-Aware Hybrid Tableaux

Labelled tableaux have been a traditional approach to define satisfiability checking procedures for Modal Logics. In many cases, they can also be used to obtained tight complexity bounds and lead to efficient implementations of reasoning tools. More recently, it has been shown that the expressive power provided by the operators characterizing Hybrid Logics (nominals and satisfiability modalities) can be used to internalize labels, leading to well-behaved inference procedures for fairly expressive logics. The resulting procedures are attractive because they do not use external mechanisms outside the language of the logic at hand, and have good logical and computational properties. Lately, many proof systems based on Hybrid Logic have been investigated, in particular related to Modal Logics featuring some form of data comparison. In this paper, we introduce an internalized tableaux calculus for XPath, arguably one of the most prominent approaches for querying semistructured data. More precisely, we define data-aware tableaux for XPath featuring data comparison operators and enriched with nominals and the satisfiability modalities from Hybrid Logic. We prove that the calculus is sound, complete and terminating. Moreover, we show that tableaux can be constructed in polynomial space, without compromising completeness, establishing in this way that the satisfiability problem for the considered logic is PSPACE-complete. Finally, we explore different extensions of the calculus, in particular to handle data trees and other frame classes.