Tableau reasoning for description logics and its extension to probabilities

Abstract

The increasing popularity of the Semantic Web drove to a widespread adoption of Description Logics (DLs) for modeling real world domains. To help the diffusion of DLs, a large number of reasoning algorithms have been developed. Usually these algorithms are implemented in procedural languages such as Java or C++. Most of the reasoners exploit the tableau algorithm which features non-determinism, that is not easily handled by those languages. Prolog directly manages non-determinism, thus is a good candidate for dealing with the tableau’s non-deterministic expansion rules. We present TRILL, for “Tableau Reasoner for descrIption Logics in proLog”, that implements a tableau algorithm and is able to return explanations for queries and their corresponding probability, and TRILLP, for “TRILL powered by Pinpointing formulas”, which is able to compute a Boolean formula representing the set of explanations for a query. Reasoning on real world domains also requires the capability of managing probabilistic and uncertain information. We show how TRILL and TRILLP can be used to compute the probability of queries to knowledge bases following DISPONTE semantics. Experiments comparing these with other systems show the feasibility of the approach

Probabilistic DL Reasoning with Pinpointing Formulas: A Prolog-based Approach

“…This special case greatly simplifies reasoning while still achieving significant expressiveness. Note that if we need the added expressiveness of BEL, as shown in (Zese et al 2018), the Bayesian network can be translated into an equivalent one where all the random variables are mutually unconditionally independent, so that the KB can be represented with DISPONTE.…”

“…TRILL (Zese et al 2018;Zese 2017) computes the probability of a query with respect to KBs that follow DISPONTE by first computing all the explanations for the query and then building a BDD that represents them. An explanation is a subset of axioms κ of a KB K such that κ |= Q.…”

“…Reasoners written in Prolog can exploit Prolog's backtracking facilities for performing the search, as has been observed in various works (Beckert and Posegga 1995;Hustadt et al 2008;Lukácsy and Szeredi 2009;Ricca et al 2009;Gavanelli et al 2015). For this reason, in Zese et al (2018) and Zese (2017) we proposed the system TRILL, a tableau reasoner implemented in Prolog. Prolog's search strategy is exploited for taking into account the non-determinism of the tableau rules.…”

“…They encode the results of the inference process in a BDD from which the probability can be computed in a time linear in the size of the diagram. We also developed TRILL P (Zese et al 2018;Zese 2017), which builds a pinpointing formula able to compactly represent the set of explanations. This formula is used to build the corresponding BDD and compute the query's probability.…”

“…This formula is used to build the corresponding BDD and compute the query's probability. In Riguzzi et al (2015), Zese et al (2018), and Zese (2017), we have extensively tested BUNDLE, TRILL, and TRILL P , showing that they can achieve significant results in terms of scalability and speed.…”