# TRILL Framework

TRILL framework is composed of three systems able to answer queries to Description Logics knowledge bases using a tableau algorithm. They return the probability of the query, which represents the degree of belief of its truth.

Prolog non-determinism is used for easily handling non-deterministic expansion rules that produce more than one tableau.

TRILL framework contains:

**TRILL**(*Tableau Reasoner for descrIption Logics in Prolog*) that implements the tableau algorithm for SHOIQ description logics, it returns the set of explanations for a query or its probability;**TRILL**(^{P}*Tableau Reasoner for*descrIption*Logics in Prolog powered by Pinpointing formula*) that implements the tableau algorithm for SHI description logics, it returns a*pinpointing formula*using the techniques proposed by Baader and Penaloza [Baader and Penaloza, 2010] representing the set of explanations for a query or its probability;**TORNADO**(*Trill powered by pinpOinting foRmulas and biNAry DecisiOn diagrams*) that implements the tableau algorithm for SHI description logics and returns a binary decision diagram modeling the pinpointing formula or the corresponding probability.

For TRILL and TRILL^{P} two versions are available, one for Yap Prolog and one for SWI-Prolog. They differ in the features offered. Differently, TORNADO is only available for SWI-Prolog.

You can try online the three systems at TRILLonSWISH, where the SWI-Prolog version is used.

The Yap version of TRILL is available here. **trill.pl** is the version that does not compute the probability of the query, it returns only the set of explanations. **trillProb.pl** is the probabilistic version, it returns also the probability of queries. At the link above TRILL^{P} is also available in its Yap version in the archive **TRILLP.zip**.

The SWI-Prolog version is distributed as a pack.

You can find the manuals at:

To signal a bug or ask for help you can write in the TRILL group, the general chat of TRILLonSWISH or open an issue on the GitHub page.

*References*:

[Baader and Penaloza, 2010] Franz Baader and Rafael Penaloza. “Axiom Pinpointing in General Tableaux”. Journal of Logic and Computation, 20(1):5–34, 2010.

**Bibliography:**

Riccardo Zese, Elena Bellodi, Giuseppe Cota, Fabrizio Riguzzi, and Evelina Lamma. Probabilistic DL reasoning with pinpointing formulas: A Prolog-based approach. *Theory and Practice of Logic Programming*, 19(3):449–476, 2019. [ bib |

DOI |

.pdf ]

Riccardo Zese. *Probabilistic Semantic Web*, volume 28 of *Studies on the Semantic Web*. IOS Press, 2017. [ bib | DOI | http ]

Elena Bellodi, Evelina Lamma, Fabrizio Riguzzi, Riccardo Zese, and Giuseppe Cota. A web system for reasoning with probabilistic OWL. *Software: Practice and Experience*, 47(1):125-142, © Wiley, January 2017. [ bib | DOI | .pdf ]

Riccardo Zese, Elena Bellodi, Fabrizio Riguzzi, Giuseppe Cota, and Evelina Lamma. Tableau reasoning for description logics and its extension to probabilities. *Annals of Mathematics and Artificial Intelligence*, 82(1):101–130, © Springer, March 2018. [ bib |

DOI |

.pdf |

http ]

Riccardo Zese, Elena Bellodi, Evelina Lamma, and Fabrizio Riguzzi. Logic programming techniques for reasoning with probabilistic ontologies. In Odile Papini, Salem Benferhat, Laurent Garcia, and Marie-Laure Mugnier, editors, *International Workshop on Ontologies and Logic Programming for Query Answering*, © by the authors, 2015.

[ bib | .pdf | http ]

Riccardo Zese, Elena Bellodi, Fabrizio Riguzzi, and Evelina Lamma. Tableau reasoners for probabilistic ontologies exploiting logic programming techniques. In Elena Bellodi and Alessio Bonfietti, editors, *Proceedings of the Doctoral Consortium (DC) co-located with the 14th Conference of the Italian Association for Artificial Intelligence (AI*IA 2015)*, volume 1485 of *CEUR Workshop Proceedings*, pages 1-6, Aachen, Germany, 2015. © by the authors, Sun SITE Central Europe.

[ bib | .pdf ]

Riccardo Zese, Elena Bellodi, Evelina Lamma, Fabrizio Riguzzi, and Fabiano Aguiari. Semantics and inference for probabilistic description logics. In Fernando Bobillo, Rommel N. Carvalho, Paulo C.G. Costa, Claudia d’Amato, Nicola Fanizzi, Kathryn B. Laskey, Kenneth J. Laskey, Thomas Lukasiewicz, Matthias Nickles, and Michael Pool, editors, *Uncertainty Reasoning for the Semantic Web III*, volume 8816 of *Lecture Notes in Computer Science*, pages 79-99. Springer International Publishing, © Springer International Publishing, 2014. The original publication is available at http://www.springerlink.com.

[ bib | DOI | .pdf ]