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Past Matters: Supporting LTL+Past in the BLACK Satisfiability Checker

Authors Luca Geatti , Nicola Gigante , Angelo Montanari , Gabriele Venturato

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Author Details

Luca Geatti
  • University of Udine, Italy
  • Fondazione Bruno Kessler, Trento, Italy
Nicola Gigante
  • Free University of Bozen-Bolzano, Italy
Angelo Montanari
  • University of Udine, Italy
Gabriele Venturato
  • University of Udine, Italy
  • KU Leuven, Belgium


The open access publication of this article was supported by the Alpen-Adria-Universität Klagenfurt, Austria. Nicola Gigante acknowledges the partial support of the TOTA project "Temporal Ontologies and Tableaux Algorithms". Gabriele Venturato acknowledges the partial support of the KU Leuven Research Fund (C14/18/062).

Cite AsGet BibTex

Luca Geatti, Nicola Gigante, Angelo Montanari, and Gabriele Venturato. Past Matters: Supporting LTL+Past in the BLACK Satisfiability Checker. In 28th International Symposium on Temporal Representation and Reasoning (TIME 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 206, pp. 8:1-8:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


LTL+Past is the extension of Linear Temporal Logic (LTL) supporting past temporal operators. The addition of the past does not add expressive power, but does increase the usability of the language both in formal verification and in artificial intelligence, e.g., in the context of multi-agent systems. In this paper, we add the support of past operators to BLACK, a satisfiability checker for LTL based on a SAT encoding of a tree-shaped tableau system. We implement two ways of supporting the past in the tool. The first one is an equisatisfiable translation that removes the past operators, obtaining a future-only formula that can be solved with the original LTL engine. The second one extends the SAT encoding of the underlying tableau to directly support the tableau rules that deal with past operators. We describe both approaches and experimentally compare the two between themselves and with the νXmv model checker, obtaining promising results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • SAT
  • LTL
  • LTL+Past
  • Tableaux


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