Enumerative Level-2 Solution Counting for Quantified Boolean Formulas (Short Paper)

Authors Andreas Plank , Sibylle Möhle , Martina Seidl

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Andreas Plank
  • Institute for Symbolic Artificial Intelligence, JKU Linz, Austria
Sibylle Möhle
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Martina Seidl
  • Institute for Symbolic Artificial Intelligence, JKU Linz, Austria

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Andreas Plank, Sibylle Möhle, and Martina Seidl. Enumerative Level-2 Solution Counting for Quantified Boolean Formulas (Short Paper). In 29th International Conference on Principles and Practice of Constraint Programming (CP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 280, pp. 49:1-49:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We lift the problem of enumerative solution counting to quantified Boolean formulas (QBFs) at the second level. In contrast to the well-explored model counting problem for SAT (#SAT), where models are simply assignments to the Boolean variables of a formula, we are now dealing with tree (counter-)models reflecting the dependencies between the variables of the first and the second quantifier block. It turns out that enumerative counting on the second level does not give the complete model count. We present the - to the best of our knowledge - first approach of counting tree (counter-)models together with a counting tool that exploits state-of-the-art QBF technology. We provide several kinds of benchmarks for testing our implementation and illustrate in several case studies that solution counting provides valuable insights into QBF encodings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • QBF
  • Second-Level Model Counting


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