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Real Equation Systems with Alternating Fixed-Points

Authors Jan Friso Groote , Tim A. C. Willemse

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Jan Friso Groote
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
Tim A. C. Willemse
  • Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands

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Jan Friso Groote and Tim A. C. Willemse. Real Equation Systems with Alternating Fixed-Points. In 34th International Conference on Concurrency Theory (CONCUR 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 279, pp. 28:1-28:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


We introduce the notion of a Real Equation System (RES), which lifts Boolean Equation Systems (BESs) to the domain of extended real numbers. Our RESs allow arbitrary nesting of least and greatest fixed-point operators. We show that each RES can be rewritten into an equivalent RES in normal form. These normal forms provide the basis for a complete procedure to solve RESs. This employs the elimination of the fixed-point variable at the left side of an equation from its right-hand side, combined with a technique often referred to as Gauß-elimination. We illustrate how this framework can be used to verify quantitative modal formulas with alternating fixed-point operators interpreted over probabilistic labelled transition systems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Modal and temporal logics
  • Theory of computation → Verification by model checking
  • Real Equation System
  • Solution method
  • Gauß-elimination
  • Model checking
  • Quantitative modal mu-calculus


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