Mathematics in Computer Science

, Volume 11, Issue 3–4, pp 483–502 | Cite as

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

  • Thomas Sturm
Open Access


Effective quantifier elimination procedures for first-order theories provide a powerful tool for generically solving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifier elimination procedures are based on a fixed set of admissible logical symbols with an implicitly fixed semantics. This admits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination for the reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifier elimination we are going to discuss recent results with a subtropical procedure for an existential fragment of the reals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems in chemistry and in the life sciences.


Real quantifier elimination and decision Satisfiability Virtual substitution Subtropical methods Real geometry Verification Reaction systems Stability analysis 

Mathematics Subject Classification




Open access funding provided by Max Planck Society.


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Authors and Affiliations

  1. 1.University of Lorraine, CNRS, Inria, and LORIANancyFrance
  2. 2.Max Planck Institute for Informatics and Saarland UniversitySaarbrückenGermany

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