Combined Decision Techniques for the Existential Theory of the Reals

  • Grant Olney Passmore
  • Paul B. Jackson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5625)


Methods for deciding quantifier-free non-linear arithmetical conjectures over ℝ are crucial in the formal verification of many real-world systems and in formalised mathematics. While non-linear (rational function) arithmetic over ℝ is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have “sweet spots” – e.g., types of problems for which they perform much better than they do in general. Such “sweet spots” can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD (“Real Algebra in High Dimensions”) is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective “sweet-spots.” We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples.


Strict Inequality Decision Method Disjunctive Normal Form Existential Theory Sweet Spot 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Grant Olney Passmore
    • 1
  • Paul B. Jackson
    • 1
  1. 1.LFCSUniversity of EdinburghUK

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