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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… (More)

MetiTarski [1] is an automatic theorem prover that can prove inequalities involving sin, cos, exp, ln, etc. During its proof search, it generates a series of subproblems in nonlinear polynomial real arithmetic which are reduced to true or false using a decision procedure for the theory of real closed fields (RCF). These calls are often a bottleneck: RCF is… (More)

High-performance SMT solvers contain many tightly integrated , hand-crafted heuristic combinations of algorithmic proof methods. While these heuristic combinations tend to be highly tuned for known classes of problems, they may easily perform badly on classes of problems not anticipated by solver developers. This issue is becoming increasingly pressing as… (More)

Hybrid systems with both discrete and continuous dynamics are an important model for real-world cyber-physical systems. The key challenge is to ensure their correct functioning w.r.t. safety requirements. Promising techniques to ensure safety seem to be model-driven engineering to develop hybrid systems in a well-defined and traceable manner, and formal… (More)

Though decidable, the theory of real closed fields (RCF) is fundamentally infeasible. This is unfortunate, as automatic proof methods for nonlinear real arithmetic are crucially needed in both formalised mathematics and the verification of real-world cyber-physical systems. Consequently, many researchers have proposed fast, sound but incomplete RCF proof… (More)

We present a complete, certificate-based decision procedure for first-order univariate polynomial problems in Isabelle [17]. It is built around an executable function to decide the sign of a univariate polynomial at a real algebraic point. The procedure relies on no trusted code except for Isabelle's kernel and code generation. This work is the first step… (More)

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn't particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenarios like… (More)

We present novel Gröbner basis algorithms based on saturation loops used by modern superposition theorem provers. We illustrate the practical value of the algorithms through an experimental implementation within the Z3 SMT solver.

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., " x n ∈ Q " and " x n ∈ Z. " Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.