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We describe contributions to algorithmic proof techniques for deciding the satisfiability of boolean combinations of many-variable nonlinear polynomial equations and inequalities over the real and complex numbers. In the first half, we present an abstract theory of Gröbner basis construction algorithms for algebraically closed fields of characteristic zero… (More)

We have constructed a tool for using SMT (SAT Modulo Theories) solvers to discharge verification conditions (VCs) from programs written in the SPARK language. The tool can drive any solver supporting the SMT-LIB standard input language and has API interfaces for some solvers. SPARK is a subset of Ada used primarily in high-integrity systems in the… (More)

- Grant Olney Passmore, Paul B. Jackson
- Calculemus/MKM
- 2009

Methods for deciding quantifier-free non-linear arithmetical conjectures over R are crucial in the formal verification of many realworld 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)

Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal… (More)

- Leonardo Mendonça de Moura, Grant Olney Passmore
- Automated Reasoning and Mathematics
- 2013

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)

- Stefan Mitsch, Grant Olney Passmore, André Platzer
- Mathematics in Computer Science
- 2014

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 functioningw.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)

- Grant Olney Passmore, Lawrence C. Paulson, Leonardo Mendonça de Moura
- AISC/MKM/Calculemus
- 2012

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)

- Leonardo de Moura, Grant Olney Passmore
- SMT '09
- 2009

Hilbert's weak Nullstellensatz guarantees the existence of algebraic proof objects certifying the unsatisfiability of systems of polynomial equations not satisfiable over any algebraically closed field. Such proof objects take the form of ideal membership identities and can be found algorithmically using Gröbner bases and cofactor-based linear algebra… (More)

Hybrid systems with both discrete and continuous dynamics are an important model for real-world physical systems. The key challenge is how 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)

- Yuri Gurevich, Grant Olney Passmore
- Bulletin of the EATCS
- 2011

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)