Grant Olney Passmore

Computation in real closed infinitesimal and transcendental extensions of the rationals

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 extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thoms Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples.

The strategy challenge in SMT solving

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 SMT solvers begin to gain the attention of practitioners in diverse areas of science and engineering. We present a challenge to the SMT community: to develop methods through which users can exert strategic control over core heuristic aspects of SMT solvers. We present evidence that the adaptation of ideas of strategy prevalent both within the Argonne and LCF theorem proving paradigms can go a long way towards realizing this goal.

Collaborative Verification-Driven Engineering of Hybrid Systems

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 verification to prove their correctness. Their combination forms the vision of verification-driven engineering. Often, hybrid systems are rather complex in that they require expertise from many domains (e. g., robotics, control systems, computer science, software engineering, and mechanical engineering). Moreover, despite the remarkable progress in automating formal verification of hybrid systems, the construction of proofs of complex systems often requires nontrivial human guidance, since hybrid systems verification tools solve undecidable problems. It is, thus, not uncommon for development and verification teams to consist of many players with diverse expertise. This paper introduces a verification-driven engineering toolset that extends our previous work on hybrid and arithmetic verification with tools for (1) graphical (UML) and textual modeling of hybrid systems, (2) exchanging and comparing models and proofs, and (3) managing verification tasks. This toolset makes it easier to tackle large-scale verification tasks.

Real algebraic strategies for metitarski proofs

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 fundamentally infeasible. However, by studying these subproblems, we can design specialised variants of RCF decision procedures that run faster and improve MetiTarskis performance.

On locally minimal Nullstellensatz proofs

Hilberts 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 techniques. However, these proof objects may contain redundant information: a proper subset of the equational assumptions used in these proofs may be sufficient to derive the unsatisfiability of the original polynomial system. For using Nullstellensatz techniques in SMT-based decision methods, a minimal proof object is often desired. With this in mind, we introduce a notion of locally minimal Nullstellensatz proofs and give ideal-theoretic algorithms for their construction.

Impugning Randomness, Convincingly

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 the lottery one. We attempt to rectify that.

Formal Verification of Financial Algorithms

Many deep issues plaguing today’s financial markets are symptoms of a fundamental problem: The complexity of algorithms underlying modern finance has significantly outpaced the power of traditional tools used to design and regulate them. At Aesthetic Integration, we have pioneered the use of formal verification for analysing the safety and fairness of financial algorithms. With a focus on financial infrastructure (e.g., the matching logics of exchanges and dark pools and FIX connectivity between trading systems), we describe the landscape, and illustrate our Imandra formal verification system on a number of real-world examples. We sketch many open problems and future directions along the way.

Deciding Univariate Polynomial Problems Using Untrusted Certificates in Isabelle/HOL

We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition. We follow the approach of untrusted certificates, separating solving from verifying: efficient external tools perform expensive real algebraic computations, producing evidence that is formally checked within Isabelle’s logic. This allows us to exploit highly-tuned computer algebra systems like Mathematica to guide our procedure without impacting the correctness of its results. We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.

Superfluous S-polynomials in Strategy-Independent Groebner Bases

Using the machinery of proof orders originally introduced by Bachmair and Dershowitz in the context of canonical equational proofs, we give an abstract, strategy-independent presentation of Groebner basis procedures and prove the correctness of two classical criteria for recognising superfluous S-polynomials, Buchbergers criteria 1 and 2, w.r.t. arbitrary fair and correct basis construction strategies. To do so, we develop a general method for proving the strategy-independent correctness of superfluous S-polynomial criteria which seems to be quite powerful. We also derive a new superfluous S-polynomial criterion which is a generalisation of Buchberger-1 and is proved to be correct strategy-independently.

Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., “\(x^n \in \mathbb {Q}\)” and “\(x^n \in \mathbb {Z}\).” Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.