Paul B. Jackson

Clause form conversions for boolean circuits

The Boolean circuits is well established as a data structure for building propositional encodings of problems in preparation for satisfiability solving. The standard method for converting Boolean circuits to clause form (naming every vertex) has a number of shortcomings. In this paper we give a projection of several well-known clause form conversions to a simplified form of Boolean circuit. We introduce a new conversion which we show is equivalent to that of Boy de la Tour in certain circumstances and is hence optimal in the number of clauses that it produces. We extend the algorithm to cover reduced Boolean circuits, a data structure used by the model checker NuSMV. We present experimental results for this and other conversion procedures on BMC problems demonstrating its superiority, and conclude that the CNF conversion has a significant role in reducing the overall solving time.

Exploring Abstract Algebra in Constructive Type Theory

I describe my implementation of computational abstract algebra in the Nuprl system. I focus on my development of multivariate polynomials. I show how I use Nuprls expressive type theory to define classes of free abelian monoids and free monoid algebras. These classes are combined to create a class of all implementations of polynomials. I discuss the issues of subtyping and computational content that came up in designing the class definitions. I give examples of relevant theory developments, tactics and proofs. I consider how Nuprl could act as an algebraic ‘oracle’ for a computer algebra system and the relevance of this work for abstract functional programming.

Verifying a Garbage Collection Algorithm

We present a case study in using the PVS interactive theorem prover to formally model and verify properties of a tricolour garbage collection algorithm. We model the algorithm using state transition systems and verify safety and liveness properties in linear temporal logic. We set up two systems, each of which models the algorithm itself, object allocation, and the behaviour of user programs. The models differ in how concretely they model the heap. We verify the properties of the more abstract system, and then, once a refinement relation is exhibited between the systems, we show the more concrete system to have corresponding properties.

A Method for Invariant Generation for Polynomial Continuous Systems

This paper presents a method for generating semi-algebraic invariants for systems governed by non-linear polynomial ordinary differential equations under semi-algebraic evolution constraints. Based on the notion of discrete abstraction, our method eliminates unsoundness and unnecessary coarseness found in existing approaches for computing abstractions for non-linear continuous systems and is able to construct invariants with intricate boolean structure, in contrast to invariants typically generated using template-based methods. In order to tackle the state explosion problem associated with discrete abstraction, we present invariant generation algorithms that exploit sound proof rules for safety verification, such as differential cut

Using SMT solvers to verify high-integrity programs

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Direct Formal Verification of Liveness Properties in Continuous and Hybrid Dynamical Systems

, and a new proof rule that we call differential divide-and-conquer

Verifying safety and persistence properties of hybrid systems using flowpipes and continuous invariants

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