Jeremy E. Dawson

Automating Open Bisimulation Checking for the Spi Calculus

We consider the problem of automating open bisimulation checking for the spi calculus, an extension of the pi-calculus with cryptographic primitives. The notion of open bisimulation considered here is indexed by a (symbolic) environment, represented as bi-traces (i.e., pairs of symbolic traces), which encode the history of interaction between the intruder with the processes being checked for bisimilarity. A crucial part of the definition of this open bisimulation, that is, the notion of consistency of bi-traces, involves infinite quantification over a certain notion of “respectful substitutions”. We show that one needs only to check a finite number of respectful substitutions in order to check bi-trace consistency. Our decision procedure uses techniques that have been well developed in the area of symbolic trace analysis for security protocols. More specifically, we make use of techniques for symbolic trace refinement, which transform a symbolic trace into a finite set of symbolic traces in a certain “solved form”. Crucially, we show that refinements of a projection of a bitrace can be uniquely extended to refinements of the bi-trace, and that consistency of all instances of the original bi-trace can be reduced to consistency of its finite set of refinements. We then give a sound and complete procedure for deciding open bisimilarity for finite spi processes.

Isabelle Theories for Machine Words

We describe a collection of Isabelle theories which facilitate reasoning about machine words. For each possible word length, the words of that length form a type, and most of our work consists of generic theorems which can be applied to any such type. We develop the relationships between these words and integers (signed and unsigned), lists of booleans and functions from index to value, noting how these relationships are similar to those between an abstract type and its representing set. We discuss how we used Isabelles bin type, before and after it was changed from a datatype to an abstract type, and the techniques we used to retain, as nearly as possible, the convenience of primitive recursive definitions. We describe other useful techniques, such as encoding the word length in the type.

Formalised Cut Admissibility for Display Logic

We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a machine-checked proof of cut-admissibility for δRA. Unlike other implementations, we explicitly formalise the structural induction in Isabelle/HOL and believe this to be the first full formalisation of cut-admissibility in the presence of explicit structural rules.

A construction for generalized hadamard matrices GH(4q, EA(q))

Abstract We give a construction for a generalized Hadamard matrix GH(4 q , EA( q )) as a 4 × 4 matrix of q × q blocks, for q an odd prime power other than 3 or 5. Each block is a GH( q , EA( q )) and certain combinations of 4 blocks form GH(2 q , EA( q )) matrices. Hence a GH(4 q , EA( q )) matrix exists for every prime power q .

A Mechanised Proof System for Relation Algebra using Display Logic

We describe an implementation of the Display Logic calculus for relation algebra as an Isabelle theory. Our implementation is the first mechanisation of any display calculus. The inference rules of Display Logic are coded directly as Isabelle theorems, thereby guaranteeing the correctness of all derivations. Our implementation generalises easily to handle other display calculi. It also provides a useful interactive proof assistant for relation algebras. We describe various tactics and derived rules developed for simplifying proof search, including an automatic cut-elimination procedure, and example theorems proved using Isabelle. We show how some relation algebraic theorems proved using our system can be put in the form of structural rules of Display Logic, facilitating later re-use. We then show how the implementation can be used to prove results comparing alternative formalizations of relation algebra from a proof-theoretic perspective.

Embedding Display Calculi into Logical Frameworks:: Comparing Twelf and Isabelle

Abstract Logical frameworks are computer systems which allow a user to formalise mathematics using specially designed languages based upon mathematical logic and Churchs theory of types. They can be used to derive programs from logical specifications, thereby guaranteeing the correctness of the resulting programs. They can also be used to formalise rigorous proofs about logical systems. We compare several methods of implementing the display (sequent) calculus δRA for relation algebra in the logical frameworks Isabelle and Twelf. We aim for an implementation enabling us to formalise, within the logical framework, proof-theoretic results such as the cut-elimination theorem for δRA, and any associated increase in proof length. We discuss issues arising from this requirement.

A New Machine-checked Proof of Strong Normalisation for Display Logic

Abstract We use a deep embedding of the display calculus for relation algebras δRA in the logical framework Isabelle/HOL to formalise a new, machine-checked, proof of strong normalisation and cut-elimination for δRA which does not use measures on the size of derivations. Our formalisation generalises easily to other display calculi and can serve as a basis for formalised proofs of strong normalisation for the classical and intuitionistic versions of a vast range of substructural logics like the Lambek calculus, linear logic, relevant logic, BCK-logic, and their modal extensions. We believe this is the first full formalisation of a strong normalisation result for a sequent system using a logical framework.

A General Theorem on Termination of Rewriting

We re-express our theorem on the strong-normalisation of display calculi as a theorem about the well-foundedness of a certain ordering on first-order terms, thereby allowing us to prove the termination of systems of rewrite rules. We first show how to use our theorem to prove the well-foundedness of the lexicographic ordering, the multiset ordering and the recursive path ordering. Next, we give examples of systems of rewrite rules which cannot be handled by these methods but which can be handled by ours. Finally, we show that our method can also prove the termination of the Knuth-Bendix ordering and of dependency pairs.

Optimizing user-level communication patterns on the Fujitsu AP3000

We present techniques and algorithms to improve the performance of various communication patterns on message passing platforms where, for reasons of safety, user level communications must be buffered in (special) memory on both the send and the receive. These algorithms can not only minimize message copying but overlap the copying to/from the special memory with the actual transfer enabling full bandwidth to be achieved. These patterns include tree broadcast and reductions, (ring based) multiple broadcasts and reductions, pipelined broadcast and buffered point-to-point sends. In each case, the messages have a simple stride. All of these patterns are used in dense linear algebra applications, although they are also used it many other contexts. These algorithms are implemented and their performance evaluated on the Fujitsu AP3000, a message passing multicomputer having many characteristics of the cluster model. Some aspects, such as the performance characteristics of the special memory are specific to the AP3000; however the algorithms still apply to any platform using a similar mode of user level communications. Worthwhile performance increases are obtained, especially for patterns involving moderate-large number of processors.

Annotation-Free Sequent Calculi for full Intuitionistic Linear Logic

The Point Hyperplane Cover problem in