# Paul BrunetUniversity College London | UCL · Department of Computer Science

Paul Brunet

Doctor

## About

29

Publications

845

Reads

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139

Citations

Introduction

You'll find more up-to-date info on my personal home page:
http://paul.brunet-zamansky.fr

Additional affiliations

January 2017 - present

October 2016 - December 2016

September 2014 - August 2016

Education

September 2011 - September 2013

September 2008 - September 2009

September 2005 - June 2007

**Lycéee Corneille, Rouen**

Field of study

- Mathematics and Physics

## Publications

Publications (29)

Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that t...

We introduce partially observable concurrent Kleene algebra (POCKA), an algebraic framework to reason about concurrent programs with variables as well as control structures, such as conditionals and loops, that depend on those variables. We illustrate the use of POCKA through concrete examples. We prove that POCKA is a sound and complete axiomatisa...

We introduce partially observable concurrent Kleene algebra (POCKA), an algebraic framework to reason about concurrent programs with control structures, such as conditionals and loops. POCKA enables reasoning about programs that can access variables and values, which we illustrate through concrete examples. We prove that POCKA is a sound and comple...

Concurrent Kleene Algebra is an elegant tool for equational reasoning about concurrent programs. An important feature of concurrent programs that is missing from CKA is the ability to restrict legal interleavings. To remedy this we extend the standard model of CKA, namely pomsets, with a new feature, called boxes, which can specify that part of the...

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and \(\mathsf {while}\)-loops. It turns out that integrating tests in CKA is sub...

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and while -loops. It turns out that integrating tests in CKA is subtle, due to t...

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and $\mathsf{while}$-loops. It turns out that integrating tests in CKA is subtle...

In this paper we present a detailed proof of an important result of algebraic logic: namely that the free commutative Kleene algebra is the space of semilinear sets. The first proof of this result was proposed by Redko in 1964, and simplified and corrected by Pilling in his 1970 thesis. However, we feel that a new account of this proof is needed no...

Concurrent Kleene Algebra is an elegant tool for equational reasoning about concurrent programs. An important feature of concurrent programs that is missing from CKA is the ability to restrict legal interleavings. To remedy this we extend the standard model of CKA, namely pomsets, with a new feature, called boxes, which can specify that part of the...

We consider algebras of languages over the signature of reversible Kleene lattices, that is the regular operations (empty and unit languages, union, concatenation and Kleene star) together with intersection and mirror image. We provide a complete set of axioms for the equational theory of these algebras. This proof was developed in the proof assist...

Kleene algebra with tests (KAT) is an algebraic framework for reasoning about the control flow of sequential programs. Generalising KAT to reason about concurrent programs is not straightforward, because axioms native to KAT in conjunction with expected axioms for concurrency lead to an anomalous equation. In this paper, we propose Kleene algebra w...

Concurrent Kleene Algebra (CKA) is a formalism to study concurrent programs. Like previous Kleene Algebra extensions, developing a correspondence between denotational and operational perspectives is important, for both foundations and applications. This paper takes an important step towards such a correspondence, by precisely relating bi-Kleene Alg...

Kleene algebra with tests (KAT) is an algebraic framework for reasoning about the control flow of sequential programs. However, when generalising KAT to reason about concurrent programs, axioms native to KAT in conjunction with expected axioms for reasoning about concurrency lead to an unexpected equation. In this paper, we propose Kleene algebra w...

Pomset automata are an operational model of weak bi-Kleene algebra, which describes program that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that th...

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the axioms for CKA with bounded parallelism are complete for the semantics proposed in the original paper; consequently, these semantics are the free model for this fragment. This result sett...

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the axioms for CKA with bounded parallelism are complete for the semantics proposed in the original paper; consequently, these semantics are the free model for this fragment. This result sett...

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the semantics proposed in the original paper is the free model of CKA with bounded parallel iteration, meaning the completeness of these axioms. This result settles a conjecture of Hoare and...

Concurrent Kleene algebras support equational reasoning about computing systems with concurrent behaviours. Their natural semantics is given by series(-parallel) rational pomset languages, a standard true concurrency semantics, which is often associated with processes of Petri nets. We use constructions on Petri nets to provide two decision procedu...

Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper...

Kleene algebra axioms are complete with respect to both language models and binary relation models. In particular, two regular expressions recognise the same language if and only if they are universally equivalent in the model of binary relations. We consider Kleene allegories, i.e., Kleene algebras with two additional operations and a constant whi...

Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper...

Algebras of relations appear naturally in many contexts, in computer science as well as in mathematics. They constitute a framework well suited to the semantics of imperative programs. Kleene algebra are a starting point: these algebras enjoy very strong decidability properties, and a complete axiomatisation. The goal of this thesis was to export k...

We present an extension of a Coq library for relation algebras, where we provide support for cardinals in a point-free way. This makes it possible to reason purely algebraically, which is well-suited for mechanisation. We discuss several applications in the area of graph theory and program verification.

An axiomatisation of Nominal Kleene Algebra has been proposed by Gabbay and Ciancia, and then shown to be complete and decidable by Kozen et al. However, one can think of at least four different formulations for a Kleene Algebra with names: Using freshness conditions or a presheaf structure (types), and with explicit permutations or not. We formall...

The equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik and Stefanescu in 1995. In particular, they obtained its decidability by using a particular automata construction.
We show that deciding this equational theory is...

Kleene algebra axioms are complete with respect to both language models and binary relation models. In particular, two regular expressions recognise the same language if and only if they are universally equivalent in the model of binary relations.
We consider Kleene allegories, i.e. Kleene algebra with two additional operations which are natural in...

Families of binary relations are important interpretations of regular expressions, and the equivalence of two regular expressions with respect to their relational interpretations is decidable: the problem reduces to the equality of the denoted regular languages. Putting together a few results from the literature, we first make explicit a generalisa...

The equational theory generated by all algebras of binary relations with operations of union, composition, converse and reflexive transitive closure was studied by Bernátsky, Bloom, Ésik, and Stefanescu in 1995. We reformulate some of their proofs in syntactic and elementary terms, and we provide a new algorithm to decide the corresponding theory....

## Projects

Project (1)