Concurrent Kleene Algebra: Free Model and Completeness

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 settles a conjecture of Hoare and collaborators. Moreover, the techniques developed along the way are reusable; in particular, they allow us to establish pomset automata as an operational model for CKA.Comment: Version 2 includes an overview section that outlines the completeness proof, as well as some extra discussion of the interpolation lemma. It also includes better typography and a number of minor fixes. Version 3 incorporates the changes by comments from the anonymous referees at ESOP. Among other things, these include a worked example of computing the syntactic closure by han

On the homomorphism order of labeled posets

Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.Comment: 14 page

Attack Trees with Sequential Conjunction

We provide the first formal foundation of SAND attack trees which are a popular extension of the well-known attack trees. The SAND attack tree formalism increases the expressivity of attack trees by introducing the sequential conjunctive operator SAND. This operator enables the modeling of ordered events. We give a semantics to SAND attack trees by interpreting them as sets of series-parallel graphs and propose a complete axiomatization of this semantics. We define normal forms for SAND attack trees and a term rewriting system which allows identification of semantically equivalent trees. Finally, we formalize how to quantitatively analyze SAND attack trees using attributes.Comment: This is an extended version of the work published at IFIP SEC 201

Exploring an interface model for CKA

Concurrent Kleene Algebras (CKAs) serve to describe general concurrent systems in a unified way at an abstract algebraic level. Recently, a graph-based model for CKA has been defined in which the incoming and outgoing edges of a graph define its input/output interface. The present paper provides a simplification and a significant extension of the original model to cover notions of states, predicates and assertions in the vein of algebraic treatments using modal semirings. Moreover, it uses the extension to set up a variant of the temporal logic CTL* for the interface model

Schedulers and finishers : on generating the behaviours of an event structure

It is well known that every trace of a transition system can be generated using a scheduler. However, this basic completeness result does not hold in event structure models. The reason for this failure is that, according to its standard definition, a scheduler chooses which action to schedule and, at the same time, finishes the one scheduled last. Thus, scheduled events will never be able to overlap. We propose to separate scheduling from finishing and introduce the dual notion of finishers which, together with schedulers, are enough to regain completeness back. We then investigate all possible interactions between schedulers and finishers, concluding that simple alternating interactions are enough to express complex ones. Finally, we show how finishers can be used to filter behaviours to the extent to which they capture intrinsic system characteristics.18 page(s