# Concurrent Kleene Algebra: Free Model and Completeness

@article{Kapp2018ConcurrentKA, title={Concurrent Kleene Algebra: Free Model and Completeness}, author={Tobias Kapp{\'e} and Paul Brunet and Alexandra Silva and Fabio Zanasi}, journal={ArXiv}, year={2018}, volume={abs/1710.02787} }

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 technique developed to this end allows us to establish a Kleene Theorem for CKA, extending…

## 26 Citations

Completeness and Incompleteness of Synchronous Kleene Algebra

- Computer ScienceMPC
- 2019

This work proposes an alternative set of axioms for SKA, based on Salomaa's axiomatisation of regular languages, and shows that these provide a sound and complete characterisation w.r.t. the original language semantics.

Kleene Algebra with Observations

- MathematicsCONCUR
- 2019

This paper proposes Kleene algebra with observations (KAO), a variant of KAT, as an alternative foundation for extending KAT to a concurrent setting, and characterises the free model of KAO, and establishes a decision procedure to establish its equational theory.

Equivalence checking for weak bi-Kleene algebra

- Computer Science, MathematicsLog. Methods Comput. Sci.
- 2021

A fragment of pomset automata is characterized that admits a decision procedure for language equivalence that corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism, and it is proved that equivalence of series- rational expressions is decidable.

Partially Observable Concurrent Kleene Algebra

- Computer ScienceCONCUR
- 2020

It is proved that POCKA is a sound and complete axiomatisation of a model of partial observations, and the semantics passes an important check for sequential consistency.

On Series-Parallel Pomset Languages: Rationality, Context-Freeness and Automata

- Computer ScienceJ. Log. Algebraic Methods Program.
- 2019

Pomsets with Boxes: Protection, Separation, and Locality in Concurrent Kleene Algebra

- Computer ScienceFSCD
- 2020

The standard model of CKA is extended, namely pomsets, with a new feature, called boxes, which can specify that part of the system is protected from outside interference, and a logic is provided, 'pomset logic', that is an assertion language for specifying such properties, and which is interpreted on pomset with boxes.

L O ] 1 6 N ov 2 01 8 Kleene Algebra with Observations ⋆

- Mathematics, Computer Science
- 2018

This paper proposes Kleene algebra with observations (KAO), a variant of KAT, as an alternative foundation for extending KAT to a concurrent setting, characterise the free model of KAO, and establish a decision procedure w.r.t. its equational theory.

A note on commutative Kleene algebra

- Mathematics, Computer ScienceArXiv
- 2019

A detailed proof is presented that the free commutative Kleene algebra is the space of semilinear sets, a key component in the completeness proofs of several algebraic models of concurrent computations.

Completeness for Identity-free Kleene Lattices

- Mathematics, Computer ScienceCONCUR
- 2018

A finite set of axioms for identity-free Kleene lattices are provided, which are proved sound and complete for the equational theory of their relational models.

On Star Expressions and Coalgebraic Completeness Theorems

- Mathematics, Computer ScienceMFPS
- 2021

This paper analyses Grabmayer and Fokkink’s proof of completeness from the perspective of universal coalgebra, and gives an abstract account of their proof method, and compares this proof method to another approach to completeness proofs from coalgebraic language theory.

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