# Kleene algebra with domain

@article{Desharnais2003KleeneAW, title={Kleene algebra with domain}, author={Jules Desharnais and Bernhard M{\"o}ller and Georg Struth}, journal={ACM Trans. Comput. Log.}, year={2003}, volume={7}, pages={798-833} }

We propose Kleene algebra with domain (KAD), an extension of Kleene algebra by simple equational axioms for a domain and a codomain operation. KAD considerably augments the expressiveness of Kleene algebra, in particular for the specification and analysis of programs and state transition systems. We develop the basic calculus, present the most interesting models and discuss some related theories. We demonstrate applicability by two examples: algebraic reconstructions of Noethericity and…

## 200 Citations

### Kleene Algebras with Domain

- Mathematics, Computer ScienceArch. Formal Proofs
- 2016

This work formalises a modular hierarchy of algebras with domain and antidomain (domain complement) operations in Isabelle/HOL that ranges from domain and antidote semigroups to modal Kleene alge bras and divergence Kleenes.

### Modal Kleene algebra and applications - a survey

- Computer Science
- 2004

Modal Kleene algebras are Kleene algebras with forward and backward modal operators, defined via domain and codomain operations. They provide a concise and convenient algebraic framework that…

### Modal Kleene Algebra and Partial Correctness

- Computer Science, MathematicsAMAST
- 2004

Modal Kleene algebra provides a unifying semantics for various program calculi and enhances efficient cross-theory reasoning in this class, often in a very concise state-free style.

### Lazy Kleene Algebra

- Mathematics, Computer ScienceMPC
- 2004

We propose a relaxation of Kleene algebra by giving up strictness and right-distributivity of composition. This allows the subsumption of Dijkstra’s computation calculus, Cohen’s omega algebra and…

### Termination in Modal Kleene Algebra

- MathematicsIFIP TCS
- 2004

The paper formalizes and compares different notions of termination, including Lob's formula, in MKA and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma.

### Cylindric Kleene Lattices for Program Construction

- Computer ScienceMPC
- 2019

Cylindric algebras have been developed as an algebraisation of equational first order logic and are adapted to cylindric Kleene lattices and their variants to derive Morgan’s refinement calculus and to derive an algebraic Hoare logic for while programs with assignment laws.

### Algebraic coherent confluence and higher globular Kleene algebras

- MathematicsLog. Methods Comput. Sci.
- 2022

We extend the formalisation of confluence results in Kleene algebras to a
formalisation of coherent confluence proofs. For this, we introduce the
structure of higher globular Kleene algebra, a…

### One-sorted Program Algebras

- MathematicsArXiv
- 2022

Kleene algebra with tests, KAT, provides a simple two-sorted algebraic framework for verifying properties of propositional while programs. Kleene algebra with domain, KAD, is a one-sorted alternative…

## References

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Modal Kleene algebra provides a unifying semantics for various program calculi and enhances efficient cross-theory reasoning in this class, often in a very concise state-free style.

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The completeness of the equational theory of Kleene algebras with tests and *-continuous Kleene algebra with tests over language-theoretic and relational models is proved.

### Termination in Modal Kleene Algebra

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The paper formalizes and compares different notions of termination, including Lob's formula, in MKA and gives calculational proofs of two fundamental termination-dependent statements from rewriting theory: the well-founded union theorem by Bachmair and Dershowitz and Newman’s lemma.

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DRA (demonic Re nement Algebra), a variation of KAT for total correctness and illustrating its modeling and reasoning power with a number of applications and examples are described.

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A purely equational proof is given, using Kleene algebra with tests and commutativity conditions, of the following classical result: every while program can be simulated by a while program with at most one while loop.

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Kleene algebra (an Horn axiomatization of Kleene's algebra of regular events) has proved to be an eeective tool for reasoning about programs. Within the algebra, we can reason succinctly about both…

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A finitary axiomatization of the algebra of regular events involving only equations and equational implications that is sound for all interpretations over Kleene algebras is given. Axioms for Kleene…

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The specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning, and it is shown that Kleene algebra with tests subsumes propositionalHoare logic.

### The Variety of Kleene Algebras with Conversion is not Finitely Based

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