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Kleene algebra with tests and commutativity conditions
[chapter]

1996
*
Lecture Notes in Computer Science
*

We give an equational proof, using

doi:10.1007/3-540-61042-1_35
fatcat:jut32r57vne4ljtlmadnxu5jre
*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 ... The proof illustrates the use of*Kleene**algebra**with*extra*conditions*in program equivalence proofs. ... Acknowledgements Ernie Cohen, David Gries, David Harel, Vaughan Pratt,*and*Fred B. Schneider provided valuable comments. ...##
###
Kleene algebra with tests

1997
*
ACM Transactions on Programming Languages and Systems
*

The proof illustrates the use of

doi:10.1145/256167.256195
fatcat:asxkldtklfefjnfjzgyjl4j36e
*Kleene**algebra**with**tests**and**commutativity**conditions*in program equivalence proofs. ... We give a purely equational proof, 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 ... Schneider, Frederick Smith, Dave Walker,*and*the anonymous referees for valuable comments. ...##
###
Algebraic Notions of Termination

2011
*
Logical Methods in Computer Science
*

To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence

doi:10.2168/lmcs-7(1:1)2011
fatcat:stigsogyszap7ng6b5mxfyl2r4
*Kleene**algebras**and*omega*algebras*. ... The study is based on modal semirings, which are additively idempotent semirings*with*forward*and*backward modal operators. ... 2004 conference*and*of LMCS for helpful comments on earlier versions. ...##
###
Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL
[chapter]

2013
*
Lecture Notes in Computer Science
*

Schematic

doi:10.1007/978-3-642-39634-2_16
fatcat:bzxg2fl7i5gvfevgx444jsuwgy
*Kleene**algebra**with**tests*(SKAT) supports the equational verification of flowchart scheme equivalence*and*captures simple while-programs*with*assignment statements. ... We extend SKAT*with*assertion statements*and*derive the inference rules of Hoare logic. ...*Kleene**Algebra**with**Tests**Kleene**algebras**with**tests*(KAT) are at the basis of our implementation. ...##
###
Kleene algebra with tests: Completeness and decidability
[chapter]

1997
*
Lecture Notes in Computer Science
*

We prove the completeness of the equational theory of

doi:10.1007/3-540-63172-0_43
fatcat:f2ztqth6angwjisiycrpn5f57u
*Kleene**algebra**with**tests**and**-continuous*Kleene**algebra**with**tests*over language-theoretic*and*relational models. We also show decidability. ...*Kleene**algebras**with**tests*provide a rigorous framework for equational speci cation*and*veri cation. ... We simplify Cohen's construction*and*generalize it to handle*Kleene**algebra**with**tests*. 2*Kleene**Algebra**with**Tests*A*Kleene**algebra**with**tests*17] is a*Kleene**algebra**with*an embedded Boolean subalgebra ...##
###
On the Complexity of Reasoning in Kleene Algebra

2002
*
Information and Computation
*

We study the complexity of reasoning in

doi:10.1006/inco.2001.2960
fatcat:b2as6cpvtzhqxp5y6aw5tfmirq
*Kleene**algebra**and** -continuous*Kleene**algebra*in the presence of extra-equational assumptions E; that is, the complexity of deciding the validity of universal ... Our main results are as follows: for * -continuous*Kleene**algebra*, (i) if E contains only*commutativity*assumptions pq = qp, the problem is 0 1 -complete; (ii) if E contains only monoid equations, the ... The views*and*conclusions contained herein are those of the author*and*should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of these ...##
###
Diagram Chase in Relational System Development

2005
*
Electronical Notes in Theoretical Computer Science
*

Diagrams are endowed

doi:10.1016/j.entcs.2004.08.049
fatcat:gwar6lxgabhinar2m2yq3zgmla
*with*a simple*algebraic*semantics that imposes a convenient balance between expressive*and*algorithmic power. ... These are similar to those for functional diagrams in category theory*and*inspired by rewriting theory. ... Acknowledgment: The authors would like to thank the anonymous referees*and*the participants of the VLFM'04 for valuable discussions. ...##
###
Abstract abstract reduction

2006
*
The Journal of Logic and Algebraic Programming
*

Section 2 introduces

doi:10.1016/j.jlap.2005.04.001
fatcat:sb642bhnqnanrohtc3cgcmwmb4
*Kleene**algebra**and*ω-*algebra*; Section 3 recalls some of their basic properties. Section 4 connects these*algebras**with*abstract reduction*and*the associated diagrammatic proofs. ... First, we calculate proofs of generalized Church-Rosser theorems in*Kleene**algebras*, which are idempotent semirings*with*an operation for finite iteration. ... In particular I am indebted to one of the referees for communicating his proof experiments in*Kleene**algebra**with*the Otter theorem prover. ...##
###
Normal design algebra

2010
*
The Journal of Logic and Algebraic Programming
*

This clarifies the

doi:10.1016/j.jlap.2009.07.002
fatcat:ytenm5azeff3heou75iqrtrpwq
*algebraic*structure of designs*and*considerably simplifies reasoning about them, for example, since they form a*Kleene**and*omega*algebra**and*a*test*semiring. ... This substantially involves*Kleene**and*omega*algebra*as well as additional*algebraic*formulations of determinacy, invariants, domain, pre-image, convergence*and*Noetherity. ... Acknowledgement We are grateful to Roland Glück, Peter Höfner*and*an anonymous referee for valuable remarks. ...##
###
Algebras for Program Correctness in Isabelle/HOL
[chapter]

2014
*
Lecture Notes in Computer Science
*

We present a reference formalisation of

doi:10.1007/978-3-319-06251-8_4
fatcat:jmwp5mqm5beyhekkirkn7oybuu
*Kleene**algebra**and*demonic refinement*algebra**with**tests*in Isabelle/HOL. It provides three different formalisations of*tests*. ... Our structured comprehensive libraries for these*algebras*extend an existing*Kleene**algebra*library. ... The authors are grateful to Jordan Milner for a preparatory implementation*and*to Peter Jipsen for joint work on the one-sorted*test*axiomatisation. They also acknowledge funding from CNPq*and*EPSRC. ...##
###
Concurrent Kleene algebra with tests and branching automata

2016
*
Journal of Logical and Algebraic Methods in Programming
*

Abstract We introduce concurrent

doi:10.1016/j.jlamp.2015.12.005
fatcat:nhu7vxfcijcwbhvixcyulxnfrq
*Kleene**algebra**with**tests*(CKAT) as a combination of*Kleene**algebra**with**tests*(KAT) of Kozen*and*Smith*with*concurrent*Kleene**algebras*(CKA), introduced by Hoare, Möller ... To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas*and*relate them to branching automata*and*to concurrent*Kleene**algebras**with**tests*. ...*Kleene**algebras**with**tests*. ...##
###
Automating Algebraic Methods in Isabelle
[chapter]

2011
*
Lecture Notes in Computer Science
*

We implement a large Isabelle/HOL repository of

doi:10.1007/978-3-642-24559-6_41
fatcat:5jpfr7dazveyddlinhogksywjq
*algebras*for application in modelling computing systems. ... They subsume computational logics such as dynamic*and*Hoare logics*and*form a basis for various software development methods. ... PHL is also a fragment of PDL [15]*and*it is subsumed by*Kleene**algebra**with**tests*[23] . ...##
###
Modal Kleene Algebra and Partial Correctness
[chapter]

2004
*
Lecture Notes in Computer Science
*

We use this modal

doi:10.1007/978-3-540-27815-3_30
fatcat:3tvtu2jbt5e43oxw3dhwoqcxku
*Kleene**algebra*for calculating soundness*and*completeness proofs for propositional Hoare logic. ... We enrich*Kleene**algebra*by domain*and*codomain operators. These abstractions of relational notions give rise to four modal operators. ... Acknowledgment: We would like to thank Jules Desharnais, Thorsten Ehm*and*Joakim von Wright for valuable discussions*and*comments. ...##
###
Concurrent Kleene Algebra with Tests
[chapter]

2014
*
Lecture Notes in Computer Science
*

*Kleene*

*algebra*

*with*

*tests*(KAT) were dened earlier by Kozen

*and*Smith [KS97] . ... Concurrent

*Kleene*

*algebras*

*with*

*tests*(CKAT) combine these concepts

*and*give a relatively simple

*algebraic*model for reasoning about operational semantics of concurrent programs. ... ) of

*Kleene*

*algebra*

*with*

*tests*. ...

##
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Generalising KAT to verify weighted computations
[article]

2019
*
arXiv
*
pre-print

*Kleene*

*algebra*

*with*

*tests*(KAT) was introduced as an

*algebraic*structure to model

*and*reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean

*tests*. ... This paper introduces two generalisations of this structure able to express programs as weighted transitions

*and*

*tests*

*with*outcomes in non necessarily bivalent truth spaces: graded

*Kleene*

*algebra*

*with*...

*algebra*

*with*

*tests*(GKAT)

*and*the idempotent graded

*Kleene*

*algebra*

*with*

*tests*(I-GKAT). ...

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