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Kleene algebra with tests and commutativity conditions [chapter]

Dexter Kozen
1996 Lecture Notes in Computer Science  
We give an 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 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.  ... 
doi:10.1007/3-540-61042-1_35 fatcat:jut32r57vne4ljtlmadnxu5jre

Kleene algebra with tests

Dexter Kozen
1997 ACM Transactions on Programming Languages and Systems  
The proof illustrates the use of 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.  ... 
doi:10.1145/256167.256195 fatcat:asxkldtklfefjnfjzgyjl4j36e

Algebraic Notions of Termination

Desharnais Jules, Bernhard Moeller, Struth Georg, Dexter Kozen
2011 Logical Methods in Computer Science  
To model infinite behaviours, idempotent semirings are extended to divergence semirings, divergence 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.  ... 
doi:10.2168/lmcs-7(1:1)2011 fatcat:stigsogyszap7ng6b5mxfyl2r4

Program Analysis and Verification Based on Kleene Algebra in Isabelle/HOL [chapter]

Alasdair Armstrong, Georg Struth, Tjark Weber
2013 Lecture Notes in Computer Science  
Schematic 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.  ... 
doi:10.1007/978-3-642-39634-2_16 fatcat:bzxg2fl7i5gvfevgx444jsuwgy

Kleene algebra with tests: Completeness and decidability [chapter]

Dexter Kozen, Frederick Smith
1997 Lecture Notes in Computer Science  
We prove the completeness of the equational theory of 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  ... 
doi:10.1007/3-540-63172-0_43 fatcat:f2ztqth6angwjisiycrpn5f57u

On the Complexity of Reasoning in Kleene Algebra

Dexter Kozen
2002 Information and Computation  
We study the complexity of reasoning in 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  ... 
doi:10.1006/inco.2001.2960 fatcat:b2as6cpvtzhqxp5y6aw5tfmirq

Diagram Chase in Relational System Development

Michael Ebert, Georg Struth
2005 Electronical Notes in Theoretical Computer Science  
Diagrams are endowed 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.  ... 
doi:10.1016/j.entcs.2004.08.049 fatcat:gwar6lxgabhinar2m2yq3zgmla

Abstract abstract reduction

Georg Struth
2006 The Journal of Logic and Algebraic Programming  
Section 2 introduces 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.  ... 
doi:10.1016/j.jlap.2005.04.001 fatcat:sb642bhnqnanrohtc3cgcmwmb4

Normal design algebra

Walter Guttmann, Bernhard Möller
2010 The Journal of Logic and Algebraic Programming  
This clarifies the 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.  ... 
doi:10.1016/j.jlap.2009.07.002 fatcat:ytenm5azeff3heou75iqrtrpwq

Algebras for Program Correctness in Isabelle/HOL [chapter]

Alasdair Armstrong, Victor B. F. Gomes, Georg Struth
2014 Lecture Notes in Computer Science  
We present a reference formalisation of 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.  ... 
doi:10.1007/978-3-319-06251-8_4 fatcat:jmwp5mqm5beyhekkirkn7oybuu

Concurrent Kleene algebra with tests and branching automata

Peter Jipsen, M. Andrew Moshier
2016 Journal of Logical and Algebraic Methods in Programming  
Abstract We introduce concurrent 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.  ... 
doi:10.1016/j.jlamp.2015.12.005 fatcat:nhu7vxfcijcwbhvixcyulxnfrq

Automating Algebraic Methods in Isabelle [chapter]

Walter Guttmann, Georg Struth, Tjark Weber
2011 Lecture Notes in Computer Science  
We implement a large Isabelle/HOL repository of 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] .  ... 
doi:10.1007/978-3-642-24559-6_41 fatcat:5jpfr7dazveyddlinhogksywjq

Modal Kleene Algebra and Partial Correctness [chapter]

Bernhard Möller, Georg Struth
2004 Lecture Notes in Computer Science  
We use this modal 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.  ... 
doi:10.1007/978-3-540-27815-3_30 fatcat:3tvtu2jbt5e43oxw3dhwoqcxku

Concurrent Kleene Algebra with Tests [chapter]

Peter Jipsen
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.  ... 
doi:10.1007/978-3-319-06251-8_3 fatcat:tnekgiex4jfsbpeesa52q2vt7a

Generalising KAT to verify weighted computations [article]

Leandro Gomes, Alexandre Madeira, Luís Soares Barbosa
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).  ... 
arXiv:1911.01146v1 fatcat:45sut6pllbfjpjca3t7nrrb4u4
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