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Kleene Algebra with Observations [article]

Tobias Kappé and Paul Brunet and Jurriaan Rot and Alexandra Silva and Jana Wagemaker and Fabio Zanasi
2019 arXiv   pre-print
In this paper, we propose Kleene algebra with observations (KAO), a variant of KAT, as an alternative foundation for extending KAT to a concurrent setting.  ...  Kleene algebra with tests (KAT) is an algebraic framework for reasoning about the control flow of sequential programs.  ...  We call this system weak Kleene algebra with observations; like KAT, its axioms are based on the axioms of Boolean algebra and Kleene algebra, as follows. Definition 4.1 (WKAO axioms).  ... 
arXiv:1811.10401v3 fatcat:3tbaptzye5gvjel3cbqhc5khni

Concurrent Kleene Algebra with Observations: From Hypotheses to Completeness [chapter]

Tobias Kappé, Paul Brunet, Alexandra Silva, Jana Wagemaker, Fabio Zanasi
2020 Lecture Notes in Computer Science  
It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO).  ...  Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs.  ...  Algebra with Observations (KAO) in previous work [13] .  ... 
doi:10.1007/978-3-030-45231-5_20 fatcat:hiblehptwnh2hboxqvqyctx5qi

Concurrent Kleene Algebra with Observations: from Hypotheses to Completeness [article]

Tobias Kappé and Paul Brunet and Alexandra Silva and Jana Wagemaker and Fabio Zanasi
2020 arXiv   pre-print
It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO).  ...  Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs.  ...  In order to fix this issue, we have presented Kleene Algebra with Observations (KAO) in previous work [13] .  ... 
arXiv:2002.09682v1 fatcat:uv4z53k755hd7kiubof4ijoazi

Kleene Algebras and Logic: Boolean and Rough Set Representations, 3-valued, Rough and Perp Semantics [article]

Arun Kumar, Mohua Banerjee
2015 arXiv   pre-print
One concludes with the observation that all the semantics for this logic are equivalent.  ...  A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets.  ...  Let us spell out the natural connections of the Kleene algebra 3 with rough sets.  ... 
arXiv:1511.07165v1 fatcat:kyjzxlniqjc2rimgy35cmpm5uq

A Free Construction of Kleene Algebras with Tests [chapter]

Hitoshi Furusawa
2004 Lecture Notes in Computer Science  
Moreover we show that our free Kleene algebra with tests is isomorphic to Kozen and Smith's Kleene algebra with tests if their construction available, that is, a generator of the Boolean algebra is finite  ...  In this paper we define Kleene algebra with tests in a slightly more general way than Kozen's definition.  ...  Due to this observation, we define a Kleene algebra with tests as a triple consisting of a Boolean algebra, a Kleene algebra, and a function from the carrier set of the Boolean algebra to the carrier of  ... 
doi:10.1007/978-3-540-27764-4_8 fatcat:u426j6wuzze4devtfy6gpg2iue

A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events

D. Kozen
1994 Information and Computation  
Unlike Salomaa's axiomatizations, the axiomatization given here is sound for all interpretations over Kleene algebras.  ...  We give a nitary axiomatization of the algebra of regular events involving only equations and equational implications.  ...  Introduction Kleene algebras are algebraic structures with operators +, , , 0, and 1 satisfying certain axioms.  ... 
doi:10.1006/inco.1994.1037 fatcat:qjketrvu25bunju4xlvfobmkse

Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets [article]

Jouni Järvinen, Sándor Radeleczki
2019 arXiv   pre-print
We introduce Kleene-Varlet spaces as partially ordered sets equipped with a polarity satisfying certain additional conditions.  ...  By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene algebra is isomorphic to a subalgebra of the rough set regular pseudocomplemented Kleene algebra defined by a tolerance  ...  Let (X, ≤, g) be a Kleene-Varlet space. Here we present some observations related to the map g.  ... 
arXiv:1806.09203v3 fatcat:ltetwjitgfhsdoay4k74l6b4by

MV-algebras freely generated by finite Kleene algebras

Stefano Aguzzoli, Leonardo M. Cabrer, Vincenzo Marra
2013 Algebra Universalis  
variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely generated--equivalently, finite.  ...  The proofs rely heavily on the Davey-Werner natural duality for Kleene algebras, on the representation of finitely presented MV-algebras by compact rational polyhedra, and on the theory of bases of MV-algebras  ...  (III) If A is an MV-algebra with a basis B, the multipliers of B are uniquely determined [17, (iii ) (Observe that ∅ ∈ B whenever 0 = 1 in A, i.e., whenever A is non-trivial.  ... 
doi:10.1007/s00012-013-0250-3 fatcat:royq43fn4rgr5cawihe6g4543u

Modal Tools for Separation and Refinement

Georg Struth
2008 Electronical Notes in Theoretical Computer Science  
Kleene modules and modal Kleene algebras are applied to automatically verify refinement laws for infinite loops and separation of termination.  ...  defined, and with ∅ and P as neutral elements is a Kleene algebra.  ...  So a Kleene algebra with (Boolean) domain is simply a (Boolean) domain semiring that is also a Kleene algebra.  ... 
doi:10.1016/j.entcs.2008.06.005 fatcat:gzvmozk35nhzncfs5p5arvwos4

Relation-Algebraic Verification of Prim's Minimum Spanning Tree Algorithm [chapter]

Walter Guttmann
2016 Lecture Notes in Computer Science  
Many existing results from relation algebras and Kleene algebras generalise from the relation model to the weighted-graph model with no or small changes.  ...  We introduce new generalisations of relation algebras and Kleene algebras, in which most of the proof can be carried out.  ...  I thank Peter Höfner and Bernhard Möller for discussing alternative approaches to minimum spanning trees in Kleene algebras.  ... 
doi:10.1007/978-3-319-46750-4_4 fatcat:7mcpqvvxgzerzfbyenz7usw4lu

Kleene algebras, adjunction and structural control [article]

Giuseppe Greco, Fei Liang, Alessandra Palmigiano
2018 arXiv   pre-print
In the present paper, we introduce a multi-type calculus for the logic of measurable Kleene algebras, for which we prove soundness, completeness, conservativity, cut elimination and subformula property  ...  In this paper, we introduce a subclass of Kleene algebras, referred to as measurable Kleene algebras, 2 which are Kleene algebras endowed with a dual Kleene star operation, associating any element with  ...  Next, we introduce a subclass of Kleene algebras endowed with both a Kleene star and a dual Kleene star.  ... 
arXiv:1805.07583v1 fatcat:egfclzr43zgx5h276ueeah4bti

Kleene algebra with tests

Dexter Kozen
1997 ACM Transactions on Programming Languages and Systems  
We introduce Kleene algebra with tests, an equational system for manipulating programs.  ...  The proof illustrates the use of Kleene algebra with tests and commutativity conditions in program equivalence proofs.  ...  In Section 4.1, we observe that the universal Horn theory of the *-continuous Kleene algebras is not recursively enumerable, therefore not finitely axiomatizable.  ... 
doi:10.1145/256167.256195 fatcat:asxkldtklfefjnfjzgyjl4j36e

Kleene posets and pseudo-Kleene posets [article]

Ivan Chajda, Helmut Länger
2020 arXiv   pre-print
The concept of a Kleene algebra (sometimes also called Kleene lattice) was already generalized by the first author for non-distributive lattices under the name pseudo-Kleene algebra.  ...  Moreover, we prove that the Dedekind-MacNeille completion of a pseudo-Kleene poset is a pseudo-Kleene algebra and that the Dedekind-MacNeille completion of a finite Kleene poset is a Kleene algebra.  ...  A Kleene algebra (see [10] ) is a distributive pseudo-Kleene algebra. Observe that a lattice with an antitone involution is a (pseudo-)Kleene poset if and only if it is a (pseudo-)Kleene algebra.  ... 
arXiv:2006.04417v1 fatcat:pj6s2wazmffennhih5j2te4pma

Kleene algebras are almost universal

M. E. Adams, H. A. Priestley
1986 Bulletin of the Australian Mathematical Society  
This paper concerns endomorphism monoids of Kleene algebras and complements [31, which deals with endomorphism monoids of de Morgan algebras.  ...  Kleene algebras form a subvariety of de Morgan algebras containing Boolean algebras.  ...  We have already observed in Section 2 that the number of non-constant endomorphisms of a finite Kleene algebra increases with the size of the algebra.  ... 
doi:10.1017/s0004972700010248 fatcat:xvthmjj3xjcynaojoe5fsr43zy

Design Methodology for the Implementation of Fuzzy Inference Systems Based on Boolean Relations

Helbert Espitia, José Soriano, Iván Machón, Hilario López
2019 Electronics  
The methodology uses Kleene algebra via truth tables for the fuzzy system design, allowing the simplification of the equations that implement the fuzzy system.  ...  In comparison with the Boolean algebra in Kleen algebra is observable that: x ∨ x ≤ 1 (7) x ∧ x ≥ 0 For the FIS-BBR design process proposed, the distributive and absorption properties are especially used  ...  , being an important aspect the manipulation done with Boolean and Kleene algebras.  ... 
doi:10.3390/electronics8111243 fatcat:q5jut64i3rfvdogh4agjmhprjq
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