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## On Kleene Algebras and Closed Semirings (1990)

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Citations: | 56 - 6 self |

### Citations

2873 |
The Design and Analysis of Computer Algori thms
- Hopcroft
- 1966
(Show Context)
Citation Context ...semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures =-=[2, 14, 15, 5, 6, 1, 10, 7]-=-. In this paper we establish some new relationships among these structures. Our main results are: ffl There is a Kleene algebra in the sense of [6] that is not *-continuous. ffl The categories of *-co... |

1297 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...ses. Strictly speaking, in order to establish an adjunction, we must also check a naturality condition that ensures that the bijection given by and 0 interacts nicely with morphisms in KA and CS; see =-=[9]-=-. Specifically, we must show that for all morphisms h : K 0 ! K, g : K ! KC, and f : C ! C 0 , Kf ffi g ffi h = f ffi g ffi Ch : Again, this can be verified by straightforward arrow-chasing, which we ... |

997 | Dynamic Logic
- Harel, Kozen, et al.
- 2000
(Show Context)
Citation Context ... 0, the identity relation for 1, and reflexive transitive closure for is a Kleene algebra. Kleene algebras of binary relations are used to model programs in Dynamic Logic and other logics of programs =-=[4, 13]-=-. All these examples are -continuous. In x3 we will construct a Kleene algebra that is not -continuous. The n \Theta n matrices over a Kleene algebra again form a Kleene algebra [6]. 2.1.2 Elementary ... |

354 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...y. The operation is the constant 0 function. As with Kleene algebras, the n \Theta n matrices over a closed semiring again form a closed semiring. 2.3 Conway's Algebras According to the definition in =-=[2]-=-, an S-algebra (S; X ; \Delta; 0; 1) is similar to a closed semiring, except that P is defined not on sequences but on multisets of elements of S; moreover, there is no cardinality restriction on the ... |

264 | Semantical considerations on Floyd-Hoare logic
- Pratt
- 1976
(Show Context)
Citation Context ... 0, the identity relation for 1, and reflexive transitive closure for is a Kleene algebra. Kleene algebras of binary relations are used to model programs in Dynamic Logic and other logics of programs =-=[4, 13]-=-. All these examples are -continuous. In x3 we will construct a Kleene algebra that is not -continuous. The n \Theta n matrices over a Kleene algebra again form a Kleene algebra [6]. 2.1.2 Elementary ... |

243 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ... !-continuous semiring morphisms, and one relating CS and SA. The functor from KA to CS can be described as a completion by countably generated -ideals. ffl We show that the axioms for Kleene algebra =-=[6]-=- are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103]. ffl We show that right-handed Kleene algebras do not have to be left-handed. Thi... |

97 |
Two Complete Axiom Systems for the Algebra of Regular Events
- Salomaa
- 1966
(Show Context)
Citation Context ...gebra is Reg \Sigma , the family of regular sets over a finite alphabet \Sigma. The equational theory of this structure is called the algebra of regular events. This theory was axiomatized by Salomaa =-=[16]-=-, but his axiomatization involved rules that were not sound in general when interpreted over other Kleene algebras. It was shown in [6] that the Kleene algebra axioms completely axiomatize the algebra... |

50 |
Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness
- Mehlhorn
- 1984
(Show Context)
Citation Context ...semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures =-=[2, 14, 15, 5, 6, 1, 10, 7]-=-. In this paper we establish some new relationships among these structures. Our main results are: ffl There is a Kleene algebra in the sense of [6] that is not *-continuous. ffl The categories of *-co... |

43 | Dynamic Algebras as a Well-Behaved Fragment of Relation Algebras
- Pratt
- 1990
(Show Context)
Citation Context ...semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains several inequivalent definitions of Kleene algebras and related algebraic structures =-=[2, 14, 15, 5, 6, 1, 10, 7]-=-. In this paper we establish some new relationships among these structures. Our main results are: ffl There is a Kleene algebra in the sense of [6] that is not *-continuous. ffl The categories of *-co... |

28 |
Relation Algebras with Transitive Closure
- Ng
- 1984
(Show Context)
Citation Context ...e the equivalence of regular expressions and finite automata [2, 1], to give fast algorithms for transitive closure and shortest paths in directed graphs [1, 10], and to axiomatize relational algebra =-=[11, 12, 15]-=-. There has been some disagreement regarding the proper definition of Kleene algebras. At least ten inequivalent definitions of Kleene algebras or related structures have appeared in the literature [2... |

24 | The Kleene and Parikh theorem in complete semirings - Kuich - 1987 |

20 |
personal communication
- Conway
(Show Context)
Citation Context ...tization of that equational theory and thus improving Salomaa's completeness theorem [16]. A similar (and in fact stronger) theorem is stated by Conway without proof [2, p. 108]; but to his knowledge =-=[3]-=- and the author's, no proof has ever appeared in the literature. A Kleene algebra is called -continuous if it satisfies the axiom ab c = X n ab n c where P refers to the supremum in the natural order ... |

20 | On induction vs. ∗-continuity
- Kozen
- 1982
(Show Context)
Citation Context |

14 |
Dynamic algebra and the nature of induction
- Pratt
- 1980
(Show Context)
Citation Context |

12 | Relation algebras with transitive closure, abstract 742-02-09 - Ng, Tarski - 1977 |

2 | Relation algebras with transitive closure,"Abstract 742-02-09 - Ng, Tarski - 1977 |