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## Myhill-Nerode relations on automatic systems and the completeness of Kleene algebra

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Venue: | In STACS 2001 (Dresden), volume 2010 of Lecture |

Citations: | 6 - 0 self |

### Citations

4745 | Introduction to Automata Theory, Languages, and Computation - Hopcroft, Ullman - 1979 |

2873 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...a has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata and formal language theory [14, 15], and the design and analysis of algorithms =-=[1, 6, 9]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [11] and references therein. There are many competing definitions and axiomatizations, and in fact there ... |

465 |
Representation of events in nerve nets and finite automata
- Kleene
- 1956
(Show Context)
Citation Context ...thods. This provides an alternative method for proving the completeness of Kleene algebra. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions. It dates to a 1956 paper of Kleene =-=[7]-=- and was further developed in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata an... |

354 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...pleteness of Kleene algebra. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions. It dates to a 1956 paper of Kleene [7] and was further developed in the 1971 monograph of Conway =-=[4]-=-. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata and formal language theory [14, 15], and the design and analysis... |

296 | Derivatives of regular expressions
- Brzozowski
- 1964
(Show Context)
Citation Context ...– Nerode relations in much the same way that finite automata can. Again, the chief property describing the relationship between the collapsed and uncollapsed systems is (1). The Brzozowski derivative =-=[3]-=- on an algebra of polynomials over a Kleene algebra gives rise to a triangular automatic system that can be solved using these methods. Completeness is proved essentially by showing that two equivalen... |

243 | A completeness theorem for Kleene algebras and the algebra of regular events
- Kozen
- 1994
(Show Context)
Citation Context ...of the algebraic theory; see [11] and references therein. There are many competing definitions and axiomatizations, and in fact there is no universal agreement on the definition of Kleene algebra. In =-=[10]-=-, a Kleene algebra was defined to be an idempotent semiring such that a b is the least solution to b + ax x and ba the least solution to b + xa x. This is a finitary universal Horn axiomatization (uni... |

178 |
Automata and Computability
- Kozen
- 1997
(Show Context)
Citation Context ...orial constructions in the theory of finite automata, including determinization via the subset construction and state minimization via the formation of a quotient modulo a Myhill–Nerode relation (see =-=[5, 12]-=-). A key theorem of Kleene algebra used in both these constructions is ax = xb ! a x = xb : (1) Intuitively, x represents a transformation between two state spaces, and a and b are transition relation... |

141 |
The design and analysis of algorithms
- Kozen
- 1991
(Show Context)
Citation Context ...a has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata and formal language theory [14, 15], and the design and analysis of algorithms =-=[1, 6, 9]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [11] and references therein. There are many competing definitions and axiomatizations, and in fact there ... |

43 | Dynamic Algebras as a Well-Behaved Fragment of Relation Algebras
- Pratt
- 1990
(Show Context)
Citation Context ... paper of Kleene [7] and was further developed in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs =-=[8, 17]-=-, automata and formal language theory [14, 15], and the design and analysis of algorithms [1, 6, 9]. Many authors have contributed over the years to the development of the algebraic theory; see [11] a... |

28 |
Relation Algebras with Transitive Closure
- Ng
- 1984
(Show Context)
Citation Context ... of regular expressions. It dates to a 1956 paper of Kleene [7] and was further developed in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra =-=[16, 20]-=-, semantics and logics of programs [8, 17], automata and formal language theory [14, 15], and the design and analysis of algorithms [1, 6, 9]. Many authors have contributed over the years to the devel... |

28 |
Representation of events in nerve nets and nite automata, in \Automata Studies
- Kleene
- 1956
(Show Context)
Citation Context ...thods. This provides an alternative method for proving the completeness of Kleene algebra. 1 Introduction Kleene algebra (KA) is the algebra of regular expressions. It dates to a 1956 paper of Kleene =-=[7]-=- and was further developed in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata an... |

24 | Typed Kleene Algebra
- Kozen
- 1998
(Show Context)
Citation Context ... left by ; thus in the above example, b = . to S, Now for any � , (e ) �[ ] = = X e � a a X e � a a�[ ] a =(e= ) [ ]�[ ] = X [ ] c c �[ ](e= ) [ ]�[ ] =( (e= )) �[ ]� therefore e = (e= ). By (1) (see =-=[13]-=-), e = (e= ) . Since c = (c= ), we have e c = e (c= )= (e= ) (c= )� which shows that the least solution e c of S is obtained by lifting the least solution (e= ) (c= ) of S= . 7 Brzozowski Derivatives ... |

24 |
The Kleene and Parikh theorem in complete semirings
- Kuich
- 1987
(Show Context)
Citation Context ...d in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata and formal language theory =-=[14, 15]-=-, and the design and analysis of algorithms [1, 6, 9]. Many authors have contributed over the years to the development of the algebraic theory; see [11] and references therein. There are many competin... |

20 | On induction vs. ∗-continuity
- Kozen
- 1982
(Show Context)
Citation Context ... paper of Kleene [7] and was further developed in the 1971 monograph of Conway [4]. Kleene algebra has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs =-=[8, 17]-=-, automata and formal language theory [14, 15], and the design and analysis of algorithms [1, 6, 9]. Many authors have contributed over the years to the development of the algebraic theory; see [11] a... |

14 |
A semiring on convex polygons and zero-sum cycle problems
- Iwano, Steiglitz
- 1990
(Show Context)
Citation Context ...a has appeared in one form or another in relational algebra [16, 20], semantics and logics of programs [8, 17], automata and formal language theory [14, 15], and the design and analysis of algorithms =-=[1, 6, 9]-=-. Many authors have contributed over the years to the development of the algebraic theory; see [11] and references therein. There are many competing definitions and axiomatizations, and in fact there ... |

12 |
Closure algorithms and the star-height problem of regular languages
- Backhouse
- 1975
(Show Context)
Citation Context ...isomorphic to Reg , the Kleene algebra of regular sets of strings over . The evaluation morphism " : 2[ ] ! 2, where "(a) =0for a 2 , corresponds to the empty word property (EWP) discussed by Salomaa =-=[2, 19]-=-. This map satisfies the property that "( )=1if 1 , 0 otherwise. 3 Generalized Triangular Matrices Let A be a set and a preorder (reflexive and transitive) on A. The preordered set A is finitary if al... |

11 | Kleene algebra with tests and commutativity conditions - Kozen - 1996 |