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## A completeness theorem for Kleene algebras and the algebra of regular events (1994)

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Venue: | Information and Computation |

Citations: | 243 - 27 self |

### Citations

4745 |
Introduction to Automata Theory, Languages and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...regular languages over an alphabet \Sigma. The relationship between these two formalisms forms the basis of a well-developed classical theory. Classical developments range from the more combinatorial =-=[20, 11]-=- to the more algebraic [29, 10, 5, 9, 27]. The approach taken in this paper must ultimately be attributed to Conway [8]. In this section we define the notion of an automaton over an arbitrary Kleene a... |

2873 |
The Design and Analysis of Computer Algori thms
- Hopcroft
- 1966
(Show Context)
Citation Context ...various guises in a number of settings: relational algebra [22, 23], semantics and logics of programs [14, 24], automata and formal language theory [18, 19], and the design and analysis of algorithms =-=[1, 21, 12]-=-. An important example of a Kleene algebra is Reg \Sigma , the family of regular sets over a finite alphabet \Sigma. The equational theory of this structure has been called the algebra of regular even... |

466 |
Representation of events in nerve nets and finite automata, automata studies
- Kleene
- 1956
(Show Context)
Citation Context ...s structure has been called the algebra of regular events. This theory was first studied by Infor. and Comput. 110:2 (May 1994), 366--390. A preliminary version of this paper appeared as [16]. Kleene =-=[13]-=-, who posed axiomatization as an open problem. Salomaa [28] gave two complete axiomatizations of the algebra of regular events in 1966, but these axiomatizations depended on rules of inference that ar... |

354 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...1964 that no finite set of equational axioms could characterize the algebra of regular events. The algebra of regular events and its axiomatization is the subject of the extensive monograph of Conway =-=[8]-=-. The bulk of Conway's treatment is infinitary. In 1981, we gave a complete infinitary equational deductive system for the algebra of regular events that is sound over all *-continuous Kleene algebras... |

314 |
C.: Elements of the Theory of Computation
- Lewis, Papadimitriou
- 1981
(Show Context)
Citation Context ...regular languages over an alphabet \Sigma. The relationship between these two formalisms forms the basis of a well-developed classical theory. Classical developments range from the more combinatorial =-=[20, 11]-=- to the more algebraic [29, 10, 5, 9, 27]. The approach taken in this paper must ultimately be attributed to Conway [8]. In this section we define the notion of an automaton over an arbitrary Kleene a... |

200 |
Rational Series and their Languages
- Berstel, Reutenauer
- 1988
(Show Context)
Citation Context ...bet \Sigma. The relationship between these two formalisms forms the basis of a well-developed classical theory. Classical developments range from the more combinatorial [20, 11] to the more algebraic =-=[29, 10, 5, 9, 27]-=-. The approach taken in this paper must ultimately be attributed to Conway [8]. In this section we define the notion of an automaton over an arbitrary Kleene algebra. In subsequent sections, we will u... |

97 |
Two Complete Axiom Systems for the Algebra of Regular Events
- Salomaa
- 1966
(Show Context)
Citation Context ...This theory was first studied by Infor. and Comput. 110:2 (May 1994), 366--390. A preliminary version of this paper appeared as [16]. Kleene [13], who posed axiomatization as an open problem. Salomaa =-=[28]-=- gave two complete axiomatizations of the algebra of regular events in 1966, but these axiomatizations depended on rules of inference that are not sound in general under nonstandard interpretations. R... |

71 |
On the complexity of !-automata
- Safra
- 1988
(Show Context)
Citation Context ... T v = u T B v = fi : 2 6 Open Problems An intriguing question is whether the techniques developed here can be extended to automata on infinite objects. An algebraic treatment of Safra's construction =-=[26] mig-=-ht conceivably be used to establish completeness of the propositional ��-calculus. Progress toward this goal has recently been made by Walukiewicz [31]. Another question is whether the axioms pres... |

56 | On Kleene Algebras and Closed Semirings
- Kozen
- 1990
(Show Context)
Citation Context ...and S-algebras are defined in terms of an infinitary summation operator P , whose sole purpose, it seems, is to define . These structures are all closely related; the precise relationship is drawn in =-=[15]-=-. 1.2 Salomaa's Axiomatizations Before one can fully appreciate the axiomatization of x2, it is important to understand Salomaa's axiomatizations [28] and their limitations. Let R \Sigma denote the in... |

50 | Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness - Mehlhorn - 1984 |

45 |
Complete systems of B-rational identities
- Krob
(Show Context)
Citation Context ...tion relies on the presence of a converse operator. Schematic equational axiomatizations for the algebra of regular events, necessarily representing infinitely many equations, have been given by Krob =-=[17]-=- and Bloom [6]. There is some disagreement regarding the definition of Kleene algebras [8, 24, 14]. The literature contains several inequivalent definitions. In this paper we introduce yet another: a ... |

43 | Dynamic Algebras as a Well-Behaved Fragment of Relation Algebras
- Pratt
- 1990
(Show Context)
Citation Context ...lgebraic structures with operators +, \Delta, , 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra [22, 23], semantics and logics of programs =-=[14, 24]-=-, automata and formal language theory [18, 19], and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma , the family of regular sets over a finite... |

34 | On defining relations for the algebra of regular events - Redko - 1964 |

28 |
Relation Algebras with Transitive Closure
- Ng
- 1984
(Show Context)
Citation Context ...ebras. 1 Introduction Kleene algebras are algebraic structures with operators +, \Delta, , 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra =-=[22, 23]-=-, semantics and logics of programs [14, 24], automata and formal language theory [18, 19], and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma... |

28 |
Representation of events in nerve nets and nite automata, in \Automata Studies
- Kleene
- 1956
(Show Context)
Citation Context ...his structure has been called the algebra ofregular events. This theory was rst studied by Infor. and Comput. 110:2 (May 1994), 366{390. A preliminary version of this paper appeared as [16]. 1Kleene =-=[13]-=-, who posed axiomatization as an open problem. Salomaa [28] gave two complete axiomatizations of the algebra of regular events in 1966, but these axiomatizations depended on rules of inference that ar... |

25 |
Peák: Algebraic Theory of Automata, Akadémiai Kiadó
- Gécseg, I
(Show Context)
Citation Context ...bet \Sigma. The relationship between these two formalisms forms the basis of a well-developed classical theory. Classical developments range from the more combinatorial [20, 11] to the more algebraic =-=[29, 10, 5, 9, 27]-=-. The approach taken in this paper must ultimately be attributed to Conway [8]. In this section we define the notion of an automaton over an arbitrary Kleene algebra. In subsequent sections, we will u... |

24 |
The Kleene and Parikh theorem in complete semirings
- Kuich
- 1987
(Show Context)
Citation Context ..., 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra [22, 23], semantics and logics of programs [14, 24], automata and formal language theory =-=[18, 19]-=-, and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma , the family of regular sets over a finite alphabet \Sigma. The equational theory of thi... |

20 | On induction vs. ∗-continuity
- Kozen
- 1982
(Show Context)
Citation Context ...lgebraic structures with operators +, \Delta, , 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra [22, 23], semantics and logics of programs =-=[14, 24]-=-, automata and formal language theory [18, 19], and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma , the family of regular sets over a finite... |

19 |
Equational axioms for regular sets
- Bloom, Ésik
(Show Context)
Citation Context ...on the presence of a converse operator. Schematic equational axiomatizations for the algebra of regular events, necessarily representing in nitely many equations, havebeengiven by Krob [17] and Bloom =-=[6]-=-. There is some disagreement regarding the de nition of Kleene algebras [8, 24, 14]. The literature contains several inequivalent de nitions. In this paper we introduce yet another: a Kleene algebra i... |

14 | A semiring on convex polygons and zero-sum cycle problems - Iwano, Steiglitz - 1990 |

12 |
Closure algorithms and the star-height problem of regular languages
- Backhouse
- 1975
(Show Context)
Citation Context ...roved by encoding the classical combinatorial constructions of the theory of finite automata, e.g. state minimization, algebraically. There is an extensive literature on the algebra of regular events =-=[8, 4, 17]-=- and much of the development of this paper is a recapitulation of previous work. For example, the construction of a transition matrix representing a finite automaton equivalent to a given regular expr... |

12 |
Une remarque sur les systèmes complets d’identités rationnelles
- Boffa
- 1990
(Show Context)
Citation Context ...e minimal deterministic finite automaton to obtain completeness. Conway states a similar theorem without proof in the latter part of his book [8, Theorem 5, p. 108]. Krob [17], based on work of Boffa =-=[7]-=-, and Archangelsky [3] have recently independently obtained similar results based on different techniques. 1.1 Examples of Kleene Algebras Kleene algebras abound in computer science. We have already m... |

12 |
Relation algebras with transitive closure, abstract 742-02-09
- Ng, Tarski
- 1977
(Show Context)
Citation Context ...ebras. 1 Introduction Kleene algebras are algebraic structures with operators +, \Delta, , 0, and 1 satisfying certain axioms. They arise in various guises in a number of settings: relational algebra =-=[22, 23]-=-, semantics and logics of programs [14, 24], automata and formal language theory [18, 19], and the design and analysis of algorithms [1, 21, 12]. An important example of a Kleene algebra is Reg \Sigma... |

7 |
and Zoltán Ésik. Equational axioms for regular sets
- Bloom
- 1993
(Show Context)
Citation Context ...the presence of a converse operator. Schematic equational axiomatizations for the algebra of regular events, necessarily representing infinitely many equations, have been given by Krob [17] and Bloom =-=[6]-=-. There is some disagreement regarding the definition of Kleene algebras [8, 24, 14]. The literature contains several inequivalent definitions. In this paper we introduce yet another: a Kleene algebra... |

7 |
Completeness of calculi for axiomatically defined classes of algebras, Algebra Universalis 2
- Selman
- 1972
(Show Context)
Citation Context ...tem for the universal Horn theory of Kleene algebras (the set of universally quantified equational implications ff 1 = fi 1s\Delta \Delta \Deltasff n = fi n ! ff = fi (1) true in all Kleene algebras) =-=[30]-=-. The main result of this paper is that this deductive system is complete for the algebra of regular events. In other words, two regular expressions ff and fi over \Sigma denote the same regular set i... |

5 |
On the algebra of regular expressions
- Anderaa
- 1965
(Show Context)
Citation Context ...a (a) = fag ; a 2 \Sigma : This is called the standard interpretation. Salomaa [28] presented two axiomatizations F 1 and F 2 for the algebra of regular events and proved their completeness. Aanderaa =-=[2]-=- independently presented a system similar to Salomaa's F 1 . Backhouse [4] gave an algebraic version of F 1 . These systems are equational except for one rule of inference in each case that is sound u... |

4 |
A new finite complete solvable quasiequational calculus for algebra of regular languages
- Archangelsky
- 1992
(Show Context)
Citation Context ...c finite automaton to obtain completeness. Conway states a similar theorem without proof in the latter part of his book [8, Theorem 5, p. 108]. Krob [17], based on work of Boffa [7], and Archangelsky =-=[3]-=- have recently independently obtained similar results based on different techniques. 1.1 Examples of Kleene Algebras Kleene algebras abound in computer science. We have already mentioned the regular s... |

4 |
Kleene's theorem revisited: A formal path from Kleene to Chomsky
- Sakarovitch
- 1987
(Show Context)
Citation Context ...y implicit in the work of Kleene [13] and appears in Conway's monograph [8]; the algebraic approach to the elimination of ffl-transitions appears in the work of Kuich and Salomaa [19] and Sakarovitch =-=[27]-=-; and the results on the closure of Kleene algebras under the formation of matrices essentially go back to Conway's monograph [8] and the thesis of Backhouse [4]. We extend this program by showing how... |

4 |
On the complexity of!-automata
- Safra
- 1988
(Show Context)
Citation Context ...B PP T v = u T B v = : 2 6 Open Problems An intriguing question is whether the techniques developed here can be extended to automata on in nite objects. An algebraic treatment of Safra's construction =-=[26]-=- might conceivably be used to establish completeness of the propositional -calculus. Progress toward this goal has recently been made by Walukiewicz [31]. Another question is whether the axioms presen... |

3 |
Elements of the Theoryof Computation
- Lewis, Papadimitriou
- 1981
(Show Context)
Citation Context ...f the regular languages over an alphabet . The relationship between these two formalisms forms the basis of a well-developed classical theory. Classical developments range from the more combinatorial =-=[20, 11]-=- to the more algebraic [29, 10, 5, 9, 27]. The approach taken in this paper must ultimately be attributed to Conway [8]. 15In this section we de ne the notion of an automaton over an arbitrary Kleene... |

3 | On de ning relations for the algebra of regular events - Redko - 1964 |

1 |
Completeness result for the propositional ��-calculus
- Walukiewicz
- 1993
(Show Context)
Citation Context ... algebraic treatment of Safra's construction [26] might conceivably be used to establish completeness of the propositional ��-calculus. Progress toward this goal has recently been made by Walukiew=-=icz [31]-=-. Another question is whether the axioms presented in x2 are complete for the universal Horn theory of the *-continuous Kleene algebras. Acknowledgements Roland Backhouse, John Horton Conway, Merrick ... |

1 |
A new nite complete solvable quasiequational calculus for algebra of regular languages
- Archangelsky
- 1992
(Show Context)
Citation Context ...stic nite automaton to obtain completeness. Conway states a similar theorem without proof in the latter part of his book [8, Theorem 5, p. 108]. Krob [17], based on work of Bo a [7], and Archangelsky =-=[3]-=- have recently independently obtained similar results based on di erent techniques. 1.1 Examples of Kleene Algebras Kleene algebras abound in computer science. We have already mentioned the regular se... |

1 |
Une remarque sur les systemes complets d'identites rationnelles. Informatique theoretique et
- a
- 1990
(Show Context)
Citation Context ... the minimal deterministic nite automaton to obtain completeness. Conway states a similar theorem without proof in the latter part of his book [8, Theorem 5, p. 108]. Krob [17], based on work of Bo a =-=[7]-=-, and Archangelsky [3] have recently independently obtained similar results based on di erent techniques. 1.1 Examples of Kleene Algebras Kleene algebras abound in computer science. We have already me... |

1 |
Completeness of calculi for axiomatically de ned classes of algebras
- Selman
- 1972
(Show Context)
Citation Context ... constitute a complete deductive system for the universal Horn theory of Kleene algebras (the set of universally quanti ed equational implications 1 = 1 ^ ^ n = n ! = (1) true in all Kleene algebras) =-=[30]-=-. The main result of this paper is that this deductive system is complete for the algebra of regular events. In other words, two regular expressions and over denote the same regular set in Reg if and ... |