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## Rewriting Extended Regular Expressions (1993)

Citations: | 24 - 1 self |

### Citations

13831 |
Computers and Intractability -- A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ...the product of the numbers of dissimilar derivatives corresponding to a and b). This is not surprising since the problem of non-equivalence of two ground REGexpressions is known to be PSPACE-complete =-=[9]-=-. On many particular examples, however, the system TR produces respectably short inferences. The use of the rewriting system SIM for simplifying regular expressions may be crucial here. Consider for e... |

815 | Rewrite Systems - Dershowitz, Jouannaud - 1990 |

295 | Derivatives of regular expressions
- Brzozowski
- 1964
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Citation Context ...representation in (18) and to provide a constructive procedure for calculating linear forms. One possible way to do this is to introduce special operations on the term algebra T & such as derivatives =-=[5, 7]-=- (also called left quotients [6] and left residuals [22]), but this would involve new equations to define these operations and would make the inference system more complicated. We prefer to resolve th... |

242 | A completeness theorem for kleene algebras and the algebra of regular events
- Kozen
- 1994
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Citation Context ... meet in order to get a complete axiomatization of the ground equational theory of Reg & [A]. 3.1 Axioms for Reg Several different axiomatizations of the algebra of regular events have been suggested =-=[25, 7, 15, 13, 4, 18, 16]-=-. Some of these are finitary, some involve infinite sets of identities presented by finite numbers of schemes. (A deep analysis of this latter kind of systems of identities has been presented by Conwa... |

230 | Order sorted algebra I. Equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science
- Goguen, Meseguer
- 1992
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Citation Context ...y is replaced by an equational implication involving the meet operation. A new collection of equations then characterizes the meet operation. Our axiomatization exploits order-sorted equational logic =-=[11]-=- by introducing A as a subsort of the sort of all regular events. In Sect. 4 we develop some term-rewriting techniques which lead to a simple and practical algebraic calculus for proving/disproving eq... |

122 |
Introducing OBJ3
- Goguen, Winkler
- 1988
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Citation Context ...oving equations between extended regular expressions -- avoiding construction of finite automata. The calculus is based on several rewrite systems; we have used the algebraic programming language OBJ =-=[12]-=- to implement and to experiment with these. We provide some examples of inferences obtained with the help of this calculus. Finally, in Sect. 5 we review a large amount of related work, and consider p... |

118 | Finite automata
- Perrin
- 1990
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Citation Context ...dure for calculating linear forms. One possible way to do this is to introduce special operations on the term algebra T & such as derivatives [5, 7] (also called left quotients [6] and left residuals =-=[22]-=-), but this would involve new equations to define these operations and would make the inference system more complicated. We prefer to resolve the problem by providing a particular strategy for applyin... |

96 |
Two Complete Axiom Systems for the Algebra of Regular Events
- Salomaa
- 1966
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Citation Context ...ts completeness for the ground equational theory of the algebra. The axiomatization is finite when A is finite. The axioms concerning the usual algebra of regular events are based on Salomaa's system =-=[25]-=-, but the inference rule depending on the negation of the empty word property is replaced by an equational implication involving the meet operation. A new collection of equations then characterizes th... |

64 | Action logic and pure induction
- Pratt
- 1990
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Citation Context ...depending on a particular alphabet and referring to the negation of e.w.p. (see also Salomaa [26]), the other one (developed by Tixier in his thesis) getting rid of e.w.p. through intersection. Pratt =-=[23]-=- considered action algebras equipped with residuations (left and right) and gave a finite equationalsaxiomatization (in the enriched signature) of the equational theory of the variety of action algebr... |

60 |
P.: Complexity of matching problems
- Benanav, Kapur, et al.
- 1985
(Show Context)
Citation Context ... regular expressions, and TR for reducing sets of equations. All these are modulo associativity /commutativity and thus based on the corresponding matching algorithm, which is known to be NP-complete =-=[2]-=-. Therefore, the most adequate complexity measure seems to be the length of inferences providing by the system TR. An analysis of the proof of Thm. 10 shows that in the worst case the length of the in... |

51 |
Unified algebras and institutions
- Mosses
- 1989
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Citation Context ...with only monotonic operations. The latter property is crucial for some applications, for instance in the algebraic specification of abstract data types in the framework of so-called unified algebras =-=[20]-=-, where sorts of values are themselves treated algebraically as values. Such specifications are used in action semantic descriptions of programming languages [21]; our extended algebra of regular even... |

44 |
On equations for regular languages, finite automata, and sequential networks
- Brzozowski, Leiss
- 1980
(Show Context)
Citation Context ...de a constructive procedure for calculating linear forms. One possible way to do this is to introduce special operations on the term algebra T & such as derivatives [5, 7] (also called left quotients =-=[6]-=- and left residuals [22]), but this would involve new equations to define these operations and would make the inference system more complicated. We prefer to resolve the problem by providing a particu... |

33 |
On defining relations for the algebra of regular events
- Redko
- 1964
(Show Context)
Citation Context ...ing family of cyclic identities C k : a = (a k ) ( + a + a 2 + : : : + a k\Gamma1 ) (38) for all k ? 0, forms a set of equations in Reg which is not derivable from any finite set of equational axioms =-=[24, 7]-=-. Consider the inference of C 3 produced by TR: i S i H i (R i ) 1: a = (a 3 ) ( + a + a 2 ) true (SPL) 2: a = a(a(a 3 ) ( + a + a 2 ) + ) +sS 1 (SPL) 3: a = a(a 3 ) ( + a + a 2 ) +sH 2sS 2 (SPL) 4: a... |

26 |
Intersection and union of regular languages and state complexity
- Birget
- 1992
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Citation Context ...e suggested leads to a better average-case complexity algorithm than known ones constructing automata to solve the word problems in Reg and Reg & ? This possibility is supported by a result of Birget =-=[3], proving that -=-the size of a minimal DFA may increase exponentially for both sums A 1 + A 2 + ::: + A k and intersections A 1 " A 2 " ::: " A k of minimal DFA A i of the same size n (i.e., the resulti... |

26 |
Action Semantics, Cambridge Tracts in Theoretical Computer Science 26
- Mosses
- 1992
(Show Context)
Citation Context ...amework of so-called unified algebras [20], where sorts of values are themselves treated algebraically as values. Such specifications are used in action semantic descriptions of programming languages =-=[21]-=-; our extended algebra of regular events has been found to be particularly appropriate in connection with the description of various operations on semantic entities, as well as with that of abstract s... |

20 |
Regular Algebra and Finite Machines (Chapman and
- Conway
- 1971
(Show Context)
Citation Context ... meet in order to get a complete axiomatization of the ground equational theory of Reg & [A]. 3.1 Axioms for Reg Several different axiomatizations of the algebra of regular events have been suggested =-=[25, 7, 15, 13, 4, 18, 16]-=-. Some of these are finitary, some involve infinite sets of identities presented by finite numbers of schemes. (A deep analysis of this latter kind of systems of identities has been presented by Conwa... |

20 | On induction vs. ∗-continuity
- Kozen
- 1982
(Show Context)
Citation Context ... meet in order to get a complete axiomatization of the ground equational theory of Reg & [A]. 3.1 Axioms for Reg Several different axiomatizations of the algebra of regular events have been suggested =-=[25, 7, 15, 13, 4, 18, 16]-=-. Some of these are finitary, some involve infinite sets of identities presented by finite numbers of schemes. (A deep analysis of this latter kind of systems of identities has been presented by Conwa... |

20 |
Theory of Automata, Pergamon
- Salomaa
- 1969
(Show Context)
Citation Context ... -term t can be represented in the following form: t = o(t) + X x2Hd(l) x \Delta r x (18) where o(t) denotes the "constant part" of t, which is equal to either ; or . (This recalls a classic=-=al result [25, 27, 26]-=- that every (extended) regular expression can be equationally characterized; note however that we do not use the sum over the whole alphabet A, which may be infinite!) Yet the representation is not un... |

14 | On action algebras
- Kozen
- 1994
(Show Context)
Citation Context ...ization (in the enriched signature) of the equational theory of the variety of action algebras (which conservatively extends the ground equational theory of Reg[A] on the countable alphabet A). Kozen =-=[17]-=- extended the above with intersection to obtain action lattices, but his axioms do not axiomatize GEq (Reg & [A]). ffl Using order-sorted algebras: In the present paper we exploit the rather natural i... |

12 |
Une remarque sur les systèmes complets d’identités rationnelles
- Boffa
- 1990
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12 |
A procedure for checking equality of regular expressions
- Ginzburg
- 1967
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Citation Context ...following work: ffl Using derivatives: Brzozowski [5] and Conway [7] showed how to use derivatives to carry on some calculations in Reg, also when extended with meet and complement. See also Ginzburg =-=[10] for "mechanica-=-l methods" for proving equivalence in Reg. ffl Calculating normal forms: Johansen [14] provided "algebraic normal forms" for Reg (actually, not unique in general). Other papers have dev... |

8 |
Complete systems of B-rational identities. Theoret
- Krob
- 1990
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2 |
Rewriting extended regular expressions (short version
- Antimirov, Mosses
- 1994
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Citation Context ...provements and extensions of our results. We also discuss a possible complexity A short version of this paper appeared in the Proceedings of Developments in Language Theory, Univ. of Turku, July 1993 =-=[1]-=-. y Current address: INRIA-Lorraine/CRIN, 615, rue du Jardin Botanique, B.P. 101, F--54602 Villers-l`es-Nancy, France z BRICS, Department of Computer Science, University of Aarhus, Ny Munkegade Bldg. ... |

2 |
An Algebraic Normal Form for Regular Events (Polyteknisk Forlag
- Johansen
- 1972
(Show Context)
Citation Context ... carry on some calculations in Reg, also when extended with meet and complement. See also Ginzburg [10] for "mechanical methods" for proving equivalence in Reg. ffl Calculating normal forms:=-= Johansen [14] provided -=-"algebraic normal forms" for Reg (actually, not unique in general). Other papers have developed normal forms for some proper subclasses of Reg (cf. further references given by Johansen [14])... |

2 | Une remarque sur les systemes complets d'identites rationnelles - Boa - 1990 |

1 |
Archangelsky, Conditional equations in an algebra of regular languages, Reports of Ukrainian Academy of Sci., series A 10
- Gorshkov, V
- 1987
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1 |
Generalized language equations with multiple solutions, Theoret
- Leiss
- 1986
(Show Context)
Citation Context ...rther references given by Johansen [14]). ffl Solving systems of equations: Brzozowski and Leiss [6] showed how to do this for linear equations in Reg extended with intersection and complement. Leiss =-=[19]-=- has subsequently demonstrated some advantages that arise in the absence of complement. 15 The method for inferring equations given in the present paper involves several rewrite systems -- LF for calc... |