## Antimirov and Mosses’s Rewrite System Revisited (2008)

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Citations: | 5 - 2 self |

### BibTeX

@MISC{Almeida08antimirovand,

author = {Marco Almeida and Nelma Moreira and Rogério Reis},

title = {Antimirov and Mosses’s Rewrite System Revisited },

year = {2008}

}

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### Abstract

Antimirov and Mosses proposed a rewrite system for deciding the equivalence of two (extended) regular expressions. In this paper we present a functional approach to that method, prove its correctness, and give some experimental comparative results. Besides an improved version of Antimirov and Mosses’s algorithm, we present a version using partial derivatives. Our preliminary results lead to the conclusion that, indeed, these methods are feasible and, generally, faster than the classical methods.

### Citations

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4178 |
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Citation Context ...s and Automata Here we recall some definitions and facts concerning regular languages, regular expressions and finite automata. For further details we refer the reader to the works of Hopcroft et al. =-=[HMU00]-=-, Kozen [Koz97] and Kuich and Salomaa [KS86]. Let Σ be an alphabet and Σ⋆ be the set of all words over Σ. Theempty word is denoted by ɛ and the length of a word w is denoted by |w|. A language is a su... |

320 |
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(Show Context)
Citation Context ...sed for regular language manipulation, regular expressions (re) provide a particularly good notation for their representation. The problem of deciding whether two re are equivalent is PSPACE-complete =-=[SM73]-=-. This is normally solved by transforming each re into an equivalent NFA, convert those automata to equivalent deterministic ones, and finally minimize both DFAs, and decide if the resulting automata ... |

230 | Derivatives of regular expressions
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(Show Context)
Citation Context ...ee grammar (abstract syntax) with initial symbol A ′ ,andA and B are as in G1: A ′ →∅|D D → A | D · B | (D + D). The set of all pre-linear re is denoted by REplin. (G2) 2.3 Derivatives The derivative =-=[Brz64]-=- of a re α with respect to a symbol a ∈ Σ, denoted a−1 (α), is defined recursively on the structure of α as follows: a −1 (∅)=∅; a −1 (α + β)=a −1 (α)+a −1 (β); a −1 (ɛ)=∅; a −1 (αβ)=a −1 (α)β + ε(α)a... |

201 | A completeness theorem for Kleene algebras and the algebra of regular events - Kozen - 1994 |

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Citation Context ... no element of head(α) occurs more than once. We denote the set of all deterministic linear re by REdet. Everyre α can be written as a disjunction of its constant part and a (deterministic) linear re =-=[Sal66]-=-. A re is said to pre-linear if it belongs to the language generated by the following context-free grammar (abstract syntax) with initial symbol A ′ ,andA and B are as in G1: A ′ →∅|D D → A | D · B | ... |

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66 | Partial derivatives of regular expressions and finite automaton constructions
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Citation Context ...α∼ β ⇒ α ′ ∼ β ′ . Theorem 4. The call equiv({(α, β)}, ∅) returns True if and only if α ∼ β.s52 M. Almeida, N. Moreira, and R. Reis 3.3 Improved Equivalence Method Using Partial Derivatives Antimirov =-=[Ant96]-=- introduced the notion of the partial derivatives set of a regular expression α and proved that its cardinality is bounded by the number of alphabetic symbols that occurs in α. He showed that this set... |

46 |
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(Show Context)
Citation Context ..., NFAs, and re represent the same set of languages, i.e., regular languages. 2.1 Succinct Regular Expressions Equivalent re do not need to have the same size. Irreducible re as defined by Ellul et.al =-=[ESW02]-=- have no redundant occurrences of ∅, ɛ, ⋆, and parentheses. A re α is uncollapsible if none of the following conditions hold: – α contains the proper sub-expression ∅,and|α| > 1; – α contains a sub-ex... |

29 |
Follow automata
- Ilie, Yu
(Show Context)
Citation Context ...fairness of the comparison for size size sizesAntimirov and Mosses’s Rewrite System Revisited 55 the method using NFAs, we tried several algorithms for computing (small) NFAs from re (c.f Ilie and Yu =-=[IY03]-=-), but the size of the NFAs seems not to affect significantly the overall performance. 5 Conclusion We presented a variant method based on a rewrite system for testing the equivalence of two re, that ... |

23 | A linear algorithm for testing equivalence of finite automata - Hopcroft, Karp - 1971 |

20 | Rewriting extended regular expressions
- Antimirov, Mosses
- 1994
(Show Context)
Citation Context ...minimize both DFAs, and decide if the resulting automata are isomorphic. The worst case complexity of the automata determinization process is exponential in the number of states. Antimirov and Mosses =-=[AM94]-=- presented a rewrite system for deciding the equivalence of extended re based on a new complete axiomatization of the extended algebra of regular sets. This axiomatization, or any other classical comp... |

16 | Enumeration and generation with a string automata representation - Almeida, Moreira, et al. |

15 |
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- 1994
(Show Context)
Citation Context ...ude thetimenecessarytoparseeachre. Each batch contains re of size 10, 50 or 100, with either 2, 5 or 10 symbols. For the uniform generation of random re we implemented the method described by Mairson =-=[Mai94]-=- for the generation of context-free languages. We used a grammar for almost irreducible re presented by Shallit [Sha04]. As the data sets were obtained with a uniform random generator, the size of eac... |

14 | Enumerating regular expressions and their languages - LEE, SHALLIT |

9 | On the Representation of Finite Automata
- Reis, Moreira, et al.
- 2005
(Show Context)
Citation Context ...ta to equivalent deterministic ones, and minimize both DFAs. Because, for a given regular language, the minimal DFA is unique up to isomorphism, these can be compared using a canonical representation =-=[RMA05]-=-, and thus checked if L(α)=L(β). In this section, we present two methods to verify the equivalence of two re. The first method is a variant of the rewrite system presented by Antimirov and Mosses [AM9... |

5 | On the performance of automata minimization algorithms
- Almeida, Moreira, et al.
- 2008
(Show Context)
Citation Context ... algorithms: Hopcroft and Brzozowski’s. On one hand, Hopcroft’s algorithm has the best known worst-case running time complexity analysis, O(knlog n). On the other, it is pointed out by Almeida et. al =-=[AMR07]-=- that when minimizing NFAs, Brzozowski’s algorithm has a better practical performance. As for the direct comparison method, we compared both the original rewriting system (AM)and our variant of the al... |

3 | R.: Testing regular languages equivalence - Almeida, Moreira, et al. - 2010 |

2 |
Regular expressions, enumeration and state complexity
- Shallit
- 2005
(Show Context)
Citation Context ...e uniform generation of random re we implemented the method described by Mairson [Mai94] for the generation of context-free languages. We used a grammar for almost irreducible re presented by Shallit =-=[Sha04]-=-. As the data sets were obtained with a uniform random generator, the size of each sample is sufficient to ensure a 95% confidence level within a 1% error margin. It is calculated with the formula n =... |

1 | Testing the equivalence of regular expressions - Almeida, Moreira, et al. - 2007 |