## A fuzzy approach to erroneous inputs in context-free language recognition (1995)

Venue: | Dept. of Comp. Sci., Twente University of Technology |

Citations: | 10 - 6 self |

### BibTeX

@INPROCEEDINGS{Asveld95afuzzy,

author = {Peter R. J. Asveld},

title = {A fuzzy approach to erroneous inputs in context-free language recognition},

booktitle = {Dept. of Comp. Sci., Twente University of Technology},

year = {1995},

pages = {14--25}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract − Using fuzzy context-free grammars one can easily describe a finite number of ways to derive incorrect strings together with their degree of correctness. However, in general there is an infinite number of ways to perform a certain task wrongly. In this paper we introduce a generalization of fuzzy context-free grammars, the so-called fuzzy context-free K-grammars, to model the situation of making a finite choice out of an infinity of possible grammatical errors during each context-free derivation step. Under minor assumptions on the parameter K this model happens to be a very general framework to describe correctly as well as erroneously derived sentences by a single generating mechanism. Our first result characterizes the generating capacity of these fuzzy context-free K-grammars. As consequences we obtain: (i) bounds on modeling grammatical errors within the framework of fuzzy context-free grammars, and (ii) the fact that the family of languages generated by fuzzy context-free K-grammars shares closure properties very similar to those of the family of ordinary context-free languages. The second part of the paper is devoted to a few algorithms to recognize fuzzy context-free languages: viz. a variant of a functional version of Cocke−Younger− Kasami’s algorithm and some recursive descent algorithms. These algorithms turn out to be robust in some very elementary sense and they can easily be extended to corresponding parsing algorithms. 1.

### Citations

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(Show Context)
Citation Context ...om [14], before we can apply the algorithms from this section. The first algorithm is a modification of Cocke−Younger−Kasami’s algorithm (or CYK-algorithm for short); cf. Algorithm 5.2 below. In e.g. =-=[1, 11, 12]-=- the CYK-algorithm is given in terms of nested for-loops that fill an upper-triangular matrix. Here we start �− 21 − from an alternative functional version from [5] which has some interesting feature... |

290 |
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(Show Context)
Citation Context ... to be familiar with the rudiments of formal language and parsing theory. So for the definitions of context-free grammar, Chomsky Normal Form, and Greibach Normal Form we refer to standard texts like =-=[1, 11, 12]-=-. As mentioned in §1 a fuzzy language L over an alphabet Σ is a fuzzy subset of Σ ∗ , i.e. L is defined by a degree of membership function µ L : Σ ∗ → [0, 1]. Actually the function µ L is the primary ... |

45 |
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(Show Context)
Citation Context ... the set P ( A) is now a finite fuzzy subset of V ∗ rather than an ordinary, or so-called crisp subset. Fuzzy context-free grammars have been introduced in a slightly different, but equivalent way in =-=[14]-=-. So using fuzzy context-free grammars, now we are able to model the situation in which the use of a single correct rule can be replaced by the application of any out of a finite number of incorrect r... |

26 |
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(Show Context)
Citation Context ...only if x ∈L 0 and µ( x; L 0) = 0 if and only if x ∉L 0. But now the notion of fuzzy set may solve this problem, since a fuzzy language over Σ ∗ is defined in terms of a membership function µ : Σ ∗ → =-=[0, 1]-=-. Note that the two-element set {0, 1} has been replaced by the continuous interval [0, 1] and µ( x; L 0) expresses the degree of membership of the element x with respect to the language L 0. Thus x m... |

16 |
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Citation Context ...n V, the value of P ( α) is a finite language over the alphabet V that contains α. The containment of α in this value allows us to interpret P as a nested finite substitution; a concept introduced in =-=[10]-=- and to be recalled in §2. Let us return to errors and their description. Wrongly applying a rule A → ω, will mean in this paper that an occurrence of A is replaced by an incorrect string ω′ instead o... |

14 | Towards robustness in parsing — Fuzzifying context-free language recognition
- Asveld
(Show Context)
Citation Context ...5, 13], and a straightforward generalization of our results from previous sections. In the next sections emphasis is on the main ideas and on concrete examples; for detailed formal proofs we refer to =-=[6, 7, 8]-=-.− 16 − 2. Preliminaries on Fuzzy Languages We assume the reader to be familiar with the rudiments of formal language and parsing theory. So for the definitions of context-free grammar, Chomsky Norma... |

13 |
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Citation Context ...language theory follows: closure under union, concatenation, Kleene ∗, homomorphism, inverse homomorphism, substitution, nondeterministic finitestate transductions, and so on; cf. [10, 2, 4] and also =-=[17]-=-. Theorem 4.6. [7] (1) Let K be a nontrivial family of fuzzy languages closed under finite fuzzy substitution and under intersection with fuzzy regular languages. Then Af( K) is a full super-AFFL, and... |

11 |
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Citation Context ...cing the finite fuzzy sets P ( α) (for each α in V) by infinite ones will not work, since in that case the language L (G) generated by the resulting grammar G might not even be recursively enumerable =-=[9]-=-. Thus we have to restrain the languages P ( A) in some, preferably natural way. The method we use here, originates from [16]; viz. we assume that a family K of fuzzy languages is given in advance, fr... |

9 |
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(Show Context)
Citation Context ...i.e. µ( x; L (G 0)) = µ( x : L (G 1)) for all x in Σ∗ . � Crisp grammars with an infinite number of rules have been considered previously; e.g. grammars in extended BNF and the grammatical devices in =-=[16, 2, 3]-=-. In the next definition we generalize the fuzzy context-free grammars from [14]. Definition 3.4. Let K be a family of fuzzy languages. A fuzzy context-free K-grammar G = (V, Σ, P, S) consists of � an... |

7 |
The utilization of fuzzy sets in the recognition of imperfect strings. Fuzzy Sets and Systems
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- 1992
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Citation Context ... and “capital blunders”. Our approach in describing grammatical errors has a global character: a righthand side ω of a grammar rule may be replaced erroneously by a completely different string ω′. In =-=[15]-=- an alternative way of describing errors — using fuzzy context-free grammars too — is given. Here ω′ is restricted to those strings that are obtainable by simple edit operations (deletion, insertion, ... |

4 |
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- 1974
(Show Context)
Citation Context ...nerated by the resulting grammar G might not even be recursively enumerable [9]. Thus we have to restrain the languages P ( A) in some, preferably natural way. The method we use here, originates from =-=[16]-=-; viz. we assume that a family K of fuzzy languages is given in advance, from which we are allowed to take whatever languages we think to be appropriate. Then replacing the finite languages P ( A) ove... |

3 |
Asveld: An algebraic approach to incomparable families of formal languages
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- 1992
(Show Context)
Citation Context ...ell known in formal language theory follows: closure under union, concatenation, Kleene ∗, homomorphism, inverse homomorphism, substitution, nondeterministic finitestate transductions, and so on; cf. =-=[10, 2, 4]-=- and also [17]. Theorem 4.6. [7] (1) Let K be a nontrivial family of fuzzy languages closed under finite fuzzy substitution and under intersection with fuzzy regular languages. Then Af( K) is a full s... |

3 | An alternative formulation of Cocke−Younger−Kasami’s algorithm
- Asveld
- 1994
(Show Context)
Citation Context ...gorithm 5.2 below. In e.g. [1, 11, 12] the CYK-algorithm is given in terms of nested for-loops that fill an upper-triangular matrix. Here we start �− 21 − from an alternative functional version from =-=[5]-=- which has some interesting features: it omits implementation details like the data structure, reference to the indices of matrix entries and to the length of the input string; cf., e.g., Algorithm 12... |

3 |
Fuzzy context-free languages — Part II: Recognition Algorithms
- Asveld
(Show Context)
Citation Context ...5, 13], and a straightforward generalization of our results from previous sections. In the next sections emphasis is on the main ideas and on concrete examples; for detailed formal proofs we refer to =-=[6, 7, 8]-=-.− 16 − 2. Preliminaries on Fuzzy Languages We assume the reader to be familiar with the rudiments of formal language and parsing theory. So for the definitions of context-free grammar, Chomsky Norma... |

3 |
The recognition of imperfect strings generated by fuzzy context-sensitive grammars, Fuzzy Sets and Systems 62
- Inui, Shoaff, et al.
- 1994
(Show Context)
Citation Context ...ations of Cock-Younger-Kasami’s algorithm and of some recursive descent algorithms. Finally, §6 contains a comparison with an alternative way of describing grammatical errors using fuzzy grammars too =-=[15, 13]-=-, and a straightforward generalization of our results from previous sections. In the next sections emphasis is on the main ideas and on concrete examples; for detailed formal proofs we refer to [6, 7,... |

1 |
Asveld: Abstract grammars based on transductions, Theoret
- J
- 1991
(Show Context)
Citation Context ...i.e. µ( x; L (G 0)) = µ( x : L (G 1)) for all x in Σ∗ . � Crisp grammars with an infinite number of rules have been considered previously; e.g. grammars in extended BNF and the grammatical devices in =-=[16, 2, 3]-=-. In the next definition we generalize the fuzzy context-free grammars from [14]. Definition 3.4. Let K be a family of fuzzy languages. A fuzzy context-free K-grammar G = (V, Σ, P, S) consists of � an... |