## On approximate reasoning with graded rules

Venue: | Fuzzy Sets and Systems |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Daňková_onapproximate,

author = {Martina Daňková},

title = {On approximate reasoning with graded rules},

journal = {Fuzzy Sets and Systems},

year = {},

volume = {158},

pages = {2007}

}

### OpenURL

### Abstract

This contribution presents a comprehensive view on problems of approximate reasoning with imprecise knowledge in the form of a collection of fuzzy IF-THEN rules formalized by approximating formulas of a special type. Two alternatives that follow from the dual character of approximating formulas are developed in parallel. The link to the theory of fuzzy control systems is also explained.

### Citations

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(Show Context)
Citation Context ... non-precise input knowledge. A generalized rule of modus ponens as a particular case of compositional rule of inference in the global concept of many-valued logics has been introduced by L. Zadeh in =-=[27]-=-. The analysis of logical aspects of Zadeh’s compositional rules of inference was done by P. Hájek in [17] or V. Novák in [19–21] (for evaluated syntax). From the other works let us mention e.g. [23,2... |

446 |
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Citation Context ... question in Section 2.2. Since the main purpose of this work is to follow the results of Perfilieva in [22] (extended in [10]), it is desirable to work in the same logical framework that is BL-logic =-=[17]-=-. Unfortunately in this setting, we are not generally able to introduce the strong disjunction keeping the duality w.r.t. the strong conjunction, because the law of double negation does not hold there... |

160 |
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(Show Context)
Citation Context ...th respect to the structure of R. In this example, we take f = maxi∈I ωi and we obtain the graded rule 〈at most 〉f /R. Finally, approximate inferences that mimic approximate reasoning in the sense of =-=[26]-=- and cope with graded rules and additional premise A ∗ will be introduced. Inference with graded GradedDisK is based on Zadeh’s compositional rule of inference that allows us to derive a consequence o... |

62 | A treatise on many-valued logics
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(Show Context)
Citation Context ...AND (IF [NOT Ak] THEN Bk). In this logic, ConK and ConK2 are equivalent and, in addition, they are dual to DisK, which is not generally the case in other logics (e.g. monoidal t-norm based logics see =-=[15]-=- for the overview). We will return to this question in Section 2.2. Since the main purpose of this work is to follow the results of Perfilieva in [22] (extended in [10]), it is desirable to work in th... |

60 |
Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions
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(Show Context)
Citation Context ...”, and the degrees 2sf1, . . . , fk belong to some structure for the truth values. In the sequel, GradedDisK and GradedConK will be called graded rules. Relating to this topic, the work introduced in =-=[12]-=- and [11] (a brief overview can be found in [14]) may be of the particular interest. Their authors distinguish between gradual rules, certainty rules and their mixture. The gradualness is connected wi... |

53 |
Fuzzy Relational Systems: Foundations and Principles
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(Show Context)
Citation Context ...ed syntax). From the other works let us mention e.g. [23,25,13]. Inference rules were also intensively studied from the algebraical point of view as special operations called compositions (see, e.g., =-=[18,14,4,6]-=-). In the sequel, we will assume JF C = Jk ∪ {A ∗ }, where A ∗ is a predicate of the type 〈s1, . . . , sp〉. Let us suppose n > 1, p ∈ N such that 1 ≤ p ≤ n − 1, and ¯xp = [x1, . . . , xp], ¯yp = [xp+1... |

45 |
Fuzzy sets in approximate reasoning, part 2: Logical approaches. Fuzzy Sets and Systems
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(Show Context)
Citation Context ...e degrees 2sf1, . . . , fk belong to some structure for the truth values. In the sequel, GradedDisK and GradedConK will be called graded rules. Relating to this topic, the work introduced in [12] and =-=[11]-=- (a brief overview can be found in [14]) may be of the particular interest. Their authors distinguish between gradual rules, certainty rules and their mixture. The gradualness is connected with the pr... |

41 |
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(Show Context)
Citation Context ...roughout the whole paper and should help the reader with an orientation. Remark 2 How to read the formulae? In the following, we will prove so called graded theorems (for the first time introduced in =-=[16]-=- as stated in [3]). It means that instead of the usual formulation we are going to find n ∈ N for which If ⊢ ϕ then ⊢ ψ, ⊢ ϕ n → ψ. Whenever we do know ⊢ ϕ n → ψ and ⊢ ϕ then we can derive ⊢ ψ easily,... |

26 | A logical approach to interpolation based on similarity relations, Internat
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(Show Context)
Citation Context ... [27]. The analysis of logical aspects of Zadeh’s compositional rules of inference was done by P. Hájek in [17] or V. Novák in [19–21] (for evaluated syntax). From the other works let us mention e.g. =-=[23,25,13]-=-. Inference rules were also intensively studied from the algebraical point of view as special operations called compositions (see, e.g., [18,14,4,6]). In the sequel, we will assume JF C = Jk ∪ {A ∗ },... |

19 |
From fuzzy logic to fuzzy mathematics: A methodological manifesto. Fuzzy Sets and Systems
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(Show Context)
Citation Context ...he results from [10] by adapting Hájek’s results to the case of graded GradedDisK and we build the dual approach for graded GradedConK in parallel. Moreover, the whole methodology is in the spirit of =-=[3]-=-, which brings the generalization to the results of [10] that create part of Section 3.1. The way of generalization is explained in Remark 2. 2 Preliminaries 2.1 Many-sorted fuzzy predicate logic(BL∀)... |

18 |
Special properties, closures and interiors of crisp and fuzzy relations,” Fuzzy Sets and Systems
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(Show Context)
Citation Context ...¯x)), (42) B ∗ CNF(¯yp) ≡df (∀ ¯xp)(A ∗ (¯xp) → CNFF,k(¯x)), (43) which define B ∗ DNF and B ∗ CNF from A ∗ using the Zadeh’s compositional rule of inference and Bandler-Kouhout’s product (BK-product =-=[2]-=-), respectively. Since BK-product can be viewed as a dual to Zadeh’s composition, and moreover, CNFF,k is dual to DNFF,k, we conclude that also B ∗ CNF is in a certain sense dual to B ∗ DNF. Let us de... |

17 | The relation between inference and interpolation in the framework of fuzzy systems - Klawonn, Novák - 1996 |

8 |
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Citation Context ...limit case, it coincide with F whenever ¯ R is reflexive (57), (58). Note that we can find semantical proofs (not in graded form and under additional conditions required from ¯ R) of (57) and (58) in =-=[5]-=- or [4]. It follows from (55) and (56), that we can efficiently approximate only extensional formulae. Below, we show some examples. Example 6 Let us consider a set D = { ¯ di| i ∈ J} of examples. The... |

7 |
The ̷LΠ and ̷LΠ 1 2 propositional and predicate logics, Fuzzy Sets and Systems
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Citation Context ...e strong disjunction is dual to strong conjunction and is definable by means of negation, as is the case of logics with the law of double negation such as ̷Lukasiewicz logic, S-logic [7], IMTL or ̷LΠ =-=[9]-=-. Assuming, e.g. ̷Lukasiewicz logic with the strong conjunction & and negation ¬, where the strong disjunction ∇ is introduced as ϕ∇ψ ≡df ¬((¬ϕ)&(¬ψ)), we obtain (ϕ∇ψ) ↔ (¬ϕ → ψ). Hence, we can rewrit... |

6 |
Fuzzy logic, control engineering and artificial intelligence,” in Fuzzy Algorithms for control
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(Show Context)
Citation Context ...me structure for the truth values. In the sequel, GradedDisK and GradedConK will be called graded rules. Relating to this topic, the work introduced in [12] and [11] (a brief overview can be found in =-=[14]-=-) may be of the particular interest. Their authors distinguish between gradual rules, certainty rules and their mixture. The gradualness is connected with the properties of implication, certainty with... |

6 | On the logical basis of approximate reasoning - Novák - 1992 |

4 |
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Citation Context ...se the set of nodes {¯yi}i∈I, hence we obtain expressions relating to DNFF,k and CNFF,k. (2) Notice that DNF ∃ F and CNF ∀ F are known in literature as image and preimage of F in relation R, see e.g. =-=[24]-=-. Here, these formulae can be viewed as some kind of “limit” case of DNFF,k and CNFF,k, respectively, when we proceed over the all elements of the universe relating to the respective realization. Let ... |

4 |
Formalized theory of general fuzzy reasoning
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- 2004
(Show Context)
Citation Context ... [27]. The analysis of logical aspects of Zadeh’s compositional rules of inference was done by P. Hájek in [17] or V. Novák in [19–21] (for evaluated syntax). From the other works let us mention e.g. =-=[23,25,13]-=-. Inference rules were also intensively studied from the algebraical point of view as special operations called compositions (see, e.g., [18,14,4,6]). In the sequel, we will assume JF C = Jk ∪ {A ∗ },... |

3 |
Babuˇska: “Fuzzy control versus conventional control.” In
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Citation Context ... (25), i.e. the formulae in the usual sense. Let us consider a very simple problem to illustrate the way of using normal forms to formalize gradual rules. Example 4 Let L be standard product algebra 〈=-=[0, 1]-=-, ∩, ∪, ⊙, →⊙, 0, 1〉 and M = 〈M = [0.1, 0.9], f, r, 0.25, 0.75〉 be an L-structure for the language J2, where f, r, 0.25, 0.75 interpret F, R1, c1, c2, respectively. Let f be a fuzzy relation on M depi... |

2 | Logical structure of fuzzy - Novák, Lehmke - 2006 |

1 |
Extensionality based approximate reasoning, Int
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- 1998
(Show Context)
Citation Context ...ed syntax). From the other works let us mention e.g. [23,25,13]. Inference rules were also intensively studied from the algebraical point of view as special operations called compositions (see, e.g., =-=[18,14,4,6]-=-). In the sequel, we will assume JF C = Jk ∪ {A ∗ }, where A ∗ is a predicate of the type 〈s1, . . . , sp〉. Let us suppose n > 1, p ∈ N such that 1 ≤ p ≤ n − 1, and ¯xp = [x1, . . . , xp], ¯yp = [xp+1... |

1 |
On triangular norm based propositional fuzzy logics, Fuzzy Sets and Systems 69
- Butnariu, Klement, et al.
- 1995
(Show Context)
Citation Context ...a logic, where the strong disjunction is dual to strong conjunction and is definable by means of negation, as is the case of logics with the law of double negation such as ̷Lukasiewicz logic, S-logic =-=[7]-=-, IMTL or ̷LΠ [9]. Assuming, e.g. ̷Lukasiewicz logic with the strong conjunction & and negation ¬, where the strong disjunction ∇ is introduced as ϕ∇ψ ≡df ¬((¬ϕ)&(¬ψ)), we obtain (ϕ∇ψ) ↔ (¬ϕ → ψ). Hen... |

1 | Basics of A Formal Theory of Fuzzy Partitions
- Cintula
(Show Context)
Citation Context ...ut and output space) of both normal forms expressed by Ck, (see (48)), which we check over the whole universe of the respective model. For the recent advances in formal theory of fuzzy partitions see =-=[8]-=-. Involving the extensionality property allows us to prove the following formulae. Theorem 7 proof: Ext ¯ R F → DNFF,k ⊆ F, (53) Ext ¯ R F → F ⊆ CNFF,k, (54) Ext ¯ R F &Ck → DNFF,k ≈ F, (55) Ext ¯ R F... |