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Fundamental Properties of Fuzzy Implications

2018
*
Formalized Mathematics
*

We develop a framework

doi:10.2478/forma-2018-0023
fatcat:5ugg7zrwajbopp5gdng4fbssn4
*of*Mizar attributes allowing us for a smooth proving*of*basic*properties**of*these*fuzzy*connectives [9]. ... In the article we continue in the Mizar system [8], [2] the formalization*of**fuzzy**implications*according to the monograph*of*Baczyński and Jayaram "*Fuzzy**Implications*" [1]. ... On the other hand, in the final section I prove the formal characterization*of**fundamental**fuzzy**implications*in terms*of*four elementary*properties*[12] expressed in Table 1 I LK + + + + I GD + + ...##
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Fuzzy Implication Classes Satisfying a Boolean-Like Law
[chapter]

2013
*
Advances in Intelligent Systems and Computing
*

We focus the investigation on the following classes

doi:10.1007/978-3-642-39165-1_41
fatcat:vl5orusj5feb5f3nwky3xlckie
*of**fuzzy**implications*: (S,N)-, R-, QL-, D-, (N,T)-, f -, gand h-*implications*. ...*Properties*that are always true in the classical theory (Boolean laws) have been extended to*fuzzy*theory and so-called Boolean-like laws. ... In*fuzzy*logics, the lack*of*a consensus on Boolean*implication*meaning entails non-equivalent acceptable definitions*of**fuzzy**implications*(see [15, 16, 27, 28, 34] as examples) and classes*of**fuzzy*...##
###
On EQ-Fuzzy Logics with Delta Connective

2011
*
Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)
*

The former is a new kind

doi:10.2991/eusflat.2011.84
dblp:conf/eusflat/DybaN11
fatcat:exx2iitj6beohn5jtrvp5nk5ny
*of*many-valued logic which based on EQ-algebra*of*truth values, i.e. the algebra in which*fuzzy*equality is the*fundamental*operation and*implication*is derived from it. ... First, we extend the EQ-algebra by the ∆ operation and then introduce axioms and inference rules*of*EQ ∆ -logic. We also prove the deduction theorem formulated using*fuzzy*equalities. ... For all a, b ∈ E we put a → b = (a ∧ b) ∼ a (1) and call this operation an*implication*. The following theorem demonstrates the*fundamental**properties**of*the*fuzzy*equality and*implication*. ...##
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On the minimization of incompletely specified fuzzy functions

1974
*
Information and Control
*

and with the mathematical

doi:10.1016/s0019-9958(74)90440-9
fatcat:xzzj2x6rrngjzkxisr4ktqweii
*properties*involved in*fuzzy*logic. ... It is our hope that this new method will stimulate more researchers to look into the algebraic*properties**of**fuzzy*structures. ... Compute the*fuzzy*prime*implicants**of*F, @ F¢ ; 2. Remove the essential*fuzzy*prime*implicants*; 3. Find a minimal complexity cover*of*the remainder. ...##
###
Openings and closures of fuzzy preorderings: theoretical basics and applications to fuzzy rule-based systems

2003
*
International Journal of General Systems
*

Firstly, a general concept

doi:10.1080/0308107031000135026
fatcat:54dhbxddobfj7h4uhitviqhini
*of*closedness*of**fuzzy*sets under*fuzzy*preorderings is proposed and investigated along with the corresponding opening and closure operators. ... The purpose*of*this paper is two-fold. ... In order to provide the reader with the basic*properties**of*residual*implications*, we will now briefly recall them. ...##
###
Fuzzy logic controllers are universal approximators

1995
*
IEEE Transactions on Systems, Man and Cybernetics
*

We try to answer this

doi:10.1109/21.370193
fatcat:4r24y54ebngmtg724xgxlwippy
*fundamental*question by proving that for each fixed*fuzzy*logic belonging to a wide class*of**fuzzy*logics, and for each fixed type*of*membership function belonging to a wide class ... In this paper, we consider a*fundamental*theoretical question, Why does*fuzzy*control have such good performance for a wide variety*of*practical problems?. ... is 3c' = P by: which in the case*of*a t-norm*implication*is the general expression d) The combination*of*all*fuzzy*inferences is made by means*of*T*: iv) The defuzzification method verifies the*property*...##
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On the Paradoxical Success of Mamdani's Minimum-Based Inference
[chapter]

2012
*
Studies in Fuzziness and Soft Computing
*

introducing a simple variant

doi:10.1007/978-3-642-24666-1_19
fatcat:q3pst3ns4rgwxbbo7qhtzdxdda
*of*the Generalized Modus Ponens (GMP) that uses standard*fuzzy**implications*. ... Yet, it is well known that it uses a min-based*implication*that does not verify classical boolean logic requirements. ... standard*fuzzy**implications*. ...##
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Page 8069 of Mathematical Reviews Vol. , Issue 2003k
[page]

2003
*
Mathematical Reviews
*

the set

*of*all infinitely A-distributive*implications*on L in detail and point out that three classes*of**fundamental**fuzzy**implications*and Yager’s*implication*on [0,1] are all infinitely A-distributive ... A series*of**properties**of*subal- gebras and generalized tautologies*of*product logical systems are given. ...##
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Page 1506 of Mathematical Reviews Vol. , Issue 84d
[page]

1984
*
Mathematical Reviews
*

Incidentally, in the diagram on page 281, the

*implication*-does- not-hold arrows ( # ) should point up instead*of*down. S. E. ... The author investigates the question*of*when a space with a*property*# has a countably compact extension with*property*?. ...##
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A short note on fuzzy relational inference systems

2018
*
Fuzzy sets and systems (Print)
*

*fuzzy*relational models

*of*

*fuzzy*rule bases, namely, the Mamdani-Asilian and the

*implicative*models. ... This paper is a short note contribution to the topic

*of*

*fuzzy*relational inference (FRI) systems and the preservation

*of*their desirable

*properties*. ... Introduction The notion

*of*

*fuzzy*inference systems (or

*fuzzy*rule based systems) denotes systems incorporating the

*fundamental*idea

*of*Zadeh to imitate the human way

*of*thinking based on

*fuzzy*rules, see ...

##
###
Formalization of Human Categorization Process Using Interpolative Boolean Algebra

2015
*
Mathematical Problems in Engineering
*

Such relations are directly constructed from a Boolean consistent

doi:10.1155/2015/620797
fatcat:6m5b2rdvqzemnliwf43wm3hguy
*fuzzy*partial order relation, which is treated by Boolean*implication*. ... The introduced formalism secures the principles*of*categorization showing that Boolean laws are*fundamental*in the categorization process. ... Conflict*of*Interests The authors declare that there is no conflict*of*interests regarding the publication*of*this paper. ...##
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Rough Approximations under Level Fuzzy Sets
[chapter]

2004
*
Lecture Notes in Computer Science
*

The combination

doi:10.1007/978-3-540-25929-9_8
fatcat:pjfy7s35zfbflasronddrouf4y
*of**fuzzy*set and rough set theories lead to various models. Functional and set approaches are two categories based on different*fuzzy*representations. ... In this paper, we study rough approximations based on the notion*of*level*fuzzy*sets. Two rough approximation models, namely α-level rough set and β-level rough set, are proposed. ... The present study examines some*of*the*fundamental*issues*of*the combination from the perspective*of*level*fuzzy*sets. ...##
###
Formal Introduction to Fuzzy Implications

2017
*
Formalized Mathematics
*

In the article we present in the Mizar system the catalogue

doi:10.1515/forma-2017-0023
fatcat:c47lvntm2rd6tl2vnlrrx3zfnq
*of*nine basic*fuzzy**implications*, used especially in the theory*of**fuzzy*sets. ...*of**fuzzy*logic (together with t-norms and t-conorms, formalized previously). ...*fuzzy**implication*has also*properties**of*a classical*implication*. ...##
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Properties of Fuzzy Implications obtained via the Interval Constructor

2007
*
TEMA
*

This work considers an interval extension

doi:10.5540/tema.2007.08.01.0033
fatcat:a6jlpee7jbbphkz2z4ozl43qqq
*of**fuzzy**implication*based on the best interval representation*of*continuous t-norms. ... Some related*properties*can be naturally extended and that extension preserves the behaviors*of*the*implications*in the interval endpoints. ... Este trabalho considera a extensão intervalar da implicação*fuzzy*baseada no conceito de melhor representação intervalar de t-normas contínuas, previamente introduzido por Bedregal e Takahashi. ...##
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Page 4447 of Mathematical Reviews Vol. , Issue 99g
[page]

1999
*
Mathematical Reviews
*

This paper reveals a

*fundamental**property**of*the mu- calculus that can be viewed as an explanation*of*the determinacy result in purely algebraic terms; it also gives a new proof*of*this result which does ... Burhan (3-TRNT-I; Toronto, ON); Kreinovich, Vladik (1-TXEP-C; El Paso, TX); Yager, Ronald R. (1-IONA-M; New Rochelle, NY) A new class*of**fuzzy**implications*. Axioms*of**fuzzy**implication*revisited. ...
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