Systemic approach to fuzzy logic formalization for approximate reasoning

Aliev, R. A.; Tserkovny, Alex

doi:10.1016/j.ins.2010.11.021

Cited by 48 publications

(34 citation statements)

References 81 publications

(105 reference statements)

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“… Section 3 gives a brief introduction on proposed fuzzy logic and discusses possible interpretations of fuzzy predicates for extended geometric primitives. The fuzzy logic from [5] is introduced as possible formalism for approximate geometric reasoning with extended objects and based on extended geometric primitives fuzzification of the incidence axioms ( I 1) – ( I 4) is investigated.…”

Section: Axiomatic Geometry and Extended Objectsmentioning

confidence: 99%

“…An approach, aimed at augmenting existent axiomatization of Euclidean geometry with grades of validity for axioms (fuzzification), is also presented in [1–4]. But in contrast with [1–4], where the Lukasiewicz logic was only proposed as the basis for “fuzzification” of axioms and no proofs were presented for both fuzzy predicates and fuzzy axiomatization of incidence geometry, we use fuzzy logic from [5] for all necessary mathematical purposes to fill up above-mentioned “gap.”…”

Section: Introductionmentioning

confidence: 99%

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Fuzzy Logic for Incidence Geometry

Tserkovny

2016

The Scientific World Journal

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The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects “as if they were points.” Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid's first postulate. Fuzzy equivalence relation “extended lines sameness” is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy “degree of indiscernibility” and “discernibility measure” of extended points.

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Section: Axiomatic Geometry and Extended Objectsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fuzzy Logic for Incidence Geometry

Tserkovny

2016

The Scientific World Journal

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“…However, the converse of (ii) may not hold (e.g., [1,3] (i) F is n-increasing if (a) F is 6-increasing (i.e., F is increasing w.r.t. the product order);…”

Section: Remarkmentioning

confidence: 99%

“…The use of t-norms and t-conorms has been considered vital for more flexible and reliable fuzzy logic controllers [62]. From them, it is possible to derive several fuzzy implication functions (e.g., S-implications, R-implications, QL-implications, D-implications, U-Implications, E-implications, H-implications) [2,[6][7][8][9]12,51,69,70,93,94,102,118], which are essential in approximate reasoning and fuzzy control to perform conditionals, to make inferences and also to take charge of the propagation of uncertainty in fuzzy reasonings [3,68].…”

Section: Introductionmentioning

confidence: 99%

“…3 Moreover, since the computational complexity of solving a problem in the interval setting might be significantly larger than solving the corresponding problem with accurate input data [72,61], the use of interval additive generators, which are one-place interval functions, to define interval t-norms and t-conorms (two-place interval functions) helps to overcome such issue.…”

Section: Introductionmentioning

confidence: 99%

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Interval additive generators of interval t-norms and interval t-conorms

Dimuro

Bedregal

Santiago

et al. 2011

Information Sciences

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Fuzzy Logic for Incidence Geometry

Aliev

Tserkovny

2020

Studies in Computational Intelligence

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The paper presents a mathematical framework for approximate geometric reasoning with extended objects in the context of Geography, in which all entities and their relationships are described by human language. These entities could be labelled by commonly used names of landmarks, water areas, and so forth. Unlike single points that are given in Cartesian coordinates, these geographic entities are extended in space and often loosely defined, but people easily perform spatial reasoning with extended geographic objects "as if they were points." Unfortunately, up to date, geographic information systems (GIS) miss the capability of geometric reasoning with extended objects. The aim of the paper is to present a mathematical apparatus for approximate geometric reasoning with extended objects that is usable in GIS. In the paper we discuss the fuzzy logic (Aliev and Tserkovny, 2011) as a reasoning system for geometry of extended objects, as well as a basis for fuzzification of the axioms of incidence geometry. The same fuzzy logic was used for fuzzification of Euclid's first postulate. Fuzzy equivalence relation "extended lines sameness" is introduced. For its approximation we also utilize a fuzzy conditional inference, which is based on proposed fuzzy "degree of indiscernibility" and "discernibility measure" of extended points.

show abstract

Systemic approach to fuzzy logic formalization for approximate reasoning

Cited by 48 publications

References 81 publications

Fuzzy Logic for Incidence Geometry

Fuzzy Logic for Incidence Geometry

Interval additive generators of interval t-norms and interval t-conorms

Fuzzy Logic for Incidence Geometry

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