# The Auto-Diagnosis of Granulation of Information Retrieval on the Web

## Abstract

**:**

## 1. Introduction

- Determining the auto-diagnosis of the concept ‘I’ for the auto-diagnosis agent,
- Whether the frequency of auto-diagnosis cycles (frequency of updating the auto-diagnosis history) should be consistent with the frequency of 40 Hz of the thalamus–cortex cycles,
- Whether the frequency of synchronization of auto-diagnosis cycles with the stability of auto-diagnosis history should correspond to brain waves: alpha (8–12 Hz), beta (above 12 Hz), theta (4–8 Hz), delta (0.5–4 Hz).

**thesaurus**[2,3]. Thus, object perception determines the weight, rank, and importance of object descriptions representing information about this object. This also applies to sources of information about objects that point to these objects, which are called, in computer science entities, signs of objects, which are different from their descriptions. Each such reference is called the

**information granule**[3,4] and its occurrence is called

**data**about the object.

**granular calculations**are made, i.e., the information about the objects is interpreted. For precisely established knowledge, granules are data sets. When an incorrect classification of objects is used to represent knowledge, granules cannot be described by abstract data sets. In this situation, the following non-standard formalities of set theory [5] are proposed for determining information granules: interval analysis [6], fuzzy sets [7,8,9], rough sets [10,11,12,13,14] and shadowed sets [5,15,16]. In the indicated papers, as well as in other papers on granular calculations, the theory of information granules was missing. The description of the information granules system was specified in [1,2]. The paper [2] also shows the structure of inducing the granule system by fuzzy algebra as a perception system. In this paper, a postulation of the relationship between the structure of the brain’s neural network memory and the semantic web is continued [1,17].

## 2. Conceiving of Assertions on the Web

**namespace**is introduced. There are standard namespaces for RDF: rdfs, rdf, defining the basic RDF classes:

- rdfs: Resource—class containing all resources,
- rdfs: Class—class containing all classes and their instances,
- rdf: Property—class containing all properties,
- and other rdfs: Literal, rdfs: Datatype, rdf: langString, rdf: HTML and rdf: XMLLiteral.

**thesaurus**. Any classes containing instances in which subjects or objects are connected to a certain predicate are called

**concepts**, and predicates are called

**roles**.

**assertions**of concept C or role R, which are written as: $x:C,(x,y):R$ and reads: x is in the concept C, $x,y$ are in the role R.

**Dual concepts**${C}^{d}$ and

**dual roles**${R}^{d}$ are defined by formulas:

**atomic**assertions. Atomic formulas are formulas of the assertion language.

**The assertion formulas**are any substitution of the sentence calculus schemes with atomic assertions: e.g., if, in the scheme $\left(\right(p\Rightarrow q)\wedge p)\Rightarrow q$, p will be substitute by the atomic assertion $x:C$, and instead of q, will be $(x,y):R$, then the formula for these assertions with the description $\varphi $ is given by the formula $(x,y):\varphi {=}_{df}((x:C\Rightarrow (x,y):R)\wedge x:C)\Rightarrow (x,y):R$ where $(x,y):\varphi $ is the

**formula assertion**$\varphi $ and it is read as: instances of $x,y$ are instances of the assertion occurring in the formula $\varphi $. The assertion occurrence is coded by assigning it a value of one and no occurrence by a value of zero. The fulfilment of the assertion connectives, such as negation ¬, conjunction ∧, disjunction ∨, implication ⇒ and exclusion ∖ (the assertion does not occur if the second one occurs), is specified in the Table 1. It is similar to propositional calculus.

**concepts of assertion formulas**and the roles between them, called

**connectives of assertion formulas**.

**positive coupling**between these assertion formulas: if one does not belong to the concept ${\mathsf{\Psi}}_{1}$, then second belongs to the concept ${\mathsf{\Psi}}_{2}$.

**negative coupling**, connection between these assertion formulas: it is not true that if one belongs to the concept ${\mathsf{\Psi}}_{1}$, then second belongs to the concept ${\mathsf{\Psi}}_{2}$.

**negative feedback**between these assertion formulas: it is not true that if one belongs to the concept ${\mathsf{\Psi}}_{1}$, then second belongs to the concept ${\mathsf{\Psi}}_{2}$ and otherwise.

**positive feedback**between these assertion formulas: if one does not belong to the concept ${\mathsf{\Psi}}_{1}$, then second belongs to the concept ${\mathsf{\Psi}}_{2}$ and otherwise.

**logical square of these concepts**(Figure 2).

- Positive coupling is satisfied if the consequent is satisfied.
- Negative coupling is satisfied if it is excluded that the antecedent is satisfied, when the consequent is satisfied.
- Positive feedback is satisfied if the consequent is satisfied.
- Negative feedback is satisfied if it is excluded that the antecedent is satisfied, when the consequent is satisfied.

**Theorem**

**1.**

**Proof**

**(Proof**

**of**

**Theorem**

**1).**

**the De Morgan logical square**.

## 3. Compatibility of the Ontology Expressions with the Thesaurus Expressions

#### 3.1. Algorithm for the Compatibility of Ontology Expressions with Thesaurus Expressions

Algorithm 1 |

#### 3.2. Deduction

**justified**, while ${0}_{R}$ is

**unjustified**. In Table 2, we present the values of the connectives for the deduction.

#### 3.3. Analysis

**problematic**, while ${0}_{A}$ is

**unproblematic**. In Table 3, we present the values of the connectives for the analysis.

#### 3.4. Reduction

**hypothetical**, while ${0}_{R}$ is

**not hypothetical**. In Table 4, we present the values of the connectives for the reduction.

#### 3.5. Synthesis

**resolved**, while ${0}_{S}$ is

**to be resolved**. In Table 5, we present the values of the connectives for the synthesis.

## 4. Assertion Perception Systems in de Morgan Algebras

#### 4.1. Triangular Norms of Deduction

#### 4.1.1. Deduction Norms

**t-norm**of deduction in $[0,1]$ or the

**triangular norm of deduction**, when it satisfies following conditions for any numbers $x,y,z\in [0,1]$:

- boundary conditions$$0{\otimes}_{D}y=0,y{\otimes}_{D}1=y,$$
- monotonicity$$x{\otimes}_{D}y\le z{\otimes}_{D}y,\phantom{\rule{4.pt}{0ex}}\mathrm{when}\phantom{\rule{4.pt}{0ex}}x\le z,$$
- commutativity$$x{\otimes}_{D}y=y{\otimes}_{D}x,$$
- associativity$$x{\otimes}_{D}\left(y{\otimes}_{D}z\right)=\left(x{\otimes}_{D}y\right){\otimes}_{D}z,$$
- there exists $x{\to}_{D}y=sup\{t\in [0,1]:x{\otimes}_{D}t\le y\}$.

**Theorem**

**2.**

- boundary conditions$$0{\otimes}_{D}y=y,y{\otimes}_{D}1=1,$$
- monotonicity$$x{\otimes}_{D}y\le z{\otimes}_{D}y,\phantom{\rule{4.pt}{0ex}}when\phantom{\rule{4.pt}{0ex}}x\le z,$$
- commutativity$$x{\otimes}_{D}y=y{\otimes}_{D}x,$$
- associativity$$x{\otimes}_{D}\left(y{\otimes}_{D}z\right)=\left(x{\otimes}_{D}y\right){\otimes}_{D}z,$$
- there is$$x{-}_{D}y=inf\{t\in L:y\le x{\otimes}_{D}t\}.$$

#### 4.1.2. Analysis Norms

#### 4.1.3. Reduction Norms

#### 4.1.4. Synthesis Norms

#### 4.1.5. De Morgan Algebra

**Theorem**

**3.**

- the algebra of deduction:$$N{S}_{D}=\langle [0,1]{,}^{\prime},{\otimes}_{D},{\oplus}_{D},{\to}_{D},{-}_{D}\rangle ,$$
- the algebra of analysis:$$N{S}_{A}=\langle [0,1]{,}^{\prime},{\otimes}_{A},{\oplus}_{A},{\to}_{A},{-}_{A}\rangle ,$$
- the algebra of reduction:$$N{S}_{R}=\langle [0,1]{,}^{\prime},{\otimes}_{R},{\oplus}_{R},{\to}_{R},{-}_{R}\rangle ,$$
- the algebra of synthesis:$$N{S}_{S}=\langle [0,1]{,}^{\prime},{\otimes}_{S},{\oplus}_{S},{\to}_{S},{-}_{S}\rangle .$$

#### 4.2. Perception Systems

**perception**determines the weight, rank and validity of object descriptions representing information about an object, pointing to objects, i.e., to descriptions of objects called

**instances**(objects belonging to the universe U). Each such reference is called an

**information granule**for the description of an object [2,3].

- Let ${\mu}_{3}{=}_{df}{\mu}_{1}^{\prime}$, then ${\mu}_{1}^{\prime}\left(x\right)={\left({\mu}_{1}\left(x\right)\right)}^{\prime}$,
- Let ${\mu}_{3}{=}_{df}{\mu}_{1}{\otimes}_{\square}{\mu}_{2}$, then $\left({\mu}_{1}{\otimes}_{\square}{\mu}_{2}\right)\left(x\right)={\mu}_{1}\left(x\right){\otimes}_{\square}{\mu}_{2}\left(x\right)$,
- Let ${\mu}_{3}{=}_{df}{\mu}_{1}{\oplus}_{\square}{\mu}_{2}$, then $\left({\mu}_{1}{\oplus}_{\square}{\mu}_{2}\right)\left(x\right)={\mu}_{1}\left(x\right){\oplus}_{\square}{\mu}_{2}\left(x\right)$,
- Let ${\mu}_{3}{=}_{df}{\mu}_{1}{\to}_{\square}{\mu}_{2}$, then $\left({\mu}_{1}{\to}_{\square}{\mu}_{2}\right)\left(x\right)={\mu}_{1}\left(x\right){\to}_{\square}{\mu}_{2}\left(x\right)$,
- Let ${\mu}_{3}{=}_{df}{\mu}_{1}{-}_{\square}{\mu}_{2}$, then $\left({\mu}_{1}{-}_{\square}{\mu}_{2}\right)\left(x\right)={\mu}_{1}\left(x\right){-}_{\square}{\mu}_{2}\left(x\right)$.

**perception systems of deduction, analysis, reduction and synthesis**. When ${P}_{s}$ is the set of all functions $\mu :U\to [0,1]$, identified in this paper with fuzzy sets, then we can talk about fuzzy set algebras.

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

- Bryniarska, A. Autodiagnosis of Information Retrieval on the Web as a Simulation of Selected Processes of Consciousness in the Human Brain. In Biomedical Engineering and Neuroscience; Hunek, W.P., Paszkiel, S., Eds.; Advances in Intelligent Systems and Computing 720; Springer: Berlin/Heidelberg, Germany, 2018; pp. 111–120. [Google Scholar]
- Bryniarska, A. Certain information granule system as a result of sets approximation by fuzzy context. Int. J. Approx. Reason.
**2019**, 111, 1–20. [Google Scholar] [CrossRef] - Peters, J.F. Discovery of perceptually near information granules. In Novel Developments in Granular Computing: Applications of Advanced Human Reasoning and Soft Computation; Yao, J.T., Ed.; Information Science Reference: Hersey, NY, USA, 2009. [Google Scholar]
- Peters, J.F.; Ramanna, S. Affinities between perceptual granules: Foundations and Perspectives. In Human-Centric Information Processing Through Granular Modelling; Bargiela, A., Pedrycz, W., Eds.; Springer: Berlin, Germany, 2009; pp. 49–66. [Google Scholar]
- Pedrycz, W. Allocation of information granularity in optimization and decision-making models: Towards building the foundations of Granular Computing. Eur. J. Oper. Res.
**2014**, 232, 137–145. [Google Scholar] [CrossRef] - Moore, R. Interval Analysis; Prentice-Hall: Englewood Clifis, NJ, USA, 1966. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef][Green Version] - Zadeh, L.A. Towards a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst.
**1997**, 90, 111–117. [Google Scholar] [CrossRef] - Zadeh, L.A. Toward a generalized theory of uncertainty (GTU) an outline. Inf. Sci.
**2005**, 172, 1–40. [Google Scholar] [CrossRef] - Pawlak, Z. Rough sets. Int. J. Comp. Inform. Sci.
**1982**, 11, 341–356. [Google Scholar] [CrossRef] - Pawlak, Z. Rough Sets: Theoretical Aspects of Reasoning about Data; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1991. [Google Scholar]
- Pawlak, Z.; Skowron, A. Rough membership function. In Advaces in the Dempster-Schafer of Evidence; Yeager, R.E., Fedrizzi, M., Kacprzyk, J., Eds.; Wiley: New York, NY, USA, 1994; pp. 251–271. [Google Scholar]
- Pawlak, Z.; Skowron, A. Rudiments of rough sets. Inf. Sci.
**2007**, 177, 3–27. [Google Scholar] [CrossRef] - Pawlak, Z.; Skowron. A. Rough sets and Boolean reasoning. Inf. Sci.
**2007**, 177, 41–73. [Google Scholar] [CrossRef][Green Version] - Pedrycz, W. Shadowed sets: Representing and processing fuzzy sets. IEEE Trans. Syst. Man Cybern. Part B
**1998**, 28, 103–109. [Google Scholar] [CrossRef] [PubMed] - Pedrycz, W. Knowledge-Based Clustering: From Data to Information Granules; John Wiley & Sons: Hoboken, NJ, USA, 2005. [Google Scholar]
- Lindsay, P.H.; Norman, D.A. Human Information Processing: An Introduction to Psychology; Academic Press, Inc.: New York, NY, USA, 1972. [Google Scholar]
- Bryniarska, A. The Paradox of the Fuzzy Disambiguation in the Information Retrieval. Int. J. Adv. Res. Artif. Intell.
**2013**, 2, 55–58. [Google Scholar] [CrossRef][Green Version] - Manola, F.; Miller, E. RDF Primer. 2004. Available online: http://www.w3.org/TR/rdf-primer/ (accessed on 15 October 2020).
- Resource Description Framework (RDF). RDF Working Group. 2004. Available online: http://www.w3.org/RDF/ (accessed on 15 October 2020).
- Lassila, O.; Swick, R.R. Resource Description Framework (RDF): Model and Syntax Specification. Rekomendacja W3C. 2003. Available online: http://www.w3.org/TR/REC-rdf-syntax/ (accessed on 15 October 2020).
- Michalewicz, Z.; Fogel, D.B. How to Solve It: Modern Heuristic; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Kowalski, R. Logic for Problem Solving; North-Holland: New York, NY, USA; Amsterdam, The Netherlands; Oxford, UK, 1979. [Google Scholar]
- Bobillo, F.; Straccia, U. A Fuzzy Description Logic with Product T-norm. In Proceedings of the 2007 IEEE International Fuzzy Systems Conference, London, UK, 23–26 July 2007; pp. 1–6. [Google Scholar] [CrossRef][Green Version]
- Sanchez, E. Fuzzy Logic and the Semantic Web; Elsevier: Amsterdam, The Netherlands, 2006; ISBN 13: 978-0-444-51948-1. [Google Scholar]

**Figure 2.**The logical square of the concepts ${\mathsf{\Psi}}_{1},{\mathsf{\Psi}}_{2},{\mathsf{\Psi}}_{3},{\mathsf{\Psi}}_{4}$ of the assertion formulas.

$\mathit{\alpha}$ | $\mathit{\beta}$ | $\neg \mathit{\alpha}$ | $\mathit{\alpha}\vee \mathit{\beta}$ | $\mathit{\alpha}\wedge \mathit{\beta}$ | $\mathit{\alpha}\Rightarrow \mathit{\beta}$ | $\mathit{\alpha}\backslash \mathit{\beta}$ |
---|---|---|---|---|---|---|

1 | 1 | 0 | 1 | 1 | 1 | 0 |

0 | 1 | 1 | 1 | 0 | 1 | 1 |

1 | 0 | 0 | 1 | 0 | 0 | 0 |

0 | 0 | 1 | 0 | 0 | 1 | 0 |

${\alpha}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${1}_{D}$ | ${0}_{D}$ |

${\beta}_{D}$ | ${1}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${0}_{D}$ |

$\neg {\alpha}_{D}$ | ${0}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${1}_{D}$ |

${\alpha}_{D}{\vee}_{D}{\beta}_{D}$ | ${1}_{D}$ | ${1}_{D}$ | ${1}_{D}$ | ${0}_{D}$ |

${\alpha}_{D}{\wedge}_{D}{\beta}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${0}_{D}$ | ${0}_{D}$ |

${\alpha}_{D}{\Rightarrow}_{D}{\beta}_{D}$ | ${1}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${1}_{D}$ |

${\alpha}_{D}{\backslash}_{D}{\beta}_{D}$ | ${0}_{D}$ | ${1}_{D}$ | ${0}_{D}$ | ${0}_{D}$ |

${\alpha}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${1}_{A}$ | ${0}_{A}$ |

${\beta}_{A}$ | ${1}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${0}_{A}$ |

$\neg {\alpha}_{A}$ | ${0}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${1}_{A}$ |

${\alpha}_{A}{\vee}_{A}{\beta}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${0}_{A}$ | ${0}_{A}$ |

${\alpha}_{A}{\wedge}_{A}{\beta}_{A}$ | ${1}_{A}$ | ${1}_{A}$ | ${1}_{A}$ | ${0}_{A}$ |

${\alpha}_{A}{\Rightarrow}_{A}{\beta}_{A}$ | ${0}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${0}_{A}$ |

${\alpha}_{A}{\backslash}_{A}{\beta}_{A}$ | ${1}_{A}$ | ${1}_{A}$ | ${0}_{A}$ | ${1}_{A}$ |

${\alpha}_{R}$ | ${1}_{R}$ | ${0}_{R}$ | ${1}_{R}$ | ${0}_{R}$ |

${\beta}_{R}$ | ${1}_{R}$ | ${1}_{R}$ | ${0}_{R}$ | ${0}_{R}$ |

$\neg {\alpha}_{R}$ | ${0}_{R}$ | ${1}_{R}$ | ${0}_{R}$ | ${1}_{R}$ |

${\alpha}_{R}{\vee}_{R}{\beta}_{R}$ | ${0}_{R}$ | ${0}_{R}$ | ${0}_{R}$ | ${1}_{R}$ |

${\alpha}_{R}{\wedge}_{R}{\beta}_{R}$ | ${0}_{R}$ | ${1}_{R}$ | ${1}_{R}$ | ${1}_{R}$ |

${\alpha}_{R}{\Rightarrow}_{R}{\beta}_{R}$ | ${0}_{R}$ | ${0}_{R}$ | ${1}_{R}$ | ${0}_{R}$ |

${\alpha}_{R}{\backslash}_{R}{\beta}_{R}$ | ${1}_{R}$ | ${0}_{R}$ | ${1}_{R}$ | ${1}_{R}$ |

${\alpha}_{S}$ | ${1}_{S}$ | ${0}_{S}$ | ${1}_{S}$ | ${0}_{S}$ |

${\beta}_{S}$ | ${1}_{S}$ | ${1}_{S}$ | ${0}_{S}$ | ${0}_{S}$ |

$\neg {\alpha}_{S}$ | ${0}_{S}$ | ${1}_{S}$ | ${0}_{S}$ | ${1}_{S}$ |

${\alpha}_{S}{\vee}_{S}{\beta}_{S}$ | ${0}_{S}$ | ${1}_{S}$ | ${1}_{S}$ | ${1}_{S}$ |

${\alpha}_{S}{\wedge}_{S}{\beta}_{S}$ | ${0}_{S}$ | ${0}_{S}$ | ${0}_{S}$ | ${1}_{S}$ |

${\alpha}_{S}{\Rightarrow}_{S}{\beta}_{S}$ | ${1}_{S}$ | ${0}_{S}$ | ${1}_{S}$ | ${1}_{S}$ |

${\alpha}_{S}{\backslash}_{S}{\beta}_{S}$ | ${0}_{S}$ | ${0}_{S}$ | ${1}_{S}$ | ${0}_{S}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bryniarska, A. The Auto-Diagnosis of Granulation of Information Retrieval on the Web. *Algorithms* **2020**, *13*, 264.
https://doi.org/10.3390/a13100264

**AMA Style**

Bryniarska A. The Auto-Diagnosis of Granulation of Information Retrieval on the Web. *Algorithms*. 2020; 13(10):264.
https://doi.org/10.3390/a13100264

**Chicago/Turabian Style**

Bryniarska, Anna. 2020. "The Auto-Diagnosis of Granulation of Information Retrieval on the Web" *Algorithms* 13, no. 10: 264.
https://doi.org/10.3390/a13100264