Difference between revisions of "Fuzzy logic language/it"

In questo capitolo parleremo della ''logica fuzzy''. Si chiama ''fuzzy'' perché è caratterizzata da una gradualità: ad un oggetto si può attribuire una qualità che può avere ''vari gradi di verità''.

*Inclusione: <span class="mw-translate-fuzzy">Rappresentato dal simbolo<math>\subset</math> (is content), - (è contenuto), - ad esempio l'intero <math>A</math> è contenuto all'interno dell'insieme più grande <math>U</math>, <math>A \subset U</math> ((in questo caso si dice che <math>A</math> è un subset di <math>U</math>)</span>

*Dimostrazione, <span lang="en" dir="ltr" class="mw-content-ltr">which is indicated by the symbol <math>\mid</math> (such that)</span>

===<span lang="en" dir="ltr" class="mw-content-ltr">Set operators</span>===

<span lang="en" dir="ltr" class="mw-content-ltr">Given the whole universe</span> <math>U</math> <span lang="en" dir="ltr" class="mw-content-ltr">we indicate with</span> <math>x</math> <span lang="en" dir="ltr" class="mw-content-ltr">its generic element so that</span> <math>x \in U</math>; <span lang="en" dir="ltr" class="mw-content-ltr">then, we consider two subsets</span> <math>A</math> and <math>B</math> <span lang="en" dir="ltr" class="mw-content-ltr">internal to</span> <math>U</math> cosicché <math>A \subset U</math> e <math>B \subset U</math>

|'''Unione:''' <span lang="en" dir="ltr" class="mw-content-ltr">represented by the symbol</span> <math>\cup</math>, <span lang="en" dir="ltr" class="mw-content-ltr">indicates the union of the two sets</span> <math>A</math> e <math>B</math> <math>(A\cup B)</math>. <span lang="en" dir="ltr" class="mw-content-ltr">It is defined by all the elements that belong to</span> <math>A</math> e <math>B</math> <span lang="en" dir="ltr" class="mw-content-ltr">or both</span>:

|'''Intersezione:''' <span lang="en" dir="ltr" class="mw-content-ltr">represented by the symbol</span> <math>\cap</math>, <span lang="en" dir="ltr" class="mw-content-ltr">indicates the elements belonging to both sets</span>:

|'''Differenza:''' <span lang="en" dir="ltr" class="mw-content-ltr">represented by the symbol</span> <math>-</math>, per esempio <math>A-B</math> <span lang="en" dir="ltr" class="mw-content-ltr">shows all elements of</span> <math>A</math> <span lang="en" dir="ltr" class="mw-content-ltr">except those shared with</span> <math>B</math>

|'''<span lang="en" dir="ltr" class="mw-content-ltr">Complementary</span>:''' <span lang="en" dir="ltr" class="mw-content-ltr">represented by a bar above the name of the collection, it indicates by</span> <math>\bar{A}</math> <span lang="en" dir="ltr" class="mw-content-ltr">the complementary of</span> <math>A</math>, cioè, <span lang="en" dir="ltr" class="mw-content-ltr">the set of elements that belong to the whole universe except those of</span> <math>A</math>, in formule: <math>\bar{A}=U-A</math><br />

<span lang="en" dir="ltr" class="mw-content-ltr">The theory of fuzzy language logic is an extension of the classical theory of sets in which, however, the principles of non-contradiction and the excluded third are not valid</span>. <span lang="en" dir="ltr" class="mw-content-ltr">Remember that in classical logic, given the set</span> <math>A</math> <span lang="en" dir="ltr" class="mw-content-ltr">and its complementary</span> <math>\bar{A}</math>, <span lang="en" dir="ltr" class="mw-content-ltr">the principle of non-contradiction states that if an element belongs to the whole</span> <math>A</math> <span lang="en" dir="ltr" class="mw-content-ltr">it cannot at the same time also belong to its complementary</span> <math>\bar{A}</math>; <span lang="en" dir="ltr" class="mw-content-ltr">according to the principle of the excluded third, however, the union of a whole</span> <math>A</math> <span lang="en" dir="ltr" class="mw-content-ltr">and its complementary</span> <math>\bar{A}</math> <span lang="en" dir="ltr" class="mw-content-ltr">constitutes the complete universe</span> <math>U</math>.  

<span lang="en" dir="ltr" class="mw-content-ltr">In other words, if any element does not belong to the whole, it must necessarily belong to its complementary</span>.

==<span lang="en" dir="ltr" class="mw-content-ltr">Fuzzy set</span> <math>\tilde{A}</math> <span lang="en" dir="ltr" class="mw-content-ltr">and membership function</span> <math>\mu_{\displaystyle {\tilde {A}}}(x)</math>==

We choose - as a formalism - to represent a fuzzy set with the 'tilde':<math>\tilde{A}</math>. A fuzzy set is a set where the elements have a 'degree' of belonging (consistent with fuzzy logic): some can be included in the set at 100%, others in lower percentages.

To mathematically represent this degree of belonging is the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> called '<nowiki/>'''Membership Function''''. The function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> is a continuous function defined in the interval <math>[0;1]</math>where it is:

*<math>\mu_ {\tilde {A}}(x) = 1\rightarrow </math>   if <math>x</math> is totally contained in <math>A</math> (these points are called 'nucleus', they indicate <u>plausible</u> predicate values).

*<math>\mu_ {\tilde {A}}(x) = 0\rightarrow </math> if <math>x</math> is not contained in <math>A</math>

*<math>0<\mu_ {\tilde {A}}(x) < 1 \;\rightarrow </math> if <math>x</math> is partially contained in <math>A</math> (these points are called 'support', they indicate the <u>possible</u> predicate values).

The graphical representation of the function <math>\mu_{\displaystyle {\tilde {A}}}(x)</math> it can be varied; from those with linear lines (triangular, trapezoidal) to those in the shape of bells or 'S' (sigmoidal) as depicted in Figure 1, which contains the whole graphic concept of the function of belonging.<ref>{{Cite book  

[[File:Fuzzy_crisp.svg|alt=|left|thumb|400px|'''Figure 1:''' Types of graphs for the membership function.]]

The '''support set''' of a fuzzy set is defined as the zone in which the degree of membership results <math>0<\mu_ {\tilde {A}}(x) < 1</math>; on the other hand, the '''core''' is defined as the area in which the degree of belonging assumes value <math>\mu_ {\tilde {A}}(x) = 1</math>

The 'Support set' represents the values of the predicate deemed '''possible''', while the 'core' represents those deemed more '''plausible'''.

If <math>{A}</math> represented a set in the ordinary sense of the term or classical language logic previously described, its membership function could assume only the values <math>1</math> or <math>0</math>, <math>\mu_{\displaystyle {{A}}}(x)= 1 \; \lor \;\mu_{\displaystyle {{A}}}(x)= 0</math> depending on whether the element <math>x</math> whether or not it belongs to the whole, as considered. Figure 2 shows a graphic representation of the crisp (rigidly defined) or fuzzy concept of membership, which clearly recalls Smuts's considerations.<ref name=":0">•SMUTS J.C. 1926, [[wikipedia:Holism_and_Evolution|Holism and Evolution]], London: Macmillan.</ref>  

Let us go back to the specific case of our Mary Poppins, in which we see a discrepancy between the assertions of the dentist and the neurologist and we look for a comparison between classical logic and fuzzy logic:

[[File:Fuzzy1.jpg|thumb|400x400px|'''Figure 2:''' Representation of the comparison between a classical and fuzzy ensemble.]]

'''Figure 2:''' Let us imagine the Science Universe <math>U</math> in which there are two parallel worlds or contexts, <math>{A}</math> and <math>\tilde{A}</math>.

<math>{A}=</math>  In the scientific context, the so-called ‘crisp’, and we have converted into ''the logic'' of ''Classic Language'', in which the physician has an absolute scientific background information <math>KB</math>  with a clear dividing line that we have named <math>KB_c</math>.  

<math>\tilde{A}=</math> In another scientific context called  ‘fuzzy logic’, and in which there is a union between the subset <math>{A}</math> in <math>\tilde{A}</math> that we can go so far as to say: union between <math>KB_c</math>.

We will remarkably notice the following deductions:

*'''Classical Logic''' in the Dental Context <math>{A}</math> in which only a logical process that gives as results <math>\mu_{\displaystyle {{A}}}(x)= 1 </math> it will be possible, or <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> being the range of data <math>D=\{\delta_1,\dots,\delta_4\}</math>reduced to basic knowledge <math>KB</math> in the set <math>{A}</math>. This means that outside the dental world there is a void and that term of set theory, it is written precisely <math>\mu_{\displaystyle {{A}}}(x)= 0 </math> and which is synonymous with a high range of:

<br />{{q2|Differential diagnostic error|}}

*'''Fuzzy logic''' in a dental context <math>\tilde{A}</math> in which they are represented beyond the basic knowledge <math>KB</math> of the dental context also those partially acquired from the neurophysiological world <math>0<\mu_ {\tilde {A}}(x) < 1</math> will have the prerogative to return a result <math>\mu_\tilde{A}(x)= 1

  </math> and a result <math>0<\mu_ {\tilde {A}}(x) < 1</math> because of  basic knowledge <math>KB</math> which at this point is represented by the union of <math>KB_c</math>  dental and neurological contexts. The result of this scientific-clinical implementation of dentistry would allow a:{{q2|Reduction of differential diagnostic error|}}

==Final considerations==

Topics that could distract the reader’s attention—was, in fact, essential for demonstrating the message. Normally, in fact, when any more or less brilliant mind allows itself to throw a stone into the pond of Science, a shockwave is generated, typical of the period of Kuhn’s extraordinary science, against which most of the members of the international scientific community row. With good faith, we can say that this phenomenon—as regards the topics we are addressing here—is well represented in the premise at the beginning of the chapter.

In these chapters, in fact, a fundamental topic for science has been approached: the re-evaluation, the specific weight that has always been given to <math>P-value</math>, awareness of scientific / clinical contexts <math>KB_c</math>, having undertaken a more elastic path of Fuzzy Logic than the Classical one, realizing the extreme importance of <math>KB</math> and ultimately the union of contexts <math>KB_c</math> to increase its diagnostic capacity.<ref>Mehrdad Farzandipour, Ehsan Nabovati, Soheila Saeedi, Esmaeil Fakharian. [https://pubmed.ncbi.nlm.nih.gov/30119845/ Fuzzy decision support systems to diagnose musculoskeletal disorders: A systematic literature review] . Comput Methods Programs Biomed. 2018 Sep;163:101-109. doi: 10.1016/j.cmpb.2018.06.002. Epub 2018 Jun 6.</ref><ref>Long Huang, Shaohua Xu, Kun Liu, Ruiping Yang, Lu Wu. [https://pubmed.ncbi.nlm.nih.gov/34257635/ A Fuzzy Radial Basis Adaptive Inference Network and Its Application to Time-Varying Signal Classification] . Comput Intell Neurosci, 2021 Jun 23;2021:5528291.

doi: 10.1155/2021/5528291.eCollection 2021.</ref>

In the next chapter we will be ready to undertake an equally fascinating path that will leads us to the context of a System Language logic and will allow us to deepen our knowledge no longer only in clinical semeiotics but in the understanding of system functions as recently it is approaching in neuromotor disciplines for Parkinson's disease.<ref>Mehrbakhsh Nilashi, Othman Ibrahim, Ali Ahani. [https://pubmed.ncbi.nlm.nih.gov/27686748/ Accuracy Improvement for Predicting Parkinson's Disease Progression.] Sci Rep. 2016 Sep 30;6:34181.

doi: 10.1038/srep34181.</ref> In Masticationpedia, of course, we will report the topic 'System Inference' in the field of the masticatory  system as we could read in the next chapter entitled 'System logic'.