Everything Is a Matter . . . Formulation of the . . . There Should be an . . . First Example of . . . Everything Second Example of . . . Third Example of . . . Is a Matter of Degree: Our Explanation of . . . First Explanation: . . . Second Explanation: . . . A New Theoretical Symmetry: Another . . . Case Study: Territory . . . Justification of Acknowledgments Title Page Zadeh’s Principle ◭◭ ◮◮ Hung T. Nguyen ◭ ◮ Department of Mathematical Sciences Page 1 of 13 New Mexico State University Go Back Vladik Kreinovich Department of Computer Science Full Screen University of Texas at El Paso Close El Paso, TX 79968 Email: vladik@utep.edu Quit
Formulation of the . . . There Should be an . . . 1. Everything Is a Matter of Degree: One of the Main Ideas Behind Fuzzy Logic First Example of . . . Second Example of . . . • One of the main ideas behind Zadeh’s fuzzy logic and Third Example of . . . its applications is that everything is a matter of degree. Our Explanation of . . . First Explanation: . . . • We are often accustomed to think that every statement about a physical world is true or false: Second Explanation: . . . Symmetry: Another . . . – that an object is either a particle or a wave, Case Study: Territory . . . – that a person is either young or not, Acknowledgments – that a person is either well or ill. Title Page • However, in reality, we sometimes encounter interme- ◭◭ ◮◮ diate situations. ◭ ◮ Page 2 of 13 Go Back Full Screen Close Quit
Formulation of the . . . There Should be an . . . 2. Formulation of the Problem First Example of . . . • That everything is a matter of degree is a convincing Second Example of . . . empirical fact. Third Example of . . . Our Explanation of . . . • A natural question is: why? First Explanation: . . . • How can we explain this fact? Second Explanation: . . . • This is what we will try to do in this talk: come up Symmetry: Another . . . with a theoretical explanation of this empirical fact. Case Study: Territory . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 13 Go Back Full Screen Close Quit
Formulation of the . . . There Should be an . . . 3. There Should be an Objective Theoretical Expla- nation for Fuzziness First Example of . . . Second Example of . . . • Most traditional examples of fuzziness come from the Third Example of . . . analysis of commonsense reasoning. Our Explanation of . . . • When we reason, we use words from natural language First Explanation: . . . like “young”, “well”. Second Explanation: . . . Symmetry: Another . . . • In many practical situations, these words do not have Case Study: Territory . . . a precise true-or-false meaning, they are fuzzy. Acknowledgments • Impression: fuzziness is subjective, it is how our brains Title Page work. ◭◭ ◮◮ • However, we are the result of billions of years of suc- ◭ ◮ cessful adjusting-to-the-environment evolution. Page 4 of 13 • Everything about us humans is not accidental. Go Back • In particular, the fuzziness in our reasoning must have Full Screen an objective explanation – in fuzziness of the real world. Close Quit
Formulation of the . . . There Should be an . . . 4. First Example of Objective “Fuzziness” – Fractals First Example of . . . • Since the ancient times, we know: Second Example of . . . Third Example of . . . – 0-dimensional objects (points), Our Explanation of . . . – 1-dimensional objects (lines), First Explanation: . . . – 2-dimensional objects (surfaces), Second Explanation: . . . – 3-dimensional objects (bodies), etc. Symmetry: Another . . . • In all these examples, dim is an integer: 0, 1, 2, 3, etc. Case Study: Territory . . . Acknowledgments • In the 19th century, mathematicians discovered sets of Title Page fractional dimension (fractals). ◭◭ ◮◮ • In the 1970s, B. Mandlebrot noticed that many real-life ◭ ◮ objects are fractals, e.g.: Page 5 of 13 – shoreline of England Go Back – shape of the clouds and mountains Full Screen – noises in electric circuits. Close Quit
Formulation of the . . . There Should be an . . . 5. Second Example of Objective “Fuzziness” – Quan- tum Physics First Example of . . . Second Example of . . . • In general, states are described by continuous variables. Third Example of . . . Our Explanation of . . . • However, the set of stable states is usually discrete . First Explanation: . . . • Example: computers use memory cells with 2 stable Second Explanation: . . . states representing 0 and 1. Symmetry: Another . . . • In quantum physics: we can have superpositions Case Study: Territory . . . c 0 · � 0 | + c 1 · � 1 | for complex c i . Acknowledgments • Resulting quantum computations are much faster: Title Page ◭◭ ◮◮ – we can search in an unsorted list of n elements in time √ n ; ◭ ◮ – we can factor large integers fast – and thus, crack Page 6 of 13 the existing codes. Go Back • What we originally thought of as an integer-valued Full Screen variable turned out to be real-valued. Close Quit
Formulation of the . . . There Should be an . . . 6. Third Example of Objective “Fuzziness” – Frac- tional Charges of Quarks First Example of . . . Second Example of . . . • Matter is seemingly continuous. Third Example of . . . • It turned out that matter is discrete: it consists of Our Explanation of . . . molecules, atoms, and elementary particles. First Explanation: . . . Second Explanation: . . . • One experimental fact: all electric charges are propor- Symmetry: Another . . . tional to a single charge. Case Study: Territory . . . • Thus, protons, etc., cannot be further decomposed. Acknowledgments • Gell-Mann discovered that we can design p , n , mesons, Title Page etc. in terms of a few quarks . ◭◭ ◮◮ • Interesting aspect: quarks have fractional electric charge. ◭ ◮ • Original idea: quarks are theoretical concepts. Page 7 of 13 • Experiments revealed 3 partons within p – actual quarks. Go Back • So, what we originally thought of as an integer-valued Full Screen variable turned out to be real-valued. Close Quit
Formulation of the . . . There Should be an . . . 7. Our Explanation of Why Physical Quantities Origi- nally Thought to Be Integer-Valued Turned out to First Example of . . . Be Real-Valued: Main Idea Second Example of . . . Third Example of . . . • In philosophical terms: what we are doing is “cogniz- Our Explanation of . . . ing” the world. First Explanation: . . . • Clarification: understanding how it works and trying Second Explanation: . . . to predict consequences of different actions. Symmetry: Another . . . Case Study: Territory . . . • Objective: select the most beneficial action. Acknowledgments • If a phenomenon is not cognizable, there is nothing we Title Page can do about it. ◭◭ ◮◮ • Our explanation: in cognizable phenomena, it is rea- ◭ ◮ sonable to expect continuous-valued variables. Page 8 of 13 • In other words: properties originally thought to be dis- Go Back crete are actually matters of degree. Full Screen Close Quit
Formulation of the . . . There Should be an . . . 8. First Explanation: Goedel’s Theorem vs. Tarski’s Algorithm First Example of . . . Second Example of . . . • Goedel’s theorem: 1st example of non-cognizability. Third Example of . . . • Formulations: Our Explanation of . . . First Explanation: . . . – variables x , y , z , etc. run over integers; Second Explanation: . . . – terms t are formed from x , . . . , and const. by +, · ; Symmetry: Another . . . – elementary formulas: t = t ′ , t < t ′ , t ≤ t ′ , etc. Case Study: Territory . . . – formulas: from elem. formulas by ∨ , &, ¬ , ∃ , ∀ . Acknowledgments • Example: ∀ x ∀ y ( x < y → ∃ z ( y = x + z )) . Title Page ◭◭ ◮◮ • Goedel’s theorem: no algorithm can tell whether a given formula is true or not. ◭ ◮ • Tarski’s theorem: if we consider variables over real Page 9 of 13 numbers, then such an algorithm is possible. Go Back • Conclusion: in cognizable situations, we must have Full Screen continuous-valued variables. Close Quit
Formulation of the . . . There Should be an . . . 9. Second Explanation: Efficient Algorithms for Lin- ear Algebra vs. NP-Hardness of Integer Program- First Example of . . . ming Second Example of . . . Third Example of . . . • Practical situation: find the values x 1 , . . . , x n from the Our Explanation of . . . results y 1 , . . . , y m of indirect measurements: First Explanation: . . . f 1 ( x 1 , . . . , x n ) = y 1 ; f m ( x 1 , . . . , x n ) = y m . Second Explanation: . . . Symmetry: Another . . . • Frequent case: we know approximate � x i values of x i . Case Study: Territory . . . • How this helps: we can linearize the system: Acknowledgments Title Page a i 1 · ∆ x 1 + . . . + a in · ∆ x n = ∆ y i , 1 ≤ i ≤ m. ◭◭ ◮◮ • Case of continuous variables: efficient algorithms solve ◭ ◮ systems of linear equations. Page 10 of 13 • Case of discrete variables: problem becomes NP-hard. Go Back • Meaning (informal): every algorithm requires un-realistic Full Screen time in some cases (unless P=NP). Close Quit
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