Fuzzy Logic Introduction1.IntroductionFuzzy Logic was initiated in 1965 , , , byLotfi A. Zadeh , professor for computer science atthe University of California in Berkeley.Basically, Fuzzy Logic (FL) is a multivaluedlogic, that allows intermediate values to bedefined between conventional evaluations liketrue/false,yes/no, high/low, etc. Notions like rathertall or very fast can be formulatedmathematically and processed by computers, inorder to apply a more human−like way of thinkingin the programming of computers.Fuzzy systems is an alternative to traditionalnotions of set membership
Fuzzy systems is an alternative to traditional notions of setmembership and logic that has its origins in ancient Greekphilosophy.The precision of mathematics owes its success in large partto the efforts of Aristotle and the philosophers whopreceded him. In their efforts to devise a concise theory of logic, andlater a, the so−called "Laws of Thought" were posited.
One of these, the "Law of the Excluded Middle," states that every proposition must either be True or False. Even whenParminedes proposed the first version of this law (around 400 B.C.) there were strong and immediate objections: for example, Heraclitus proposed that things could be simultaneously True and not True. It was Plato who laid thefoundation for what would become fuzzy logic, indicating that there was a third region (beyond True and False) where theseopposites "tumbled about." Other, more modern philosophersechoed his sentiments, notably Hegel, Marx, and Engels. But itwas Lukasiewicz who first proposed a systematic alternative to the bi−valued logic of Aristotle .
Even in the present time some Greeks are still outstanding examples for fussiness and fuzziness, (note the connection to logic got lost somewhere during the last 2 mileniums ). Fuzzy Logic has emerged as a a profitable tool for the controlling and steering of systems and complex industrial processes, as well as for household and entertainmentelectronics, as well as for other expert systems and applications like the classification of SAR data.
Fuzzy Sets In contrast to crisp sets, fuzzy sets have membershipDegrees That means, in addition to the values 1 (belongs to)and 0 (does not belong to) an element can have any value in between (“kind of belongs to”) Formally speaking, themembership function μA(x) for a fuzzy set A maps elements x to any value in the interval [0, 1]: μA : X → [0, 1]
Comparing Fuzzy Sets Equality: Crisp sets: two sets A and B are equal, if they contain the same elements Fuzzy sets: all membership degrees have to be equal, i.e. μA(x) = μB(x) for all x ∈ X Containment: Crisp sets: A is contained in B (A ⊆ B), if all elements in A are also elements of B Fuzzy sets: again membership degrees have to be considered, i.e. A ⊆ B ⇔ μA(x) ≤ μB(x) for all x ∈ X
Example: Build a Fuzzy Controller FROM Fuzzy Thinkring, Bart KoskoGoal: Design a motor speed controller for airConditioner.Step 1: assign input and output variablesLet X be the temperature in FahrenheitLet Y be the motor speed of the air conditioner
Step 2: Pick fuzzy setsDefine linguistic terms of the linguisticvariablestemperature (X) and motor speed (Y) andassociate them with fuzzy setsFor example, 5 linguistic terms / fuzzysets on X• Cold, Cool, Just Right, Warm, and HotSay 5 linguistic terms / fuzzy sets on Y• Stop, Slow, Medium, Fast, and Blast
Step 3: Assign a motor speed set toeachtemperature set• If temperature is cold then motor speed is stop• If temperature is cool then motor speed is slow• If temperature is just right then motor speed isMedium• If temperature is warm then motor speed is fast• If temperature is hot then motor speed is blast
• Rather than ealing with probability (0.0-1.0), FuzzyLogic deals with fuzzy set membership – an entity canbe in two or more sets at the same time, to differentdegrees.• Key concept is thus the degree of set membership(0-1) The degree of set membership can also be referredto as degree of truth of the proposition, X is in set Y.• Degree of membership often calculatedalgebraically, e.g., if height= 185, then degree = (185-170)/(200-170) = 0.5
• Degree can also be given in tables (which wouldgive a stepped membership graph):
OPERATIONS:• Subset: A fuzzy set A is a subset of fuzzy set B if for allelements of A, the degreeof membership in A is less than in B.• Complement: 2 fuzzy sets are complementary if for allelements of A, the degree ofmembership in B is 1-A(s).• Union: The union of two fuzzy sets will assign degree ofmembership equal to thehighest of the two sets.• Intersection: The intersection of two fuzzy sets willassign degree of membershipequal to the lowest of the two sets.
MEMBERSHIP IN COMPLEX SETS:• To what degree is X in SET1 and SET2 ? – takethe lowest of the truth values (e.g.,as intraditional logic A&B is false if any one is false).• To what degree is X in SET1 or SET2 ? – take thehighest of the truth values (e.g., as in traditionallogic AvB is true if any one is true)
ReferencesL.A. Zadeh, Fuzzy Sets, Information and Control, 1965L.A. Zadeh, Outline of A New Approach to the Analysis of of ComplexSystems and Decision Processes, 1973 L.A. Zadeh, "Fuzzy algorithms," Info. & Ctl., Vol. 12, 1968, pp. 94 102. L.A. Zadeh, Making computers think like people, IEEE.Spectrum, 8/1984,pp. 26 32. S. Korner, Laws of thought, Encyclopedia of Philosophy, Vol.4, MacMillan,NY: 1967, pp. 414 417. C. Lejewski, Jan Lukasiewicz, Encyclopedia of Philosophy, Vol. 5,MacMillan, NY: 1967, pp. 104 107. A. Reigber, My life with Kostas, unpublished report, Neverending StoryPress , 1999