* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download x - WordPress.com
Statistical inference wikipedia , lookup
Model theory wikipedia , lookup
Structure (mathematical logic) wikipedia , lookup
Willard Van Orman Quine wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Modal logic wikipedia , lookup
Axiom of reducibility wikipedia , lookup
List of first-order theories wikipedia , lookup
History of the function concept wikipedia , lookup
Jesús Mosterín wikipedia , lookup
Propositional calculus wikipedia , lookup
Combinatory logic wikipedia , lookup
Naive set theory wikipedia , lookup
Lorenzo Peña wikipedia , lookup
Curry–Howard correspondence wikipedia , lookup
Quantum logic wikipedia , lookup
History of logic wikipedia , lookup
Interpretation (logic) wikipedia , lookup
Natural deduction wikipedia , lookup
Intuitionistic logic wikipedia , lookup
Mathematical logic wikipedia , lookup
Truth-bearer wikipedia , lookup
Laws of Form wikipedia , lookup
Law of thought wikipedia , lookup
Principia Mathematica wikipedia , lookup
Fuzzy Logic Introduction: In Artificial Intelligence (AI) the ultimate goal is to create machines that think like humans. Human beings make decisions based on rules. Although, we may not be aware of it, all the decisions we make are all based on computer like if-then statements. If the weather is fine, then we may decide to go out. If the forecast stays the weather will be bad today, but fine tomorrow, then we make a decision not to go today, and postpone it till tomorrow. By abandoning the rigid idea of true or false, Lofti Zadeh, redefined how we think about logic. Constant researches lead to the invention of Fuzzy Logic. For instance, consider the given three statements Dinosaurs’ have ruled on this planet for a long time (for a million of years) It hasn’t rained since a long time (for a couple of months) He waited for his turn for a long time (for a couple of hours) In these statements we cannot infer the true value of ‘for a long time’. There is no way to represent this concept in standard binary set theory. In a standard set theory an object is either a member of a set or it is not. Either the value for that object is true or false. There is no in-between. This amounts to the use of a characteristic function f for a set A, where f(A)=1 if the element belongs to A, otherwise it is 0; Definition: Fuzzy logic is a form of many valued logic, superset of Boolean logic that has been extended to handle the concept of partial truth- truth values between "completely true" and "completely false" to deal with reasoning. Consider a Universal set U of which a subset called fuzzy subset A(bar) is defined by function f. If uA(x)=1 or, f(x)=1 signify that x is completely contained in A(bar). If f(x)=0 signify that x is not a member of A(bar). Values of 0<f(x)<1 signify that x is a partial member of A(bar). To illustrate this let's talk about people and "youthness". In this case the set U (the universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined, which answers the question "to what degree is person x young?" To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. The easiest way to do this is with a membership function based on the person's age. young(x) = { 1, if age(x) <= 20, (30-age(x))/10, if 20 < age(x) <= 30, 0, if age(x) > 30 } A graph of this looks like: Given this definition, here are some example values: Person Karim Rafiq Safiq Rafi Rabbi Ratul Age 10 21 25 26 28 83 Degree of youth 1.00 0.90 0.50 0.40 0.20 00 Importance: Fuzzy sets and fuzzy logic is an important and a practical mathematical tool for the processing of uncertain and vague information. In our earlier work, a fuzzy logic with similarity was proposed and the soundness and completeness were proved (Jiabing Wang et al. 2002). In this paper, some properties of fuzzy inference based on the resolution principle and paramodulation are discussed: first, the significance of fuzzy logical inference by the resolution principle is discussed in the context of fuzzy first-order logic; second, it is shown that the fuzzy logical inference by the paramodulation is always significant; finally, the relation between a resolvent or paramodulant and ambiguity is discussed. The following rules which are common in classical set theory also apply to Fuzzy set theory. 𝐴̅ U (𝐵̅ ∩ 𝐶̅ ) = (𝐴̅ U 𝐵̅) ∩ (𝐴̅ U 𝐶̅ ) distributivity 𝐴̅ ∩ (𝐵̅ U 𝐶̅ ) = (𝐴̅ ∩ 𝐵̅) U (𝐴̅ ∩ 𝐶̅ ) (𝐴̅ U 𝐵̅ ) U 𝐶̅ = 𝐴̅ U ( 𝐵̅ U 𝐶̅ ) associativity (𝐴̅ ∩ 𝐵̅) ∩ 𝐶̅ = 𝐴̅ ∩ ( 𝐵̅ ∩ 𝐶̅ ) 𝐴̅ ∩ 𝐵̅ = 𝐵̅ ∩ 𝐴̅ , 𝐴̅ U 𝐵̅ = 𝐵̅ U 𝐴̅ commutativity 𝐴̅ ∩ 𝐴̅ = 𝐴̅ , 𝐴̅ U 𝐵̅ = 𝐴̅ idempotency There is also a form of DeMorgan’s laws: 𝑢(𝐴∩𝐵)′ (x) = 𝑢𝐴′∩𝐵′ (x) 𝑢(𝐴𝑈𝐵)′ (x) = 𝑢𝐴′𝑈𝐵′ (x) A number of operation that are unique to fuzzy sets have been defined. A few of the more common operation include. 1. Dilation 2. Concentration 3. Normalization Figure: (a) Normalization; (b) Concentration ; (c) Dilation Dilation: The dilation 𝐴̅ defined as DIL(𝐴̅)= [uA(x)]1/2 for all x in U Dilation tends to increase the degree of membership of all particular member x by spreading out the characteristic function curve. In the Figure c we illustrated Dilation. Concentration: The concentration 𝐴̅ defined as CON(𝐴̅)= [uA(x)]2 for all x in U The concentration is the opposite of dilation. It tends to decrease the degree of membership of all partial member and concentrates the characteristic function curve. In the Figure b we illustrated Concentration. Normalization: The normalization 𝐴̅ defined as NORM(𝐴̅)= uA(x)/ maxx{ uA(x)} for all x in U we illustrated normalization in the figure a. Normalization provides a means of normalizing all characteristic functions to the same base much the same as vectors can be normalized.