Fuzzy Logic: Reminder Distributivity Need to Go Beyond [0 , 1] Need for 2-D Extensions From 1-D to 2-D Fuzzy: 2-D Extensions Should . . . A Proof that Interval-Valued Are There Other . . . 2-D Logic: Set of . . . and Complex-Valued Are the Addition (“Or”- . . . Multiplication . . . Only Distributive Options Home Page Title Page Christian Servin 1 , Vladik Kreinovich 2 , and Olga Kosheleva 2 ◭◭ ◮◮ ◭ ◮ 1 Information Technology Dept., El Paso Community College 919 Hunter, El Paso, Texas 79915, USA, cservin@gmail.com Page 1 of 18 2 University of Texas at El Paso, El Paso, TX 79968, USA Go Back vladik@utep.edu, olgak@utep.edu Full Screen Close Quit
Fuzzy Logic: Reminder Distributivity 1. Fuzzy Logic: Reminder Need to Go Beyond [0 , 1] • In the traditional two-valued logic, every statement is Need for 2-D Extensions either true or false. 2-D Extensions Should . . . Are There Other . . . • In the computer these values are represented as, corre- 2-D Logic: Set of . . . spondingly 1 and 0. Addition (“Or”- . . . • These two values cannot capture a situation when an Multiplication . . . expert is not 100% sure about his/her statement. Home Page • To capture such expert uncertainty, L. Zadeh came up Title Page with an idea of fuzzy logic, where for each statement: ◭◭ ◮◮ – instead of two possible truth values 0 and 1, ◭ ◮ – we can have degrees of certainty that can take any Page 2 of 18 values from 0 to 1. Go Back • We now need to extend propositional operations from Full Screen { 0 , 1 } to [0 , 1]. Close Quit
Fuzzy Logic: Reminder Distributivity 2. Fuzzy Logic (cont-d) Need to Go Beyond [0 , 1] • We need to extend propositional operations from { 0 , 1 } Need for 2-D Extensions to [0 , 1]. 2-D Extensions Should . . . Are There Other . . . • From the purely mathematical viewpoint, there are 2-D Logic: Set of . . . many such extensions. Addition (“Or”- . . . • It is desirable to preserve as many properties of the Multiplication . . . 2-valued logic as possible. Home Page • Usually, “and”- and “or”-operations are selected to be Title Page commutative and associative. ◭◭ ◮◮ • This still leaves us with plenty of different choices. ◭ ◮ • It is therefore desirable, among all such operations, to Page 3 of 18 select those that satisfy additional properties. Go Back Full Screen Close Quit
Fuzzy Logic: Reminder Distributivity 3. Distributivity Need to Go Beyond [0 , 1] • One of such additional natural properties is distributiv- Need for 2-D Extensions ity, that A & ( B ∨ C ) is equivalent to ( A & B ) ∨ ( A & C ): 2-D Extensions Should . . . Are There Other . . . f & ( a, f ∨ ( b, c )) = f ∨ ( f & ( a, b ) , f & ( a, c )) . 2-D Logic: Set of . . . Addition (“Or”- . . . • If we require this for all a , b , and c , then Multiplication . . . f ∨ ( a, b ) = max( a, b ) . Home Page Title Page • It is known that sometimes, the expert’s use of “or” is better described by other “or”-operations. ◭◭ ◮◮ • It is reasonable to restrict the above equality to cases ◭ ◮ when f ∨ ( b, c ) < 1. Page 4 of 18 • Then, “and”- and “or”-operations are equivalent to Go Back f & ( a, b ) = a · b and f ∨ ( a, b ) = min( a + b, 1). Full Screen Close Quit
Fuzzy Logic: Reminder Distributivity 4. Need to Go Beyond [0 , 1] Need to Go Beyond [0 , 1] • The [0 , 1]-based fuzzy logic captures many features of Need for 2-D Extensions expert uncertainty. 2-D Extensions Should . . . Are There Other . . . • However, in some situations, it is not fully adequate to 2-D Logic: Set of . . . distinguish between different situations; e.g.: Addition (“Or”- . . . – if we have no information about a given statement, Multiplication . . . – then it makes sense to describe this uncertainty by Home Page the midpoint 0.5. Title Page • On the other hand: ◭◭ ◮◮ – if have exactly as many arguments supporting S as ◭ ◮ supporting ¬ S , Page 5 of 18 – then it also makes sense to describe this uncertainty by the value 0.5. Go Back Full Screen • In both situations, the truth value is the same, but the uncertainty is different. Close Quit
Fuzzy Logic: Reminder Distributivity 5. Need for 2-D Extensions Need to Go Beyond [0 , 1] • If we add an argument in support of S , then: Need for 2-D Extensions 2-D Extensions Should . . . – in the first case, we now have an argument support- Are There Other . . . ing S and no arguments supporting ¬ S , 2-D Logic: Set of . . . – so the truth value of S should drastically increase; Addition (“Or”- . . . – in the second case, the numbers of statement sup- Multiplication . . . porting S and ¬ S remains almost equal; Home Page – so, the truth value should not change much. Title Page • To distinguish between such situations, it is desirable: ◭◭ ◮◮ – to supplement the [0 , 1]-valued degree of belief ◭ ◮ – with an additional number (or numbers). Page 6 of 18 • The simplest case: use one additional number. Go Back • Thus, we use two numbers to describe our degree of Full Screen certainty in a given statement. Close Quit
Fuzzy Logic: Reminder Distributivity 6. 2-D Extensions Should Be Commutative, As- Need to Go Beyond [0 , 1] sociative, and Distributive Need for 2-D Extensions • From the commonsense viewpoint, logical operations 2-D Extensions Should . . . are commutative, associative, and distributive. Are There Other . . . 2-D Logic: Set of . . . • It is thus reasonable to require that the 2-D extensions Addition (“Or”- . . . of satisfy these three properties. Multiplication . . . • The most widely used 2-D extension is interval-valued Home Page fuzzy logic. Title Page • There, our degree of certainty in a statement is de- ◭◭ ◮◮ scribed by an interval [ d, d ] ⊆ [0 , 1]. ◭ ◮ • This enables us to clearly distinguish between the above two situations: Page 7 of 18 – the case of complete uncertainty is naturally de- Go Back scribed by the interval [0 , 1], while Full Screen – the case when equally many arguments for S and Close for ¬ S is described by [0 , 5 , 0 . 5] = { 0 . 5 } . Quit
Fuzzy Logic: Reminder Distributivity 7. Distributive 2-D Extensions Need to Go Beyond [0 , 1] • In principle, we can extend different t-norms and t- Need for 2-D Extensions conorms to the interval-valued case: 2-D Extensions Should . . . def Are There Other . . . f ([ a, a ] , [ b, b ]) = { f ( a, b ) : a ∈ [ a, a ] and b ∈ [ b, b ] } . 2-D Logic: Set of . . . • In particular, for a · b and a + b , we get Addition (“Or”- . . . [ a, a ] · [ b, b ] = [ a · b, a · b ]; [ a, a ] + [ b, b ] = [ a + b, a + b ] . Multiplication . . . Home Page • The resulting interval-valued logic is distributive. Title Page • Another useful 2-D distributive extension of the usual ◭◭ ◮◮ fuzzy logic is the complex-valued fuzzy logic. ◭ ◮ • In this logic, degrees of belief can take any complex = √− 1. def Page 8 of 18 values a + b · i, with i Go Back • The complex-valued logic lacks a clear justification and clear interpretation. Full Screen • Thus, it is not as widely used an interval-valued one. Close Quit
Fuzzy Logic: Reminder Distributivity 8. Are There Other Extension? Need to Go Beyond [0 , 1] • At first glance, it looks like: Need for 2-D Extensions 2-D Extensions Should . . . – the above two extensions have been rather arbitrar- Are There Other . . . ily chosen, and 2-D Logic: Set of . . . – in principle, there are many other extensions. Addition (“Or”- . . . • We show that interval-valued and complex-valued are Multiplication . . . the only possible 2-D distributive extensions. Home Page • This result elevates complex-valued fuzzy logic: Title Page ◭◭ ◮◮ – from the status of one of the mathematically pos- sible extensions ◭ ◮ – to a much higher status of one of the two possible Page 9 of 18 extensions. Go Back • This will, hopefully lead to a more frequent use of Full Screen complex-valued fuzzy logic. Close Quit
Fuzzy Logic: Reminder Distributivity 9. 2-D Logic: Set of Possible Values Need to Go Beyond [0 , 1] • Let ⊙ and ⊕ be 2-D extensions of · and +. Need for 2-D Extensions 2-D Extensions Should . . . • Let x be a 2-D element different from real numbers. Are There Other . . . • On this extended set, we want to allow multiplication. 2-D Logic: Set of . . . • Thus, we need to consider elements of the type b ⊙ x Addition (“Or”- . . . for arbitrary real numbers b . Multiplication . . . Home Page • We also want to allow addition between real numbers a and the products b ⊙ x : a ⊕ ( b ⊙ x ) . Title Page • The set of all such elements depends on two parameters ◭◭ ◮◮ a and b and is, thus, 2-dimensional. ◭ ◮ • We are interested in 2-D extensions. Page 10 of 18 • Thus, the desired extension cannot contain any other Go Back elements. Full Screen • So, each extension is the set of all the elements of the Close type a ⊕ ( b ⊙ x ) . Quit
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