How to Deal with Context in How to Implement the . . . A Seemingly - PowerPoint PPT Presentation

Type-2-Motivated . . . Challenge How to Implement the . . . How to Deal with Context in How to Implement the . . . A Seemingly Natural . . . Computing with Words: Limitations of a . . . Towards a Possible . . . A Type-2-Motivated This Idea Does Lead . . . Approach Home Page Title Page Vladik Kreinovich ◭◭ ◮◮ Department of Computer Science ◭ ◮ University of Texas at El Paso, USA Page 1 of 9 vladik@utep.edu Go Back Full Screen Close Quit

Type-2-Motivated . . . 1. Type-2-Motivated Approach Challenge How to Implement the . . . • An expert describes his/her opinion about a quantity How to Implement the . . . by using imprecise (“fuzzy”) natural language words. A Seemingly Natural . . . • Example: “small”, “large”, etc. Limitations of a . . . Towards a Possible . . . • Each of these words provides a rather crude description This Idea Does Lead . . . of the corresponding quantity. Home Page • A natural way to refine this description is to assign Title Page degrees to which the observed quantity fits each word. ◭◭ ◮◮ • For example, an expert can say that the value is rea- ◭ ◮ sonable small, but to some extent it is medium. Page 2 of 9 • In this refined description, we represent each quantity by a tuple of the corresponding degrees. Go Back Full Screen Close Quit

Type-2-Motivated . . . 2. Challenge Challenge How to Implement the . . . • Need for data processing: How to Implement the . . . – we have such a tuple-based information about sev- A Seemingly Natural . . . eral quantities x 1 , . . . , x m , and Limitations of a . . . – we know that another quantity y is related to x i by Towards a Possible . . . a known relation y = f ( x 1 , . . . , x m ); This Idea Does Lead . . . Home Page – it is desirable to come up with a resulting tuple- based description of y . Title Page • It turns out that a seemingly natural idea for comput- ◭◭ ◮◮ ing such a tuple does not work. ◭ ◮ • This idea cane be modified so that it can be used. Page 3 of 9 Go Back Full Screen Close Quit

Type-2-Motivated . . . 3. How to Implement the Above Approach Challenge How to Implement the . . . • We have degree d i assigned to the i -th word, with mem- How to Implement the . . . bership function µ i ( x ). A Seemingly Natural . . . • Based on this information, what is then the degree Limitations of a . . . µ d ( x ) to which x is a possible value? Towards a Possible . . . This Idea Does Lead . . . – either the quantity is described by the 1st word, Home Page and this word is adequate for x ; degree min( d 1 , µ 1 ( x )); Title Page – here, we interpret “and” as min; – or the quantity is described by the 2nd word, and ◭◭ ◮◮ this word is adequate for x ; the degree min( d 2 , µ 2 ( x )); ◭ ◮ – etc. Page 4 of 9 • We interpret “or” as max . Go Back • So, the resulting degree is µ d ( x ) = max min( d i , µ i ( x )). Full Screen i Close Quit

Type-2-Motivated . . . 4. How to Implement the Above Approach Challenge (cont-d) How to Implement the . . . How to Implement the . . . • Simplest membership functions: A Seemingly Natural . . . Limitations of a . . . µ i ( x ) ✻ ❅ � ❅ � ❅ � ❅ � Towards a Possible . . . ❅ � ❅ � ❅ � ❅ � This Idea Does Lead . . . ❅ � � ❅ ❅ � ❅ � � ❅ � ❅ � ❅ � ❅ Home Page � ❅ � ❅ � ❅ � ❅ ✲ x Title Page ◭◭ ◮◮ • The resulting degree µ d ( x ) = max min( d i , µ i ( x )): i ◭ ◮ µ d ( x ) = max min( d i , µ i ( x )) ✻ Page 5 of 9 i Go Back � ❅ � ❅ � ❅ Full Screen � ❅ ✲ x Close Quit

Type-2-Motivated . . . 5. A Seemingly Natural Implementation Challenge How to Implement the . . . • We want to be able to transform a general membership How to Implement the . . . function µ ( x ) into a tuple of degrees d = ( d 1 , . . . , d n ). A Seemingly Natural . . . • Our hope is that for the f-n µ d ( x ) = max min( d i , µ i ( x )), Limitations of a . . . i we get back the degrees d i . Towards a Possible . . . This Idea Does Lead . . . • Seemingly natural idea: µ ( x ) corresponds to the i -th Home Page word if: Title Page – either a value x is in agreement with µ ( x ) and with ◭◭ ◮◮ this word; – or a value x ′ is in agreement with µ ( x ) and with ◭ ◮ this word; Page 6 of 9 – etc. Go Back • The resulting degree is max min( µ ( x ) , µ i ( x )) . Full Screen x Close Quit

Type-2-Motivated . . . 6. Limitations of a Seemingly Natural Implemen- Challenge tation How to Implement the . . . How to Implement the . . . • Idea: estimate d i as max min( µ ( x ) , µ i ( x )) . x A Seemingly Natural . . . • Our hope is that for the f-n µ d ( x ) = max min( d i , µ i ( x )), Limitations of a . . . i we get back the degrees d i . Towards a Possible . . . This Idea Does Lead . . . • Problem: for the basic function µ ( x ) = µ 1 ( x ) corr. to Home Page d = (1 , 0 . . . , 0), we do not get back (1 , 0 , . . . , 0): Title Page µ 1 ( x ), µ 2 ( x ) ✻ ◭◭ ◮◮ �❅ �❅ � ❅ � ❅ ◭ ◮ � ❅ � ❅ � � ❅ ❅ � � ❅ ❅ ✲ Page 7 of 9 x Go Back • Specifically, for µ ( x ) = µ 1 ( x ), we get Full Screen max min( µ ( x ) , µ 2 ( x )) = 0 . 5 � = d 2 = 0 . Close x Quit

Type-2-Motivated . . . 7. Towards a Possible Solution Challenge How to Implement the . . . • Intersection leads to max min( µ ( x ) , µ i ( x )) � = d i . How to Implement the . . . x • So let us remove the intersecting parts from the mem- A Seemingly Natural . . . bership function before applying the above formula: Limitations of a . . . Towards a Possible . . . – we compute “reduced” basic functions This Idea Does Lead . . . µ ′ i ( x ) = max(0 , µ i ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))); Home Page Title Page – we also compute the “reduced” membership func- tion ◭◭ ◮◮ µ ′ ( x ) = max(0 , µ ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))); ◭ ◮ Page 8 of 9 – then, we compute the degrees based on these re- Go Back duced functions, as Full Screen � x (min( µ ′ ( x ) , µ ′ d i = max i ( x ))) . Close Quit

8. This Idea Does Lead to a Possible Solution Type-2-Motivated . . . Challenge • Reminder: we compute � d i = max x (min( µ ′ ( x ) , µ ′ i ( x ))), How to Implement the . . . How to Implement the . . . where A Seemingly Natural . . . Limitations of a . . . µ ′ i ( x ) = max(0 , µ i ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))); Towards a Possible . . . This Idea Does Lead . . . µ ′ i ( x ) ✻ Home Page ✁❆ � ❅ � ❅ � ❅ ✁ ❆ � ❅ � ❅ � ❅ Title Page ✁ ❆ � ❅ � ❅ � ❅ ✁ ❆ � � ❅ � ❅ ❅ ✁ ❆ � � ❅ � ❅ ❅ ✲ ◭◭ ◮◮ x ◭ ◮ Page 9 of 9 µ ′ ( x ) = max(0 , µ ( x ) − max( µ i − 1 ( x ) , µ i +1 ( x ))); Go Back • Interesting fact: if we apply this to the function µ d ( x ) = Full Screen max min( d i , µ i ( x )), we do get back the degrees d i : i Close � d i = d i . Quit