Fuzzy Logic: Brief . . . Towards General . . . Need for Product . . . Natural Questions How to Tell When a Product Similar Questions in . . . Definitions of Two Partially Ordered Main Result Auxiliary Results Spaces Has a Certain Proof of the Main Result Property? Home Page Title Page Francisco Zapata, Olga Kosheleva ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ 500 W. University El Paso, TX 79968, USA Page 1 of 17 fazg74@gmail.com, olgak@utep.edu Go Back Karen Villaverde Full Screen Department of Computer Science New Mexico State University Close Las Cruces, New Mexico 88003, USA Email: kvillave@cs.nmsu.edu Quit
Fuzzy Logic: Brief . . . Towards General . . . 1. Fuzzy Logic: Brief Reminder Need for Product . . . • In the traditional 2-valued logic, every statement is Natural Questions either true or false. Similar Questions in . . . Definitions • Thus, the set of possible truth values consists of two Main Result elements: true (1) and false (0). Auxiliary Results • Fuzzy logic takes into account that people have differ- Proof of the Main Result ent degrees of certainty in their statements. Home Page • Traditionally, fuzzy logic uses values from the interval Title Page [0 , 1] to describe uncertainty. ◭◭ ◮◮ • In this interval, the order is total ( linear ) in the sense ◭ ◮ that for every a, a ′ ∈ [0 , 1], either a ≤ a ′ or a ′ ≤ a . Page 2 of 17 • However, often, partial orders provide a more adequate Go Back description of the expert’s degree of confidence. Full Screen Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 2. Towards General Partial Orders Need for Product . . . • For example, an expert cannot describe her degree of Natural Questions certainty by an exact number. Similar Questions in . . . Definitions • Thus, it makes sense to describe this degree by an in- Main Result terval [ d, d ] of possible numbers. Auxiliary Results • Intervals are only partially ordered; e.g., the intervals Proof of the Main Result [0 . 5 , 0 . 5] and [0 , 1] are not easy to compare. Home Page • More complex sets of possible degrees are also some- Title Page times useful. ◭◭ ◮◮ • Not to miss any new options, in this paper, we consider ◭ ◮ general partially ordered spaces. Page 3 of 17 Go Back Full Screen Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 3. Need for Product Operations Need for Product . . . • Often, two (or more) experts evaluate a statement S . Natural Questions Similar Questions in . . . • Then, our certainty in S is described by a pair ( a 1 , a 2 ), Definitions where a i ∈ A i is the i -th expert’s degree of certainty. Main Result • To compare such pairs, we must therefore define a par- Auxiliary Results tial order on the set A 1 × A 2 of all such pairs. Proof of the Main Result • One example of a partial order on A 1 × A 2 is a Cartesian Home Page product: ( a 1 , a 2 ) ≤ ( a ′ 1 , a ′ 2 ) ⇔ (( a 1 ≤ a ′ 1 ) & ( a 2 ≤ a ′ 2 )) . Title Page • This is a cautious approach, when our confidence in S ′ ◭◭ ◮◮ is higher than in S ⇔ it is higher for both experts. ◭ ◮ • Lexicographic product: ( a 1 , a 2 ) ≤ ( a ′ 1 , a ′ 2 ) ⇔ Page 4 of 17 (( a 1 ≤ a ′ 1 ) & a 1 � = a ′ 1 ) ∨ (( a 1 = a ′ 1 ) & ( a 2 ≤ a ′ 2 ))) . Go Back • Here, we are absolutely confident in the 1st expert – Full Screen and only use the 2nd when the 1st is not sure. Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 4. Natural Questions Need for Product . . . • Question: when does the resulting partially ordered set Natural Questions A 1 × A 2 satisfy a certain property? Similar Questions in . . . Definitions • Examples: is it a total order? is it a lattice order? Main Result • It is desirable to reduce the question about A 1 × A 2 to Auxiliary Results questions about properties of component spaces A i . Proof of the Main Result Home Page • Some such reductions are known ; e.g.: Title Page – A Cartesian product is a total order ⇔ one of A i is a total order, and the other has only one element. ◭◭ ◮◮ – A lexicographic product is a total order if and only ◭ ◮ if both components are totally ordered. Page 5 of 17 • In this paper, we provide a general algorithm for such Go Back reduction. Full Screen Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 5. Similar Questions in Other Areas Need for Product . . . • Similar questions arise in other applications of ordered Natural Questions sets. Similar Questions in . . . Definitions • Example: in space-time geometry, a ≤ b means that an Main Result event a can influence the event b . Auxiliary Results • Our algorithm does not use the fact that the original Proof of the Main Result relations are orders. Home Page • Thus, our algorithm is applicable to a general binary Title Page relation – equivalence, similarity, etc. ◭◭ ◮◮ • Moreover, this algorithm can be applied to the case ◭ ◮ when we have a space with several binary relations. Page 6 of 17 • Example: we may have an order relation and a simi- Go Back larity relation. Full Screen Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 6. Definitions Need for Product . . . • By a space , we mean a set A with m binary relations Natural Questions P 1 ( a, a ′ ) , . . . , P m ( a, a ′ ) . Similar Questions in . . . Definitions • By a 1st order property , we mean a formula F obtained Main Result from P i ( x, x ′ ) by using logical ∨ , & , ¬ , → , ∃ x and ∀ x . Auxiliary Results • Note: most properties of interest are 1st order; e.g. to Proof of the Main Result be a total order means ∀ a ∀ a ′ (( a ≤ a ′ ) ∨ ( a ′ ≤ a )) . Home Page • By a product operation , we mean a collection of m Title Page propositional formulas that ◭◭ ◮◮ – describe the relation P i (( a 1 , a 2 ) , ( a ′ 1 , a ′ 2 )) between the ◭ ◮ elements ( a 1 , a 2 ) , ( a ′ 1 , a ′ 2 ) ∈ A 1 × A 2 Page 7 of 17 – in terms of the relations between the components a 1 , a ′ 1 ∈ A 1 and a 2 , a ′ 2 ∈ A 2 of these elements. Go Back • Note: both Cartesian and lexicographic order are prod- Full Screen uct operations in this sense. Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 7. Main Result Need for Product . . . • Main Result. There exists an algorithm that, given Natural Questions Similar Questions in . . . • a product operation and Definitions • a property F , Main Result generates a list of properties F 11 , F 12 , . . . , F p 1 , F p 2 s.t.: Auxiliary Results Proof of the Main Result F ( A 1 × A 2 ) ⇔ (( F 11 ( A 1 ) & F 12 ( A 2 )) ∨ . . . ∨ ( F p 1 ( A 1 ) & F p 2 ( A 2 ))) . Home Page • Example: For Cartesian product and total order F , we Title Page have ◭◭ ◮◮ F ( A 1 × A 2 ) ⇔ (( F 11 ( A 1 ) & F 12 ( A 2 )) ∨ ( F 21 ( A 1 ) & F 22 ( A 2 ))) : ◭ ◮ • F 11 ( A 1 ) means that A 1 is a total order, Page 8 of 17 • F 12 ( A 2 ) means that A 2 is a one-element set, Go Back • F 21 ( A 1 ) means that A 1 is a one-element set, and Full Screen • F 22 ( A 2 ) means that A 2 is a total order. Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 8. Auxiliary Results Need for Product . . . • Generalization: Natural Questions Similar Questions in . . . – A similar algorithm can be formulated for a product Definitions of three or more spaces. Main Result – A similar algorithm can be formulated for the case Auxiliary Results when we allow ternary and higher order operations. Proof of the Main Result • Specifically for partial orders: Home Page – The only product operations that always leads to Title Page a partial order on A 1 × A 2 for which ◭◭ ◮◮ ( a 1 ≤ 1 a ′ 1 & a 2 ≤ 2 a ′ 2 ) → ( a 1 , a 2 ) ≤ ( a ′ 1 , a ′ 2 ) ◭ ◮ Page 9 of 17 are Cartesian and lexicographic products. Go Back Full Screen Close Quit
Fuzzy Logic: Brief . . . Towards General . . . 9. Proof of the Main Result Need for Product . . . • The desired property F ( A 1 × A 2 ) uses: Natural Questions – relations P i ( a, a ′ ) between elements a, a ′ ∈ A 1 × A 2 ; Similar Questions in . . . Definitions – quantifiers ∀ a and ∃ a over elements a ∈ A 1 × A 2 . Main Result • Every element a ∈ A 1 × A 2 is, by definition, a pair Auxiliary Results ( a 1 , a 2 ) in which a 1 ∈ A 1 and a 2 ∈ A 2 . Proof of the Main Result Home Page • Let us explicitly replace each variable with such a pair. Title Page • By definition of a product operation: ◭◭ ◮◮ – each relation P i (( a 1 , a 2 ) , ( a ′ 1 , a ′ 2 )) ◭ ◮ – is a propositional combination of relations betw. el- ements a 1 , a ′ 1 ∈ A 1 and betw. elements a 2 , a ′ 2 ∈ A 2 . Page 10 of 17 • Let us perform the corresponding replacement. Go Back • Each quantifier can be replaced by quantifiers corre- Full Screen sponding to components: e.g., ∀ ( a 1 , a 2 ) ⇔ ∀ a 1 ∀ a 2 . Close Quit
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