Partial Orders for Representing Uncertainty, Causality, and Decision Making: General Properties, Operations, and Algorithms

Partial Orders are . . .

What We Plan to Do

Uncertainty is . . .

Properties of Ordered . . . Partial Orders for Towards Combining . . . Representing Uncertainty, Main Result Auxiliary Results Causality, and Decision Proof of the Main Result My Publications Making: General Properties, Home Page Operations, and Algorithms Title Page JJ II Francisco Zapata Department of Computer Science J I

University of Texas at El Paso Page1 of 45 500 W. University El Paso, TX 79968, USA Go Back [email protected] Full Screen

Quit Partial Orders are . . .

What We Plan to Do 1. Partial Orders are Important Uncertainty is . . . • One of the main objectives of science and engineering Properties of Ordered . . . is to select the most beneficial decisions. For that: Towards Combining . . . – we must know people’s preferences, Main Result Auxiliary Results – we must have the information about different events Proof of the Main Result (possible consequences of different decisions), and My Publications

– since information is never absolutely accurate, we Home Page must have information about uncertainty. Title Page • All these types of information naturally lead to partial orders: JJ II – For preferences, a ≤ b means that b is preferable to J I a. This relation is used in decision theory. Page2 of 45

– For events, a ≤ b means that a can inﬂuence b. This Go Back

causality relation is used in space-time physics. Full Screen – For uncertain statements, a ≤ b means that a is Close less certain than b (fuzzy logic etc.). Quit Partial Orders are . . .

What We Plan to Do 2. What We Plan to Do Uncertainty is . . . • In each of the three areas, there is a lot of research Properties of Ordered . . . about studying the corresponding partial orders. Towards Combining . . .

Main Result • This research has revealed that some ideas are common in all three applications of partial orders. Auxiliary Results Proof of the Main Result

• In our research, we plan to analyze: My Publications – general properties, operations, and algorithms Home Page – related to partial orders for representing uncertainty, Title Page

causality, and decision making. JJ II

• In our analysis, we will be most interested in uncer- J I tainty – the computer-science aspect of partial orders. Page3 of 45

• In our presentation: Go Back

– we ﬁrst give a general outline, Full Screen

– then present two results in detail (if time allows). Close

Quit Partial Orders are . . .

What We Plan to Do 3. Uncertainty is Ubiquitous in Applications of Partial Orders Uncertainty is . . . Properties of Ordered . . .

• Uncertainty is explicitly mentioned only in the computer- Towards Combining . . . science example of partial orders. Main Result • However, uncertainty is ubiquitous in describing our Auxiliary Results knowledge about all three types of partial orders. Proof of the Main Result My Publications

• For example, we may want to check what is happening Home Page exactly 1 second after a certain reaction. Title Page • However, in practice, we cannot measure time exactly. JJ II • So, we can only observe an event which is close to b – e.g., that occurs 1 ± 0.001 sec after the reaction. J I Page4 of 45 • In general, we can only guarantee that the observed Go Back event is within a certain neighborhood Ub of the event b. • In decision making, we similarly know the user’s pref- Full Screen erences only with some accuracy. Close

Quit Partial Orders are . . .

What We Plan to Do 4. Uncertainty-Motivated Experimentally Conﬁrmable Relation Uncertainty is . . . Properties of Ordered . . .

• Because of the uncertainty: Towards Combining . . . – the only possibility to experimentally confirm that Main Result a precedes b (e.g., that a can causally influence b) Auxiliary Results Proof of the Main Result – is when for some neighborhood Ub of the event b, My Publications we have a ≤ eb for all eb ∈ Ub. Home Page • In topological terms, this “experimentally confirmable” relation a ≺ b means that: Title Page + JJ II – the element b is contained in the future cone Ca = {c : a ≤ c} of the event a J I

– together with some neighborhood. Page5 of 45 + • In other words, b belongs to the interior Ka of the Go Back closed cone C+. a Full Screen • Such relation, in which future cones are open, are called Close open. Quit Partial Orders are . . .

What We Plan to Do 5. Uncertainty-Motivated Experimentally Conﬁrmable Relation (cont-d) Uncertainty is . . . Properties of Ordered . . .

• In usual space-time models: Towards Combining . . . + Main Result – once we know the open cone Ka , Auxiliary Results – we can reconstruct the original cone C+ as the clo- a Proof of the Main Result sure of K+: C+ = K+. a a a My Publications • A natural question is: vice versa, Home Page – can we uniquely reconstruct an open order Title Page

– if we know the corresponding closed order? JJ II

• In our paper (Zapata Kreinovich to appear), we show J I

that this reconstruction is possible. Page6 of 45

• This result provides a partial solution to a known open Go Back problem. Full Screen

Quit Partial Orders are . . .

What We Plan to Do 6. From Potentially Experimentally Conﬁrmable (EC) Relation to Actually EC One Uncertainty is . . . Properties of Ordered . . .

• It is also important to check what can be conﬁrmed Towards Combining . . . when we only have observations with a given accuracy. Main Result • For example: Auxiliary Results Proof of the Main Result

– instead of the knowing the exact time location of My Publications

an an event a, Home Page

– we only know an event a that preceded a and an Title Page event a that follows a. JJ II • In this case, the only information that we have about the actual event a is that it belongs to the interval J I Page7 of 45 [a, a] def= {a : a ≤ a ≤ a}. Go Back

• It is desirable to describe possible relations between Full Screen such intervals. Close

Quit Partial Orders are . . .

What We Plan to Do 7. From Potentially Experimentally Conﬁrmable (EC) Relation to Actually EC One (cont-d) Uncertainty is . . . Properties of Ordered . . .

• It is desirable to describe possible relations between Towards Combining . . . such intervals. Main Result • Such a description has already been done for intervals Auxiliary Results on the real line. Proof of the Main Result My Publications

• The resulting description is known as Allen’s algebra. Home Page

• In these terms, what we want is to generalize Allen’s Title Page algebra to intervals over an arbitrary poset. JJ II • We are currently working on a paper about intervals. J I

• Instead of intervals, we can also consider more general Page8 of 45 sets. Go Back • Our preliminary results about general sets are described Full Screen in a paper (Zapata Ramirez et al. 2011). Close

Quit Partial Orders are . . .

What We Plan to Do 8. Properties of Ordered Spaces Uncertainty is . . . • Once a new ordered set is deﬁned, we may be interested Properties of Ordered . . . in its properties. Towards Combining . . . • For example, we may want to know when such an order Main Result is a lattice, i.e., when: Auxiliary Results Proof of the Main Result – for every two elements, My Publications

– there is the greatest lower bound and the least up- Home Page per bound. Title Page • If this set is not a lattice, we may want to know: JJ II – when the order is a semi-lattice, i.e., e.g., J I – when every two elements have the least upper bound. Page9 of 45 • For the class of all subsets, we prove the lattice property in (Zapata Ramirez et al. 2011). Go Back • We also describe when special relativity-type ordered Full Screen spaces are lattices (K¨unziet al. 2011). Close

Quit Partial Orders are . . .

What We Plan to Do 9. Towards Combining Ordered Spaces: Fuzzy Logic Uncertainty is . . . • In the traditional 2-valued logic, every statement is Properties of Ordered . . . either true or false. Towards Combining . . .

Main Result • Thus, the set of possible truth values consists of two elements: true (1) and false (0). Auxiliary Results Proof of the Main Result

• Fuzzy logic takes into account that people have diﬀer- My Publications

ent degrees of certainty in their statements. Home Page

• Traditionally, fuzzy logic uses values from the interval Title Page [0, 1] to describe uncertainty. JJ II • In this interval, the order is total (linear) in the sense J I that for every a, a0 ∈ [0, 1], either a ≤ a0 or a0 ≤ a. Page 10 of 45 • However, often, partial orders provide a more adequate Go Back description of the expert’s degree of conﬁdence. Full Screen

Quit Partial Orders are . . .

What We Plan to Do 10. Towards General Partial Orders Uncertainty is . . . • For example, an expert cannot describe her degree of Properties of Ordered . . . certainty by an exact number. Towards Combining . . .

Main Result • Thus, it makes sense to describe this degree by an interval [d, d] of possible numbers. Auxiliary Results Proof of the Main Result

• Intervals are only partially ordered; e.g., the intervals My Publications

[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. JJ II • Not to miss any new options, in this research, we con- J I sider general partially ordered spaces. Page 11 of 45

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Quit Partial Orders are . . .

What We Plan to Do 11. Need for Product Operations Uncertainty is . . . • Often, two (or more) experts evaluate a statement S. Properties of Ordered . . . Towards Combining . . . • Then, our certainty in S is described by a pair (a1, a2), Main Result where ai ∈ Ai is the i-th expert’s degree of certainty. Auxiliary Results

• To compare such pairs, we must therefore deﬁne a par- Proof of the Main Result

tial order on the set A1 × A2 of all such pairs. My Publications

Home Page • One example of a partial order on A1×A2 is a Cartesian 0 0 0 0 product: (a1, a2) ≤ (a1, a2) ⇔ ((a1 ≤ a1)&(a2 ≤ a2)). Title Page 0 • This is a cautious approach, when our conﬁdence in S JJ II is higher than in S ⇔ it is higher for both experts. J I 0 0 • Lexicographic product: (a1, a2) ≤ (a1, a2) ⇔ Page 12 of 45 0 0 0 0 ((a1 ≤ a1)& a1 6= a1) ∨ ((a1 = a1)&(a2 ≤ a2))). 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 Partial Orders are . . .

What We Plan to Do 12. Natural Questions Uncertainty is . . . • Question: when does the resulting partially ordered set Properties of Ordered . . . A1 × A2 satisfy a certain property? Towards Combining . . . Main Result • Examples: is it a total order? is it a lattice order? Auxiliary Results

• It is desirable to reduce the question about A1 × A2 to Proof of the Main Result

questions about properties of component spaces Ai. My Publications • Some such reductions are known; e.g.: Home Page

Title Page – A Cartesian product is a total order ⇔ one of Ai is a total order, and the other has only one element. JJ II

– A lexicographic product is a total order if and only J I if both components are totally ordered. Page 13 of 45

• In this talk, we provide a general algorithm for such Go Back reduction. Full Screen

Quit Partial Orders are . . .

What We Plan to Do 13. Similar Questions in Other Areas Uncertainty is . . . • Similar questions arise in other applications of ordered Properties of Ordered . . . sets. Towards Combining . . .

Main Result • Example: in space-time geometry, a ≤ b means that an event a can inﬂuence the event b. Auxiliary Results Proof of the Main Result

• Our algorithm does not use the fact that the original My Publications

relations are orders. Home Page

• Thus, our algorithm is applicable to a general binary Title Page relation – equivalence, similarity, etc. JJ II • Moreover, this algorithm can be applied to the case J I when we have a space with several binary relations. Page 14 of 45 • Example: we may have an order relation and a simi- Go Back larity relation. Full Screen

Quit Partial Orders are . . .

What We Plan to Do 14. Deﬁnitions Uncertainty is . . . • By a space, we mean a set A with m binary relations Properties of Ordered . . . 0 0 P1(a, a ),..., Pm(a, a ). Towards Combining . . . • By a 1st order property, we mean a formula F obtained Main Result 0 Auxiliary Results from Pi(x, x ) by using logical ∨, &, ¬, →, ∃x and ∀x. Proof of the Main Result

• Note: most properties of interest are 1st order; e.g. to My Publications 0 0 0 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 JJ II 0 0 – describe the relation Pi((a1, a2), (a1, a2)) between the 0 0 J I elements (a1, a2), (a1, a2) ∈ A1 × A2 Page 15 of 45 – in terms of the relations between the components 0 0 Go Back a1, a1 ∈ A1 and a2, a2 ∈ A2 of these elements. • Note: both Cartesian and lexicographic order are prod- Full Screen

uct operations in this sense. Close

Quit Partial Orders are . . .

What We Plan to Do 15. Main Result Uncertainty is . . . • Main Result. There exists an algorithm that, given Properties of Ordered . . . Towards Combining . . . • a product operation and Main Result • a property F , Auxiliary Results

generates a list of properties F11, F12,..., Fp1, Fp2 s.t.: Proof of the Main Result

My Publications F (A ×A ) ⇔ ((F (A )& F (A ))∨...∨(F (A )& F (A ))). 1 2 11 1 12 2 p1 1 p2 2 Home Page

• Example: For Cartesian product and total order F , we Title Page have JJ II F (A1×A2) ⇔ ((F11(A1)& F12(A2))∨(F21(A1)& F22(A2))) : J I

• F11(A1) means that A1 is a total order, Page 16 of 45

• F12(A2) means that A2 is a one-element set, Go Back • F (A ) means that A is a one-element set, and 21 1 1 Full Screen • F22(A2) means that A2 is a total order. Close

Quit Partial Orders are . . .

What We Plan to Do 16. Auxiliary Results Uncertainty is . . . • Generalization: Properties of Ordered . . . Towards Combining . . . – A similar algorithm can be formulated for a product Main Result of three or more spaces. Auxiliary Results – A similar algorithm can be formulated for the case Proof of the Main Result when we allow ternary and higher order operations. My Publications • Speciﬁcally for partial orders: Home Page – The only product operations that always leads to Title Page

a partial order on A1 × A2 for which JJ II 0 0 0 0 (a1 ≤1 a1 & a2 ≤2 a2) → (a1, a2) ≤ (a1, a2) J I are Cartesian and lexicographic products. Page 17 of 45

Go Back

Full Screen

Quit Partial Orders are . . .

What We Plan to Do 17. Proof of the Main Result Uncertainty is . . .

Properties of Ordered . . . • The desired property F (A1 × A2) uses: 0 0 Towards Combining . . . – relations Pi(a, a ) between elements a, a ∈ A1 ×A2; Main Result – quantiﬁers ∀a and ∃a over elements a ∈ A1 × A2. Auxiliary Results

• Every element a ∈ A1 × A2 is, by deﬁnition, a pair Proof of the Main Result

(a1, a2) in which a1 ∈ A1 and a2 ∈ A2. My Publications • Let us explicitly replace each variable with such a pair. Home Page

Title Page • By deﬁnition of a product operation: 0 0 JJ II – each relation Pi((a1, a2), (a1, a2)) – is a propositional combination of relations betw. el- J I

0 0 Page 18 of 45 ements a1, a1 ∈ A1 and betw. elements a2, a2 ∈ A2. • Let us perform the corresponding replacement. Go Back • Each quantiﬁer can be replaced by quantiﬁers corre- Full Screen

sponding to components: e.g., ∀(a1, a2) ⇔ ∀a1∀a2. Close

Quit Partial Orders are . . .

What We Plan to Do 18. Proof of the Main Result (cont-d) Uncertainty is . . . • So, we get an equivalent reformulation of F s.t.: Properties of Ordered . . . Towards Combining . . . – elementary formulas are relations between elements Main Result of A1 or between A2, and Auxiliary Results – quantiﬁers are over A1 or over A2. Proof of the Main Result

• We use induction to reduce to the desired form My Publications

Home Page ((F11(A1)& F12(A2)) ∨ ... ∨ (Fp1(A1)& Fp2(A2))). Title Page • Elementary formulas are already of the desired form – provided, of course, that we allow free variables. JJ II J I • We will show that: Page 19 of 45 – if we apply a propositional connective or a quanti- ﬁer to a formula of this type, Go Back – then we can reduce the result again to the formula Full Screen

of this type. Close

Quit Partial Orders are . . .

What We Plan to Do 19. Applying Propositional Connectives Uncertainty is . . . • We apply propositional connectives to formulas of the Properties of Ordered . . . type Towards Combining . . .

Main Result ((F11(A1)& F12(A2)) ∨ ... ∨ (Fp1(A1)& Fp2(A2))). Auxiliary Results

• We thus get a propositional combination of the formu- Proof of the Main Result las of the type F (A ). ij j My Publications • An arbitrary propositional combination can be described Home Page

as a disjunction of conjunctions (DNF form). Title Page • Each conjunction combines properties related to A 1 JJ II and properties related to A2, i.e., has the form J I G1(A1)& ... & Gp(A1)& Gp+1(A2)& ... & Gq(A2). Page 20 of 45 0 • Thus, each conjunction has the from G(A1)& G (A2), Go Back where G(A1) ⇔ (G1(A1)& ... & Gp(A1)). Full Screen • Thus, the disjunction of such properties has the desired form. Close

Quit Partial Orders are . . .

What We Plan to Do 20. Applying Existential Quantiﬁers Uncertainty is . . .

Properties of Ordered . . . • When we apply ∃a1, we get a formula Towards Combining . . . ∃a1 ((F11(A1)& F12(A2)) ∨ ... ∨ (Fp1(A1)& Fp2(A2))). Main Result

• It is known that ∃a (A∨B) is equivalent to ∃a A∨∃a B. Auxiliary Results

• Thus, the above formula is equivalent to a disjunction Proof of the Main Result

My Publications ∃a1 (F11(A1)& F12(A2))∨...∨∃a1 (Fp1(A1)& Fp2(A2)). Home Page • Thus, it is suﬃcient to prove that each formula Title Page ∃a1 (Fi1(A1)& Fi2(A2)) has the desired form. JJ II • The term Fi2(A2) does not depend on a1 at all, it is all about elements of A2. J I • Thus, the above formula is equivalent to Page 21 of 45

Go Back (∃a1 Fi1(A1)) & Fi2(A2). 0 Full Screen • So, it is equivalent to the formula Fi1(A1)& Fi2(A2), 0 where Fi1 ⇔ ∃a1 Fi1(A1). Close

Quit Partial Orders are . . .

What We Plan to Do 21. Applying Universal Quantiﬁers Uncertainty is . . .

Properties of Ordered . . . • When we apply a universal quantiﬁer, e.g., ∀a1, then we can use the fact that ∀a1 F is equivalent to ¬∃a1 ¬F . Towards Combining . . . • We assumed that the formula F is of the desired type Main Result Auxiliary Results

(F11(A1)& F12(A2)) ∨ ... ∨ (Fp1(A1)& Fp2(A2)). Proof of the Main Result

My Publications • By using the propositional part of this proof, we con- Home Page clude that ¬F can be reduced to the desired type. Title Page • Now, by applying the ∃ part of this proof, we conclude JJ II that ∃a1 (¬F ) can also be reduced to the desired type. • By using the propositional part again, we conclude that J I ¬(∃a1 ¬F ) can be reduced to the desired type. Page 22 of 45 • By induction, we can now conclude that the original Go Back

formula can be reduced to the desired type. Full Screen

• The main result is proven. Close

Quit Partial Orders are . . .

What We Plan to Do 22. Example of Applying the Algorithm Uncertainty is . . . • Let us apply our algorithm to checking whether a Carte- Properties of Ordered . . . sian product is totally ordered. Towards Combining . . . • In this case, F has the form ∀a∀a0 ((a ≤ a0)∨(a0 ≤ a)). Main Result Auxiliary Results 0 • We ﬁrst replace each variable a, a ∈ A1 × A2 with the Proof of the Main Result

corresponding pair: My Publications

0 0 0 0 0 0 Home Page ∀(a1, a2)∀(a1, a2) (((a1, a2) ≤ (a1, a2))∨((a1, a2) ≤ (a1, a2))). • Replacing the ordering relation on the Cartesian prod- Title Page

uct with its deﬁnition, we get JJ II 0 0 0 0 0 0 ∀(a1, a2)∀(a1, a2) ((a1 ≤ a1 & a2 ≤ a2)∨(a1 ≤ a1 & a2 ≤ a2)). J I

• Replacing ∀a over pairs with individual ∀ai, we get: Page 23 of 45 0 0 0 0 0 0 ∀a1∀a2∀a1∀a2 ((a1 ≤ a1 & a2 ≤ a2))∨((a1 ≤ a1 & a2 ≤ a2))). Go Back

• By using the ∀ ⇔ ¬∃¬, we get an equivalent form Full Screen

0 0 0 0 0 0 Close ¬∃a1∃a2∃a1∃a2 ¬((a1 ≤ a1 & a2 ≤ a2)∨(a1 ≤ a1 & a2 ≤ a2))).

Quit Partial Orders are . . .

What We Plan to Do 23. Example (cont-d) Uncertainty is . . . • So far, we got: Properties of Ordered . . . 0 0 0 0 0 0 Towards Combining . . . ¬∃a1∃a2∃a1∃a2 ¬((a1 ≤ a1 & a2 ≤ a2)∨(a1 ≤ a1 & a2 ≤ a2))). Main Result

• Moving ¬ inside the propositional formula, we get Auxiliary Results 0 0 0 0 0 0 ¬∃a1∃a1∃a1∃a2 ((a1 6≤ a1 ∨a2 6≤ a2)&(a1 6≤ a1 ∨a2 6≤ a2))). Proof of the Main Result

0 0 0 0 My Publications • The formula (a1 6≤ a1 ∨ a2 6≤ a2)) & (a1 6≤ a1 ∨ a2 6≤ a2) must now be transformed into a DNF form. Home Page 0 0 0 0 Title Page • The result is (a1 6≤ a1 & a1 6≤ a1)∨(a1 6≤ a1 & a2 6≤ a2)∨ 0 0 0 0 (a2 6≤ a2 & a1 6≤ a1) ∨ (a2 6≤ a2 & a2 6≤ a2). JJ II

• Thus, our formula is ⇔ ¬(F1 ∨ F2 ∨ F3 ∨ F4), where J I

0 0 0 0 Page 24 of 45 F1 ⇔ ∃a1∃a2∃a1∃a2 (a1 6≤ a1 & a1 6≤ a1),

0 0 0 0 Go Back F2 ⇔ ∃a1∃a2∃a1∃a2 (a1 6≤ a1 & a2 6≤ a2), 0 0 0 0 F3 ⇔ ∃a1∃a2∃a1∃a2 (a2 6≤ a2 & a1 6≤ a1), Full Screen 0 0 0 0 F4 ⇔ ∃a1∃a2∃a1∃a2 (a2 6≤ a2 & a2 6≤ a2). Close

Quit Partial Orders are . . .

What We Plan to Do 24. Example (cont-d) Uncertainty is . . .

Properties of Ordered . . . • So far, we got ⇔ ¬(F1 ∨ F2 ∨ F3 ∨ F4), where Towards Combining . . . 0 0 0 0 F1 ⇔ ∃a1∃a2∃a ∃a (a1 6≤ a & a 6≤ a1), 1 2 1 1 Main Result 0 0 0 0 F2 ⇔ ∃a1∃a2∃a1∃a2 (a1 6≤ a1 & a2 6≤ a2), Auxiliary Results 0 0 0 0 Proof of the Main Result F3 ⇔ ∃a1∃a2∃a1∃a2 (a2 6≤ a2 & a1 6≤ a1), My Publications F ⇔ ∃a ∃a ∃a0 ∃a0 (a 6≤ a0 & a0 6≤ a ). 4 1 2 1 2 2 2 2 2 Home Page

• By applying the quantiﬁers to the corresponding parts Title Page of the formulas, we get JJ II 0 0 0 F1 ⇔ ∃a1∃a1 (a1 6≤ a1 & a1 6≤ a1), J I 0 0 0 0 F2 ⇔ (∃a1∃a1 a1 6≤ a1)&(∃a2∃a2 a2 6≤ a2), Page 25 of 45 0 0 0 0 F3 ⇔ (∃a1∃a1 a1 6≤ a1)&(∃a2∃a2 a2 6≤ a2), Go Back 0 0 0 0 F4 ⇔ ∃a2∃a1∃a2 (a2 6≤ a2 & a2 6≤ a2). Full Screen

• Then, we again reduce ¬(F1 ∨ F2 ∨ F3 ∨ F4) to DNF. Close

Quit Partial Orders are . . .

What We Plan to Do 25. Products of Ordered Sets: What Is Known Uncertainty is . . . • At present, two product operations are known: Properties of Ordered . . . Towards Combining . . . • Cartesian product Main Result 0 0 0 0 (a1, a2) ≤ (a1, a2) ⇔ (a1 ≤1 a1 & a2 ≤2 a2); Auxiliary Results

Proof of the Main Result and My Publications

• lexicographic product Home Page 0 0 (a1, a2) ≤ (a1, a2) ⇔ Title Page 0 0 0 0 ((a1 ≤1 a1 & a1 6= a1) ∨ (a1 = a1 & a2 ≤2 a2). JJ II • Question: what other operations are possible? J I

Page 26 of 45

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Quit Partial Orders are . . .

What We Plan to Do 26. Possible Physical Meaning of Lexicographic Order Uncertainty is . . . Properties of Ordered . . .

Idea: Towards Combining . . .

Main Result • A1 is macroscopic space-time, Auxiliary Results

• A2 is microscopic space-time: Proof of the Main Result

My Publications

Home Page (a1, a2) - '$a t 1 Title Page 0 - (a1, a2) t JJ II &% J I

Page 27 of 45

'$0 Go Back 0 - a1 (a1, a2) t Full Screen &% Close

Quit Partial Orders are . . .

What We Plan to Do 27. Possible Logical Meaning of Diﬀerent Orders Uncertainty is . . . • Reminder: our certainty in S is described by a pair Properties of Ordered . . . (a1, a2) ∈ A1 × A2. Towards Combining . . . Main Result • We must therefore deﬁne a partial order on the set Auxiliary Results A1 × A2 of all pairs. Proof of the Main Result

• Cartesian product: our conﬁdence in S is higher than My Publications 0 in S if and only if it is higher for both experts. Home Page

• Meaning: a maximally cautious approach. Title Page

• Lexicographic product: means that we have absolute JJ II conﬁdence in the ﬁrst expert. J I

• We only use the opinion of the 2nd expert when, to the Page 28 of 45 1st expert, the degrees of certainty are equivalent. Go Back

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Quit Partial Orders are . . .

What We Plan to Do 28. Main Theorem Uncertainty is . . . • By a product operation, we mean a Boolean function Properties of Ordered . . . Towards Combining . . . P : {T,F }4 → {T,F }. Main Result

Auxiliary Results • For every two partially ordered sets A1 and A2, we Proof of the Main Result deﬁne the following relation on A1 × A2: My Publications 0 0 def (a1, a2) ≤ (a1, a2) = Home Page

0 0 0 0 Title Page P (a1 ≤1 a1, a1 ≤1 a1, a2 ≤2 a2, a2 ≤2 a2). • We say that a product operation is consistent if ≤ is JJ II always a partial order, and J I

0 0 0 0 Page 29 of 45 (a1 ≤1 a1 & a2 ≤2 a2) ⇒ (a1, a2) ≤ (a1, a2). Go Back • Theorem: Every consistent product operation is the Cartesian or the lexicographic product. Full Screen Close

Quit Partial Orders are . . .

What We Plan to Do 29. Auxiliary Results: General Idea and First Ex- ample Uncertainty is . . . Properties of Ordered . . .

• For each property of intervals in an ordered set A, we Towards Combining . . . analyze: Main Result

Auxiliary Results – which properties need to be satisﬁed for A1 and A2 – so that the corresponding property is satisﬁes for Proof of the Main Result My Publications intervals in A1 × A2. Home Page • Connectedness property (CP): for every two points a, a0 ∈ A, there exists an interval that contains a and a0: Title Page ∀a ∀a0 ∃a− ∃a+ (a− ≤ a, a0 ≤ a+). JJ II • This property is equivalent to two properties: J I Page 30 of 45 – A is upward-directed: ∀a ∀a0 ∃a+ (a, a0 ≤ a+); – A is downward-directed: ∀a ∀a0 ∃a− (a− ≤ a, a0). Go Back • Cartesian product: A is upward-(downward-) directed Full Screen ⇔ both A1 and A2 are upward-(downward-) directed. Close

Quit Partial Orders are . . .

What We Plan to Do 30. Connectedness Property Illustrated Uncertainty is . . . Connectedness property (CP): for every two points a, a0 ∈ Properties of Ordered . . . A, there exists an interval that contains a and a0: Towards Combining . . .

Main Result ∀a ∀a0 ∃a− ∃a+ (a− ≤ a, a0 ≤ a+). Auxiliary Results

a+ Proof of the Main Result @ My Publications t @ @ Home Page @ @ Title Page a @ @ t @ JJ II @ 0 @ a J I @ t @ Page 31 of 45 @ @ @ Go Back @ a− Full Screen t

Quit Partial Orders are . . .

What We Plan to Do 31. Upward and Downward Directed: Illustrated Uncertainty is . . . Upward-directed: ∀a ∀a0 ∃a+ (a, a0 ≤ a+); Properties of Ordered . . . Towards Combining . . . + a Main Result @ t @ Auxiliary Results @ @ Proof of the Main Result @ a @ My Publications @ t a0 @ Home Page

t Title Page 0 − − 0 Downward-directed: ∀a ∀a ∃a (a ≤ a, a ). JJ II a J I @ @ t a0 Page 32 of 45 @ @ t Go Back @ @ @ Full Screen @ a− Close t

Quit Partial Orders are . . .

What We Plan to Do 32. First Example, Case of Cartesian Product: Proof Uncertainty is . . . • Part 1: Properties of Ordered . . . Towards Combining . . . – Let us assume that A1 × A2 is upward-directed. Main Result – We want to prove that A1 is upward-directed. Auxiliary Results – For any a , a0 ∈ A , take any a ∈ A , then 1 1 1 2 2 Proof of the Main Result + + + 0 + ∃a = (a1 , a2 ) such that (a1, a2), (a1, a2) ≤ a . My Publications 0 + Home Page – Hence a1, a1 ≤1 a1 , i.e., A1 is upward-directed. Title Page • Part 2: JJ II – Assume that both Ai are upward-directed. J I – We want to prove that A1 × A2 is upward-directed. 0 0 0 Page 33 of 45 – For any a = (a1, a2) and a = (a1, a2), for i = 1, 2,

+ 0 + Go Back ∃ ai (ai, ai ≤i ai ). 0 0 + + Full Screen – Hence (a1, a2), (a1, a2) ≤ (a1 , a2 ), i.e., A1 × A2 is upward-directed. Close

Quit Partial Orders are . . .

What We Plan to Do 33. First Example: Case of Lexicographic Prod- uct Uncertainty is . . . Properties of Ordered . . .

• A1 × A2 is upward-directed ⇔ the following two con- Towards Combining . . . ditions hold: Main Result

Auxiliary Results – the set A1 is upward-directed, and Proof of the Main Result – if A1 has a maximal element a1 (= for which there My Publications are no a with a ≺ a ), then A is upward-directed. 1 1 1 1 2 Home Page

• A1 ×A2 is downward-directed ⇔ the following two con- Title Page ditions hold: JJ II – the set A1 is downward-directed, and J I – if A1 has a minimal element a1 (= for which there Page 34 of 45 are no a1 for which a1 ≺1 a1), then A2 is downward- directed. Go Back

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What We Plan to Do 34. Case of Lexicographic Product: Proof Uncertainty is . . .

Properties of Ordered . . . • Let us assume that A1 × A2 is upward-directed. Towards Combining . . . • Part 1: Main Result

– We want to prove that A1 is upward-directed. Auxiliary Results 0 – For any a1, a1 ∈ A1, take any a2 ∈ A2, then Proof of the Main Result + + + 0 + My Publications ∃a = (a1 , a2 ) for which (a1, a2), (a1, a2) ≤ a . Home Page 0 + – Hence a1, a1 ≤1 a1 , i.e., A1 is upward-directed. Title Page • Part 2: JJ II

– Let a1 be a maximal element of A1. J I 0 – For any a2, a2 ∈ A2, we have Page 35 of 45 + + + 0 + ∃a = (a1 , a2 ) for which (a1, a2), (a1, a2) ≤ a . Go Back + + – Here, a1 ≤1 a1 and since a1 is maximal, a1 = a1. Full Screen 0 + – Hence a2, a2 ≤2 a2 , i.e., A2 is upward-directed. Close

Quit Partial Orders are . . .

What We Plan to Do 35. Proof (cont-d) Uncertainty is . . .

Properties of Ordered . . . • Let us assume that A1 is upward-directed. Towards Combining . . . • Let us assume that if A1 has a maximal element, then Main Result A2 is upward-directed. Auxiliary Results

• We want to prove that A1 × A2 is upward-directed. Proof of the Main Result 0 0 0 • Take any a = (a1, a2) and a = (a1, a2) from A1 × A2. My Publications + 0 + Home Page • Since A1 is upward-directed, ∃a1 (a1, a1 ≤1 a1 ). + 0 0 + 0 Title Page • If a1 ≺1 a1 , then (a1, a2), (a1, a2) ≤ (a1 , a2). 0 + 0 0 + JJ II • If a1 ≺1 a1 , then (a1, a2), (a1, a2) ≤ (a1 , a2). + 0 J I • If a1 = a1 = a1, and a1 is not a maximal element, then 00 00 0 0 00 Page 36 of 45 ∃a1 (a1 ≺1 a1), hence (a1, a2), (a1, a2) ≤ (a1, s2).

+ 0 Go Back • If a1 = a1 = a1, and a1 is a maximal element, then A2 + 0 + is upward-directed, hence ∃a2 (a2, a2 ≤2 a2 ) and Full Screen 0 + (a1, a2), (a1, a2) ≤ (a1, a2 ). Close

Quit Partial Orders are . . .

What We Plan to Do 36. Second Example: Intersection Property Uncertainty is . . . • The intersection of every two intervals is an interval. Properties of Ordered . . . Towards Combining . . . • Comment: this is true for intervals on the real line. Main Result

• This can be similarly reduced to two properties: Auxiliary Results – the intersection of every two future cones Proof of the Main Result + def My Publications Qa = {b : a ≤ b} is a future cone; Home Page – the intersection of every two past cones − def Title Page Qa = {b : b ≤ a} is a past cone. • If both properties hold, then the intersection of every JJ II + − J I two intervals [a, b] = Qa ∩ Qb is an interval. • Ordered sets with Q+ and Q− properties are called Page 37 of 45 upper and lower semi-lattices. Go Back

• For Cartesian product: A1 × A2 is an upper (lower) Full Screen

semi-lattice ⇔ both Ai are upper (lower) semi-lattices. Close

Quit Partial Orders are . . .

What We Plan to Do 37. Intersection Property Illustrated Uncertainty is . . . Intersection property for intervals: Properties of Ordered . . . Towards Combining . . . @ @ @ @ Main Result @ @ @ @ Auxiliary Results ?@ @ @ @ Proof of the Main Result @ @ @ @ My Publications @ @ Home Page @ @

Title Page Upper and lower semi-lattice properties: a a0 JJ II @ @ @ @ J I @ @ - @ @ @ @ @ @ Page 38 of 45 @ @ @ @ @ @ @ @ a a0 Go Back Full Screen

Quit Partial Orders are . . .

What We Plan to Do 38. My Publications Uncertainty is . . . • H.-P. K¨unzi,F. Zapata, and V. Kreinovich, When Is Properties of Ordered . . . Busemann Product a Lattice? A Relation Between Met- Towards Combining . . . ric Spaces and Corresponding Space-Time Models, Uni- Main Result versity of Texas at El Paso, Department of Computer Auxiliary Results

Science, Technical Report UTEP-CS-11-24, 2011. Proof of the Main Result • F. Zapata, O. Kosheleva, and K. Villaverde, “Prod- My Publications ucts of Partially Ordered Sets (Posets) and Intervals in Home Page Such Products, with Potential Applications to Uncer- Title Page tainty Logic and Space-Time Geometry”, Abstracts of JJ II the 14th GAMM-IMACS International Symposium on Scientiﬁc Computing, Computer Arithmetic and Vali- J I dated Numerics SCAN’2010, Lyon, France, September Page 39 of 45

27–30, 2010, pp. 142–144. Go Back

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Quit Partial Orders are . . .

What We Plan to Do 39. My Publications (cont-d) Uncertainty is . . . • F. Zapata, O. Kosheleva, and K. Villaverde, “Prod- Properties of Ordered . . . uct of Partially Ordered Sets (Posets), with Potential Towards Combining . . . Applications to Uncertainty Logic and Space-Time Ge- Main Result ometry”, International Journal of Innovative Manage- Auxiliary Results ment, Information & Production (IJIMIP), to appear. Proof of the Main Result • F. Zapata, O. Kosheleva, and K. Villaverde, “How to My Publications Tell When a Product of Two Partially Ordered Spaces Home Page Has a Certain Property: General Results with Ap- Title Page plication to Fuzzy Logic”, Proc. of the 30th Annual JJ II Conf. of the North American Fuzzy Information Pro- cessing Society NAFIPS’2011, El Paso, Texas, March J I 18–20, 2011. Page 40 of 45 • F. Zapata, O. Kosheleva, and K. Villaverde, “How to Go Back

Tell When a Product of Two Partially Ordered Spaces Full Screen Has a Certain Property?”, Journal of Uncertain Sys- Close tems, 2012, Vol. 6, No. 2, to appear. Quit Partial Orders are . . .

What We Plan to Do 40. My Publications (cont-d) Uncertainty is . . . • F. Zapata and V. Kreinovich, “Reconstructing an Open Properties of Ordered . . . Order from Its Closure, with Applications to Space- Towards Combining . . . Time Physics and to Logic”, Studia Logica, to appear. Main Result

Auxiliary Results • F. Zapata, E. Ramirez, J. A. Lopez, and O. Koshel- eva, “Strings lead to lattice-type causality”, Journal of Proof of the Main Result Uncertain Systems, 2011, Vol. 5, No. 2, pp. 154–160. My Publications Home Page

Title Page

JJ II

J I

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What We Plan to Do 41. Space-Time Geometry: Physical References Uncertainty is . . . • H. Busemann, Timelike spaces, PWN: Warszawa, 1967. Properties of Ordered . . . Towards Combining . . . • E. H. Kronheimer and R. Penrose, “On the structure Main Result of causal spaces”, Proc. Cambr. Phil. Soc., Vol. 63, No. 2, pp. 481–501, 1967. Auxiliary Results Proof of the Main Result

• C. W. Misner, K. S. Thorne, and J. A. Wheeler, Grav- My Publications

itation, New York: W. H. Freeman, 1973. Home Page

• R. I. Pimenov, Kinematic spaces: Mathematical The- Title Page ory of Space-Time, N.Y.: Consultants Bureau, 1970. JJ II

J I

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What We Plan to Do 42. Space-Time Geometry: Mathematical and Com- putational References Uncertainty is . . . Properties of Ordered . . .

• V. Kreinovich and O. Kosheleva, “Computational com- Towards Combining . . . plexity of determining which statements about causal- Main Result ity hold in diﬀerent space-time models”, Theoretical Auxiliary Results

Computer Science, 2008, Vol. 405, No. 1–2, pp. 50–63. Proof of the Main Result • A. Levichev and O. Kosheleva, “Intervals in space- My Publications time”, Reliable Computing, 1998, Vol. 4, No. 1, pp. 109– Home Page 112. Title Page

• P. G. Vroegindeweij, V. Kreinovich, and O. M. Koshel- JJ II

eva. “From a connected, partially ordered set of events J I to a ﬁeld of time intervals”, Foundations of Physics, Page 43 of 45 1980, Vol. 10, No. 5/6, pp. 469–484. Go Back

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What We Plan to Do 43. References: Uncertainty Logic Uncertainty is . . . • G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: The- Properties of Ordered . . . ory and Applications, Upper Saddle River, New Jersey: Towards Combining . . . Prentice Hall, 1995. Main Result • J. M. Mendel, Uncertain Rule-Based Fuzzy Logic Sys- Auxiliary Results tems: Introduction and New Directions, Prentice-Hall, Proof of the Main Result 2001. My Publications Home Page • H. T. Nguyen, V. Kreinovich, and Q. Zuo, “Interval- Title Page valued degrees of belief: applications of interval com- putations to expert systems and intelligent control”, JJ II

International Journal of Uncertainty, Fuzziness, and J I Knowledge-Based Systems (IJUFKS), 1997, Vol. 5, No. 3, Page 44 of 45 pp. 317–358. Go Back • H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, Chapman & Hall/CRC, Boca Raton, Florida, Full Screen 2006. Close

Quit 44. Acknowledgments Partial Orders are . . . What We Plan to Do Uncertainty is . . . The author is greatly thankful: Properties of Ordered . . . Towards Combining . . . • to members of my committee for their help: Main Result Auxiliary Results – to Dr. Martine Ceberio, Proof of the Main Result – to Dr. Vladik Kreinovich, My Publications

– to Dr. Luc Longpr´e,and Home Page

– to Dr. Piotr Wojciechowski. Title Page

• to Dr. Eric Freudenthal for his mentorship; JJ II

• to CONACyT for their ﬁnancial support; J I • last but not the least, to my family for their love and Page 45 of 45 support. Go Back

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