Products of Partially Ordered Sets Posets in Uncertainty . . . Main - PowerPoint PPT Presentation

Posets in Space-Time . . . Intervals in Space- . . . Products of Space- . . . Products of Partially Ordered Sets Posets in Uncertainty . . . Main Theorem (Posets) and Intervals in Such Products, Auxiliary Results: . . . with Potential Applications Second Example: . . . Intersection Property . . . to Uncertainty Logic and Space-Time . . . Space-Time Geometry Title Page Francisco Zapata 1 , Olga Kosheleva 1 , ◭◭ ◮◮ and Karen Villaverde 2 ◭ ◮ 1 University of Texas at El Paso Page 1 of 23 El Paso, TX 79968, USA olgak@utep.edu Go Back Full Screen 2 Department of Computer Science New Mexico State University Close Las Cruces, NM 88003, USA kvillave@cs.nmsu.edu Quit

Posets in Space-Time . . . 1. Posets in Space-Time Geometry Intervals in Space- . . . Products of Space- . . . • Starting from general relativity, space-time models are Posets in Uncertainty . . . usually formulated in terms of physical fields. Main Theorem • Typical example: a metric field g ij ( x ). Auxiliary Results: . . . • These fields assume that the space-time is smooth . Second Example: . . . Intersection Property . . . • However, there are important situations of non-smoothness : Space-Time . . . • singularities like the Big Bang or a black hole, and Title Page • quantum fluctuations . ◭◭ ◮◮ • According to modern physics, a proper description of ◭ ◮ the corresponding non-smooth space-time models means: Page 2 of 23 • that we no longer have a metric field, Go Back • that we only have a causality relation � between Full Screen events – a partial order. Close Quit

Posets in Space-Time . . . 2. Intervals in Space-Time Posets Intervals in Space- . . . Products of Space- . . . • Due to measurement inaccuracy, we rarely know the Posets in Uncertainty . . . exact space-time location of an event e . Main Theorem • Often, we only know: Auxiliary Results: . . . • an event e that precedes e : Second Example: . . . Intersection Property . . . e � e, Space-Time . . . and Title Page • an event e that follows e : ◭◭ ◮◮ e � e. ◭ ◮ Page 3 of 23 • In this case, we only know that e belongs to the interval Go Back def [ e, e ] = { e : e � e � e } . Full Screen • Comment: In the 1-D case, we get standard intervals Close on the real line. Quit

Posets in Space-Time . . . 3. Products of Space-Time Posets Intervals in Space- . . . Products of Space- . . . • Sometimes, we need to consider pairs of events. Posets in Uncertainty . . . • Example: situations like quantum entanglement, situ- Main Theorem ations of importance to quantum computing. Auxiliary Results: . . . • Question: how to extend partial orders on posets A 1 Second Example: . . . and A 2 to a partial order on the set A 1 × A 2 of all pairs? Intersection Property . . . Space-Time . . . • Reasonable assumption: the validity of ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ) Title Page depends only on: ◭◭ ◮◮ • whether a 1 � 1 a ′ 1 , ◭ ◮ • whether a ′ 1 � 1 a 1 , • whether a 2 � 2 a ′ Page 4 of 23 2 , and/or • whether a ′ 2 � 2 a 2 . Go Back • It is also reasonable to assume that: Full Screen Close if a 1 � 1 a ′ 1 and a 2 � 2 a ′ 2 then ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ). Quit

Posets in Space-Time . . . 4. Posets in Uncertainty Logic: Need for Intervals and Intervals in Space- . . . Products Products of Space- . . . Posets in Uncertainty . . . • A similar partial order � is useful in describing degrees Main Theorem of expert’s certainty, where Auxiliary Results: . . . a � a ′ ⇔ a corresponds to less certainty than a ′ . Second Example: . . . Intersection Property . . . • Often, we cannot determine the exact value a of the Space-Time . . . expert’s degree of certainty. Title Page • In many cases, we can only determine the interval [ a, a ] ◭◭ ◮◮ of possible values of a . ◭ ◮ • Sometimes, two (or more) experts evaluate a state- Page 5 of 23 ment S . Go Back • Then, our certainty in S is described by a pair ( a 1 , a 2 ), Full Screen where a i ∈ A i is the i -th expert’s degree of certainty. Close Quit

Posets in Space-Time . . . 5. Products of Ordered Sets: What Is Known Intervals in Space- . . . Products of Space- . . . • At present, two product operations are known: Posets in Uncertainty . . . • Cartesian product Main Theorem ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ) ⇔ ( a 1 � 1 a ′ 1 & a 2 � 2 a ′ Auxiliary Results: . . . 2 ); Second Example: . . . and Intersection Property . . . • lexicographic product Space-Time . . . Title Page ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ) ⇔ ◭◭ ◮◮ (( a 1 � 1 a ′ 1 & a 1 � = a ′ 1 ) ∨ ( a 1 = a ′ 1 & a 2 � 2 a ′ 2 ) . ◭ ◮ • Question: what other operations are possible? Page 6 of 23 Go Back Full Screen Close Quit

Posets in Space-Time . . . 6. Possible Physical Meaning of Lexicographic Order Intervals in Space- . . . Products of Space- . . . Idea: Posets in Uncertainty . . . • A 1 is macroscopic space-time, Main Theorem • A 2 is microscopic space-time: Auxiliary Results: . . . Second Example: . . . ✬✩ Intersection Property . . . t ✲ ( a 1 , a 2 ) Space-Time . . . a 1 t Title Page ✫✪ ( a 1 , a ′ ✲ 2 ) ◭◭ ◮◮ ◭ ◮ ✬✩ Page 7 of 23 t a ′ ✲ ( a ′ 1 , a 2 ) Go Back 1 ✫✪ Full Screen Close Quit

Posets in Space-Time . . . 7. Possible Logical Meaning of Different Orders Intervals in Space- . . . Products of Space- . . . • Reminder: our certainty in S is described by a pair Posets in Uncertainty . . . ( a 1 , a 2 ) ∈ A 1 × A 2 . Main Theorem • We must therefore define a partial order on the set Auxiliary Results: . . . A 1 × A 2 of all pairs. Second Example: . . . • Cartesian product: our confidence in S is higher than Intersection Property . . . in S ′ if and only if it is higher for both experts. Space-Time . . . Title Page • Meaning: a maximally cautious approach. ◭◭ ◮◮ • Lexicographic product: means that we have absolute ◭ ◮ confidence in the first expert. Page 8 of 23 • We only use the opinion of the 2nd expert when, to the 1st expert, the degrees of certainty are equivalent. Go Back Full Screen Close Quit

Posets in Space-Time . . . 8. Main Theorem Intervals in Space- . . . Products of Space- . . . • By a product operation , we mean a Boolean function Posets in Uncertainty . . . P : { T, F } 4 → { T, F } . Main Theorem Auxiliary Results: . . . • For every two partially ordered sets A 1 and A 2 , we Second Example: . . . define the following relation on A 1 × A 2 : Intersection Property . . . def ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ) = Space-Time . . . Title Page P ( a 1 � 1 a ′ 1 , a ′ 1 � 1 a 1 , a 2 � 2 a ′ 2 , a ′ 2 � 2 a 2 ) . ◭◭ ◮◮ • We say that a product operation is consistent if � is ◭ ◮ always a partial order, and Page 9 of 23 ( a 1 � 1 a ′ 1 & a 2 � 2 a ′ 2 ) ⇒ ( a 1 , a 2 ) � ( a ′ 1 , a ′ 2 ) . Go Back • Theorem: Every consistent product operation is the Full Screen Cartesian or the lexicographic product. Close Quit

Posets in Space-Time . . . 9. Auxiliary Results: General Idea and First Example Intervals in Space- . . . Products of Space- . . . • For each property of intervals in an ordered set A , we Posets in Uncertainty . . . analyze: Main Theorem – which properties need to be satisfied for A 1 and A 2 Auxiliary Results: . . . – so that the corresponding property is satisfies for Second Example: . . . intervals in A 1 × A 2 . Intersection Property . . . • Connectedness property (CP): for every two points a, a ′ ∈ Space-Time . . . A , there exists an interval that contains a and a ′ : Title Page ∀ a ∀ a ′ ∃ a − ∃ a + ( a − � a, a ′ � a + ) . ◭◭ ◮◮ ◭ ◮ • This property is equivalent to two properties: Page 10 of 23 – A is upward-directed : ∀ a ∀ a ′ ∃ a + ( a, a ′ � a + ); Go Back – A is downward-directed : ∀ a ∀ a ′ ∃ a − ( a − � a, a ′ ). Full Screen • Cartesian product : A is upward-(downward-) directed Close ⇔ both A 1 and A 2 are upward-(downward-) directed. Quit

Posets in Space-Time . . . 10. Connectedness Property Illustrated Intervals in Space- . . . Connectedness property (CP): for every two points a, a ′ ∈ Products of Space- . . . Posets in Uncertainty . . . A , there exists an interval that contains a and a ′ : Main Theorem ∀ a ∀ a ′ ∃ a − ∃ a + ( a − � a, a ′ � a + ) . Auxiliary Results: . . . t Second Example: . . . a + Intersection Property . . . � ❅ � ❅ Space-Time . . . � ❅ � ❅ Title Page � t ❅ � a ❅ ◭◭ ◮◮ � ❅ � ❅ t ❅ � ◭ ◮ a ′ ❅ � ❅ � Page 11 of 23 ❅ � ❅ � ❅ � Go Back ❅ t � a − ❅ � Full Screen Close Quit