Towards a General Computation-Oriented Simplicial Complexes . . . - PowerPoint PPT Presentation

An Approach to . . . Towards . . . Towards . . . A More Adequate . . . Towards a General Computation-Oriented Simplicial Complexes . . . Description of Physical Quantities: How to Describe . . . From Intervals to Graphs Actual Values: . . . Unusual Property: . . . to Simplicial Complexes Functions and Their Projective Limits Summary Vladik Kreinovich Acknowledgments Further Reading Department of Computer Science Title Page University of Texas at El Paso 500 W. University ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ vladik@utep.edu Page 1 of 15 http://www.cs.utep.edu/vladik Go Back Full Screen Close

An Approach to . . . Towards . . . 1. Main Problem: Introduction Towards . . . A More Adequate . . . • One of the main objectives of physics: predict the fu- Simplicial Complexes . . . ture behavior of real-world systems. How to Describe . . . • Fact: in modern physics, models for space, time, causal- Actual Values: . . . ity, and physical processes in general are very complex. Unusual Property: . . . • Example: Functions – physical phenomenon: a simple space-time; Summary Acknowledgments – formalism: quantum physics; Further Reading – mathematical description: a wave function ψ ( M ) Title Page defined on all pseudo-Riemannian manifolds M . ◭◭ ◮◮ • Corollary: prediction-related computations are often extremely time-consuming. ◭ ◮ • Sometimes: by the time we finish prediction computa- Page 2 of 15 tions, the predicted event has already occurred. Go Back • Problem: how can we speed up these computations? Full Screen Close

An Approach to . . . Towards . . . 2. An Approach to Solving the Main Problem: Oper- Towards . . . ationalism A More Adequate . . . Simplicial Complexes . . . • Fact: in modern physics, many quantities used in the How to Describe . . . corresponding equations are not directly observable. Actual Values: . . . • Example: the wave function ψ ( x ). Unusual Property: . . . • Related idea: restrict ourselves to only computing di- Functions rectly observable quantities. Summary • Hope: by not computing other quantities, we can save Acknowledgments computation time. Further Reading Title Page • Reason for this hope: a similar operationalistic ap- proach has been very successful in physics: ◭◭ ◮◮ – special relativity: started with Einstein’s analysis ◭ ◮ of simultaneity; Page 3 of 15 – general relativity: Einstein’s equivalence principle; Go Back – equations of quantum physics: Heisenberg’s matrix Full Screen equations (motivated by operationalism). Close

An Approach to . . . Towards . . . 3. Towards Operationalistic Approach to Computa- Towards . . . tional Physics: Binary Domains A More Adequate . . . Simplicial Complexes . . . • General idea: in any real-life measurement, we have a How to Describe . . . finite set X of possible measurement results. Actual Values: . . . • Description of measurement uncertainty: a ∼ b ↔ the Unusual Property: . . . same object can lead to both a and b . Functions • Physical example: temperature t ◦ ; values 0, 1, . . . , 100; Summary measurement accuracy: ± 0 . 5 ◦ . Acknowledgments Further Reading • X = { 0 , 1 , 2 , 3 , . . . , 100 } ; a ∼ b ↔ | a − b | ≤ 1. Title Page . . . . . . ❅ � � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ ◭◭ ◮◮ • Modified example: measurement accuracy ± 1 ◦ . ◭ ◮ • X = { 0 , 1 , 2 , 3 , . . . , 100 } ; a ∼ b ↔ | a − b | ≤ 2. Page 4 of 15 Go Back ✘✘❳ ✘❳❳ ✘✘ ❳ ✘✘❳❳ ✘ ❳ ✘✘❳❳ ✘❳❳ ✘ ❳ ❳ ✘ ❳ ✘ . . . . . . ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � ❅ � Full Screen Close

An Approach to . . . Towards . . . 4. Towards Operationalistic Approach to Computa- Towards . . . tional Physics: Binary Domains (continued) A More Adequate . . . Simplicial Complexes . . . • Counting up to n : X = { 1 , 2 , . . . , n − 1 , many } : How to Describe . . . . . . . . . 0 1 ( n − 1) many k Actual Values: . . . • Binary questions: “yes” (1), “no” (0), “unknown” ( U ); Unusual Property: . . . X = { 0 , 1 , U } , 0 ∼ U ∼ 1. Functions Summary 0 U 1 Acknowledgments • Repeated “yes”-“no” measurements: 5 possible out- Further Reading comes: 0 1 , 1 1 , U 1 0 2 , U 1 1 2 , and U 1 U 2 . Title Page – If the actual value is 0, we can get 0 1 , U 1 0 2 , U 1 U 2 ; ◭◭ ◮◮ – if the actual value is 1, we can get 1 1 , U 1 1 2 , U 1 U 2 . ◭ ◮ ✥❵❵❵❵ ✥❵❵❵❵ ✥✥✥✥ ❵ ✥✥✥✥ Page 5 of 15 ❵ 0 1 U 1 0 2 U 1 1 2 1 1 U 1 U 2 Go Back • General case: graph ( web ) � X, ∼� . Full Screen Close

An Approach to . . . Towards . . . 5. A More Adequate Description: Simplicial Com- Towards . . . plexes A More Adequate . . . Simplicial Complexes . . . • Previously: we only considered compatibility of pairs How to Describe . . . of measurement results. Actual Values: . . . • Natural idea: consider compatibility of triples, etc. Unusual Property: . . . • Formalization: Functions Summary – a set S ⊆ X is compatible Acknowledgments – if for some object, all values from S are possible Further Reading after measurement. Title Page • Simplicial complex: a pair � X, S� , where X ⊆ S ⊆ 2 X ◭◭ ◮◮ is the class of all compatible sets. ◭ ◮ Page 6 of 15 Go Back Full Screen Close

An Approach to . . . Towards . . . 6. Simplicial Complex: Example 1 Towards . . . A More Adequate . . . • Example 1: X i ∩ X j � = ∅ but X 1 ∩ X 2 ∩ X 3 = ∅ . Simplicial Complexes . . . • Corresponding simplicial complex: empty triangle How to Describe . . . • X = { x 1 , x 2 , x 3 } , Actual Values: . . . Unusual Property: . . . • S = {{ x 1 } , { x 2 } , { x 3 } , { x 1 , x 2 } , { x 2 , x 3 } , { x 1 , x 3 }} . Functions • Illustration: Summary �❅ Acknowledgments � ❅ Further Reading � ❅ � ❅ Title Page � ❅ � ❅ ◭◭ ◮◮ � ❅ � ❅ � ❅ ◭ ◮ � ❅ Page 7 of 15 Go Back Full Screen Close

An Approach to . . . Towards . . . 7. Simplicial Complex: Example 2 Towards . . . A More Adequate . . . • Example 2: X 1 ∩ X 2 ∩ X 3 � = ∅ . Simplicial Complexes . . . ✬✩ ✬✩ How to Describe . . . ✬✩ Actual Values: . . . ✫✪ ✫✪ Unusual Property: . . . ✫✪ Functions Summary • Corresponding simplicial complex: filled triangle Acknowledgments Further Reading X = { x 1 , x 2 , x 3 } , Title Page S = {{ x 1 } , { x 2 } , { x 3 } , { x 1 , x 2 } , { x 2 , x 3 } , { x 1 , x 3 } , { x 1 , x 2 , x 3 }} . ◭◭ ◮◮ �❅ ✁ � ✁ ❅ ◭ ◮ � ✁ ✁ ✁ ❅ � ✁ ✁ ✁ ❅ ✁ Page 8 of 15 � ✁ ✁ ✁ ✁ ❅ ✁ � ✁ ✁ ✁ ✁ ✁ ❅ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❅ Go Back � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❅ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❅ ✁ Full Screen � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ❅ Close

An Approach to . . . Towards . . . 8. How to Describe Actual Values of Measured Quan- Towards . . . tities A More Adequate . . . Simplicial Complexes . . . • Objective: describe actual values. How to Describe . . . • Problem: single measurement leads to approximate value. Actual Values: . . . • Solution: consider a sequence of more and more accu- Unusual Property: . . . rate measuring instruments. Functions Summary • Relation: let X describes results of first k measure- ments and X ′ results of l > k measurements. Acknowledgments Further Reading • The forgetful functor π lk : X ′ → X is a projection: Title Page – if a ′ ∼ ′ b ′ , then π ( a ′ ) ∼ π ( b ′ ); ◭◭ ◮◮ – if a ∼ b , then ∃ a ′ , b ′ s.t. π ( a ′ ) = a , π ( b ′ ) = b , and ◭ ◮ a ′ ∼ ′ b ′ . Page 9 of 15 π 2 , 1 π 3 , 2 π 4 , 3 • Definition: X 1 ← X 2 ← X 3 ← . . . Go Back • Actual values: x = ( x 1 , x 2 , . . . ) s.t. π 21 ( x 2 ) = x 1 , . . . Full Screen Close

An Approach to . . . Towards . . . 9. Actual Values: Properties and Examples Towards . . . A More Adequate . . . • Equivalence: a ∼ b iff a i ∼ i b i for all i . Simplicial Complexes . . . • Neighborhoods: N n ( a ) = { b | b ∼ n a } . How to Describe . . . • Limit: a ( k ) → a iff ∀ n ∃ m ∀ k > m ( a ( k ) Actual Values: . . . ∼ n a ). n Unusual Property: . . . • Real numbers: naturally come from intervals. Functions • Actually: we also get −∞ and + ∞ . Summary Acknowledgments • R n : naturally comes from n -dimensional boxes. Further Reading • “yes”-“no” questions: Title Page – X 1 : 0 ∼ U ∼ 1; ◭◭ ◮◮ – X 2 : 0 ∼ U 0 ∼ UU ∼ U 1 ∼ 1, 0 ∼ UU ∼ 1; ◭ ◮ – . . . Page 10 of 15 – projective limit: 0 ∼ U ∼ 1. Go Back Full Screen Close