1. Kevin H. Knuth Departments of Physics and Informatics University at Albany Intelligent Science Platforms

2. 10 September 2008Kevin H Knuth CIDU 2008 Encoding Knowledge With Lattices

3. 10 September 2008Kevin H Knuth CIDU 2008 States describe Systems Antichain State Space applebananacherry

4. 10 September 2008Kevin H Knuth CIDU 2008 Exp and Log a b c exp log

5. 10 September 2008Kevin H Knuth CIDU 2008 Exp and Log a b c exp log

6. 10 September 2008Kevin H Knuth CIDU 2008 Exp and Log a b c exp log StatesStatements (sets of states) (potential states)

7. 10 September 2008Kevin H Knuth CIDU 2008 Three Spaces a b c exp log

8. 10 September 2008Kevin H Knuth CIDU 2008 Three Spaces a b c exp log StatesQuestions (sets of statements) (potential statements) Statements (sets of states) (potential states)

9. 10 September 2008Kevin H Knuth CIDU 2008 States describe Systems Antichain State Space applebananacherry

10. 10 September 2008Kevin H Knuth CIDU 2008 Statements are sets of States Boolean Lattice Hypothesis Space implies

11. 10 September 2008Kevin H Knuth CIDU 2008 Questions are sets of Statements Free Distributive Lattice Inquiry Space answers

12. 10 September 2008Kevin H Knuth CIDU 2008 answers Central Issue “Is it an Apple, Banana, or Cherry?” “Is it an Apple?” “Is it an Apple or Cherry, or is it a Banana or Cherry?” Relevance Decreases Relevance

13. 10 September 2008Kevin H Knuth CIDU 2008 The Central Issue I = “Is it an Apple, Banana, or Cherry?” This question is answered by the following set of statements: I = { a = “It is an Apple!”, b = “It is a Banana!”, c = “It is a Cherry!” }

14. 10 September 2008Kevin H Knuth CIDU 2008 Some Questions Answer Others Now consider the binary question B = “Is it an Apple?” B = { a = “It is an Apple!”, ~ a = “It is not an Apple!”} As the defining set of I is exhaustive,

15. 10 September 2008Kevin H Knuth CIDU 2008 Ordering Questions B = “Is it an Apple?” I = “Is it an Apple, Banana, or Cherry?” I answers B B includes I

16. Valuations on Lattices

17. 10 September 2008Kevin H Knuth CIDU 2008 Valuations Valuations are functions that take lattice elements to real numbers Valuation: ℝ Bi-Valuation: ℝ

18. 10 September 2008Kevin H Knuth CIDU 2008 Valuations Valuations are functions that take lattice elements to real numbers Valuation: ℝ Bi-Valuation: ℝ By assigning valuations in accordance to the lattice structure, this generalizes lattice inclusion to degrees of inclusion. The valuation inherits meaning from the ordering relation!

19. 10 September 2008Kevin H Knuth CIDU 2008 Context The context of a bi-valuation can be made implicit Valuation Bi-Valuation Measure of x with respect to Context y is implicit Context y is explicit

20. 10 September 2008Kevin H Knuth CIDU 2008 Consistency Valuation assignments must be consistent with lattice structure

21. 10 September 2008Kevin H Knuth CIDU 2008 Consistency Valuation assignments must be consistent with lattice structure In general

22. 10 September 2008Kevin H Knuth CIDU 2008 Associativity of Join or more specifically Associativity leads to a Sum Rule…

23. 10 September 2008Kevin H Knuth CIDU 2008 Distributivity leads to a Product Rule… or more specifically

24. 10 September 2008Kevin H Knuth CIDU 2008 Commutativity leads to a Bayes Theorem… Note that Bayes Theorem involves a change of context. Valuations are not sufficient… need bi-valuations.

25. 10 September 2008Kevin H Knuth CIDU 2008 Inclusion-Exclusion (The Sum Rule) The Sum Rule for Lattices

26. 10 September 2008Kevin H Knuth CIDU 2008 Inclusion-Exclusion (The Sum Rule) The Sum Rule for Probability

27. 10 September 2008Kevin H Knuth CIDU 2008 Inclusion-Exclusion (The Sum Rule) Definition of Mutual Information

28. 10 September 2008Kevin H Knuth CIDU 2008 Inclusion-Exclusion (The Sum Rule) Polya’s Min-Max Rule for Integers

29. 10 September 2008Kevin H Knuth CIDU 2008 Inclusion-Exclusion (The Sum Rule) “Measuring Integers”, Knuth 2008 The Sum Rule derives from the Möbius function of the lattice, And is related to its Zeta function

30. 10 September 2008Kevin H Knuth CIDU 2008 Probability Probabilities are degrees of implication! Constraint Equations!

31. 10 September 2008Kevin H Knuth CIDU 2008 Relevance Relevance quantifies the degree to which one question answers another Constraint Equations

32. 10 September 2008Kevin H Knuth CIDU 2008 Probability and Relevance The degree to which one question answers another must depend on the probabilities of the possible answers. Relevance is a function of probability

33. 10 September 2008Kevin H Knuth CIDU 2008 Relevance

34. 10 September 2008Kevin H Knuth CIDU 2008 Normalization Conditions We may choose to normalize relevance between zero and one. Relevance is maximized when Relevance is minimized when

35. 10 September 2008Kevin H Knuth CIDU 2008 Relevance and Entropy

36. 10 September 2008Kevin H Knuth CIDU 2008 Higher-Order Informations This relevance is related to the mutual information. In this way one can obtain higher-order informations.

37. 10 September 2008Kevin H Knuth CIDU 2008 Higher-Order Informations The Sum Rule can be applied in larger lattices to obtain even higher-order informations as introduced by McGill (1954) and as co-informations by Bell (2003). However, it has been noted that these informations suffer from the strange properties that they can become negative. Problem Solved! Normalize properly!

38. 10 September 2008Kevin H Knuth CIDU 2008 EXAMPLE

39. 10 September 2008Kevin H Knuth CIDU 2008 Guessing Game applebananacherry Can only ask binary (YES or NO) questions!

40. 10 September 2008Kevin H Knuth CIDU 2008 Which Question to Ask? A  VMV  AM AV  AMAV  VM If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana, which question do you ask? Is it or is it not an Apple? Is it or is it not a Banana? Is it or is it not a Cherry?

41. 10 September 2008Kevin H Knuth CIDU 2008 Relevance Depends on Probability aaa b bb ccc A  BCB  ACC  AB AV  AMAV  VM Is it an Apple?Is it a Banana? Is it a Cherry? If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana, which question do you ask?

42. 10 September 2008Kevin H Knuth CIDU 2008 aaa b bb ccc A  BCB  ACC  AB Is it an Apple?Is it a Banana? Is it a Cherry? Relevance Depends on Probability AV  AMAV  VMAM  VM If you believe that there is a 75% chance that it is an Apple, and a 10% chance that it is a Banana, which question do you ask?

43. 10 September 2008Kevin H Knuth CIDU 2008 Results a a aa aa b bb bbb ccc ccc A  BCB  ACC  AB AC  ABAB  BCAC  BC

44. 10 September 2008Kevin H Knuth CIDU 2008 EXPERIMENTAL DESIGN

45. 10 September 2008Kevin H Knuth CIDU 2008 Doppler Shift PROBLEM: Determine the relative radial velocity relative to a Sodium lamp. We can measure light intensities near the doublet at 589 nm and 589.6 nm We can take ONE MEASUREMENT Which wavelength shall we examine? Recall, we don’t know the Doppler shift!

46. 10 September 2008Kevin H Knuth CIDU 2008 What Can We Ask? The question that can be asked is: “What is the intensity at wavelength λ ?” There are many questions to choose from, each corresponding to a different wavelength λ

47. 10 September 2008Kevin H Knuth CIDU 2008 What are the Possible Answers? Say that the intensity can be anywhere between 0 and 1.

48. 10 September 2008Kevin H Knuth CIDU 2008 Given Possible Doppler Shifts… Say we have information about the velocity. The Doppler shift is such that the shift in wavelength has zero mean with a standard deviation of 0.1 nm.

49. 10 September 2008Kevin H Knuth CIDU 2008 Probable Answers for Each Question We now look at the set of probable answers for each question

50. 10 September 2008Kevin H Knuth CIDU 2008 Entropy of Distribution of Probable Results Red shows the entropy of the distribution of probable results.

51. 10 September 2008Kevin H Knuth CIDU 2008 Where to Measure??? Measure where the entropy is highest!

52. 10 September 2008Kevin H Knuth CIDU 2008 Professor Keith Earle UAlbany (SUNY) ACERT Simulation Workshop 2007

53. 10 September 2008Kevin H Knuth CIDU 2008 AUTOMATED INQUIRY

54. 10 September 2008Kevin H Knuth CIDU 2008 More and more are our instruments required to perform science operations further from the intervention of humans. Remote Science Dust devils whip across Gusev Crater on Mars Opportunity on Mars

55. 10 September 2008Kevin H Knuth CIDU 2008 Underwater Robotic Explorers At the Monterey Bay Aquarium Research Institute (MBARI), researchers are employing robotic submarines to explore the Deep Pacific Ocean.

56. 10 September 2008Kevin H Knuth CIDU 2008 The LEGO Mindstorms NXT System 1 The NXT Brick is the brain of the system. 2Touch Sensor 3 4 5 6 Microphone Light Sensor Ultrasonic Rangefinder Servo Motors $250!

57. 10 September 2008Kevin H Knuth CIDU 2008 Lego teams with HiTecnic Accelerometer NEW! Color Sensor Digital Compass Prototype Board Sensor and Motor Multiplexers

58. 10 September 2008Kevin H Knuth CIDU 2008 This robot is equipped with a light sensor. It is to locate and characterize a white circle on a black playing field with as few measurements as possible. Robotic Scientists

59. 10 September 2008Kevin H Knuth CIDU 2008 The Robot’s “Thoughts” Set of Hypothesized CirclesMean Circle Area within Robot’s Reach colored according to ENTROPY Past Measurement DARK Past Measurement LIGHT Next Measurement Robot Center

60. 10 September 2008Kevin H Knuth CIDU 2008 'Am I already in the shadow of the Coming Race? and will the creatures who are to transcend and finally supersede us be steely organisms, giving out the effluvia of the laboratory, and performing with infallible exactness more than everything that we have performed with a slovenly approximativeness and self-defeating inaccuracy?' George Eliot (Mary Anne Evans), The Impressions of Theophrastus Such, 1879.

61. 10 September 2008Kevin H Knuth CIDU 2008 Special Thanks to: John Skilling Ariel Caticha Janos Aczél Keith Earle Philip Erner Deniz Gencaga Philip Goyal Steve Gull Jeffrey Jewell Carlos Rodriguez Funded in part by: University at Albany Faculty Research Development Award NASA AIST-QRS-07-0001 NASA 05-AISR05-0143