1. 1 Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? TS Eliot, 1934

2. 2 Information: The Markov Group 1. Networks 2. Logical & Numeric Uncertainty 3. Quantum Theory of Measurement Joseph E. Johnson, PhD Department of Physics University of South Carolina jjohnson@sc.edu March 2, 2006 ©

3. 3 I acknowledge useful conversations, calculations, & computer simulations on related material with a number of people including: V. Gudkov, S. Nusinoff, G. McNulty, Oskolkov, F. Wu, J. McHugh, V. Skormin, H. Farach, W. Full, D. Buell J Lala, as well as W. Campbell, M. Rabon, R. Shelley, E. Yu, N. Jeong, B. Full, T. Spearman, J Sidoran. The work that I will present today is mine.

4. 4 Presentation – 4 parts 1. Background: 2. A Proposal for Network Metrics 3. A Proposal for Logical & Numerical Uncertainty 4. Applications to quantum mechanics

5. 5 INFORMATION How can we nail this concept down. (Image for Markov Group Theory)

6. 6 1.Background: Diffusion & Entropy Many body classical systems increase entropy and disorder irreversibly This even defines our direction of time.

7. 7 Irreversibility & Markov Theory “Markov Type Lie Groups in GL(n,R)”, Joseph E Johnson, 1985, J Math. Phys. GL(n,R) = M(n,R) + A(n,R) M(n,R) = MM(n,R) + their inverses where MM refers to the Markov Monoid MM, are isomorphic (1-1) with all networks. MM contain diffusion, entropy, & irreversibility The MM representations can be used to build a new type of mathematics.

8. 8 The Objective of that Work Objective: Find continuous Markov lie group Markov transformations – are not a group But study the group that contains them, then restrict to the allowable Markov tranformation. These form the Markov Monoids.

9. 9 We begin with diffusion & irreversible processes – go back 100 years: Einstein – random walk, diffusion theory 1905 Markov (1906) transformations (80% probable to stay put and 10% probable to move to the two adjacent cells): |X’> = M |X> can be written as (note columns sum to unity):

10. 10 Define the Lie Algebra for Markov Transformations The Markov Lie Group is defined by transformations that preserve the sum of the elements of a vector ie  x i ’ =  x i The generators are defined by ‘rob Peter to pay Paul’ for M = e tL where L ij is defined by having a ‘1’ in row i and column j along with a ‘-1’ in the diagonal position row j and column j: L 12 = L 21 = One gets closure: [L 12, L 21 ] = L 12 - L 21 Note that this is the smallest possible Lie Algebra!

11. 11 Lie group defined: And the Markov group transformation then takes the form: M(t) = e s = One notes that the column sums are unity as is required for a Markov transformation. This transformation gradually transforms the x 2 value into the x 1 value preserving x 1 + x 2 = constant.

12. 12 Higher Dimensions: This Markov Lie Algebra can be defined on spaces of all integral dimensions (2, 3,….) and has n 2 - n generators for n dimensions representing basis elements with a ‘1’ in the ij position and a ‘-1’ in the jj position. This makes this basis a complete basis for all off- diagonal matrices. E.g. in 5 dimensions: L 14 =

13. 13 Graphically, what is happening? These transformations define the group of motions on a straight line in two dimensions. The Markov Lie group is a mapping of this line back into itself – but is NOT a translation group.

14. 14 Where is Irreversibility? (An Important Result) But if these components are to be interpreted as probabilities then, non-negative values must be transformed into nonnegative values! This removes all the inverse transformations. Allowable transformations thus are those and only those with non-negative linear combinations of the Lie basis elements. This removes the inverse of all transformations and gives us a Lie Monoid and Irreversibility.

15. 15 Abelian Scaling Group Consider the Abelian scaling group generated by L ii = 1: When adjoined to the Markov group, one obtains the entire general linear group, GL(n,R). It is obvious that the combined Lie algebra spans all n x n matrices.

16. 16 Conclusion on Markov Lie Monoids: One is now able to utilize all the power of Lie groups and Lie algebras to study Markov processes in a continuous way by linking these two areas of mathematics. Furthermore one can see Markov theory as that portion of the general linear group that leaves the hyperplane perpendicular to (1,1,…) as invariant i.e. <1| M = <1|. Finally, by adding the scaling transformations, one obtains all possible continuous linear transformations.

17. 17 Networks Nothing is less real than realism Details are confusing It is only by selection by elimination by emphasis That we get at the real meaning of things Georgia O'Keefe

18. 18 2. NETWORKS Networks are important and ubiquitous. They embody some very difficult unsolved mathematical problems. A network is a set of points (nodes) some of which are joined by values (weights). They are defined by a connection matrix, C ij of non-negative values off of the diagonal.

19. 19 Examples of Networks: Communication Networks The Internet Phone (wired & wireless) Mail, Fed-Ex, UPS Transportation Networks Air Traffic Highways Waterway Railroads Pipelines

20. 20 More Examples: Financial Networks Banking & Fund Transfers Accounting Flows Ownership & Investments Input-Output Economic Flows Utility & Energy Networks Electrical Power Grids Electrical Circuits & Devices Water & Sewer Flows Natural Gas Distribution Biological Networks esp. Disease Neural Nets Blood Flow

21. 21 The Network Problem For extensive networks of hundreds or thousands of nodes there of billions of connections that constantly change – not unlike the movement of gas molecules. We would like to know generally what is happening without knowing the details. Thus we seek “network metrics” – a few variables f k (C ij ) that ‘represent’ the network. This is similar to the roles that thermodynamic variables (temperature etc) fulfill for gasses.

22. 22 Problems: There is no concept of distance, thus we cannot define ‘pressure’ or ‘volume’. There is no concept of energy, thus we cannot define ‘temperature’, ‘heat’, or ‘internal energy’. There is no obvious probability distribution, so we cannot define ‘entropy’ or ‘information metrics’ of order and disorder.

23. 23 Realization 1 Since the connection matrix is exactly and totally defined by off-diagonal non-negative real numbers, then each connection matrix is exactly isomorphic to an element of the Lie monoid of Markov transformations. We only need to use the freedom to set the diagonals of C to the negatives of the sum of the other elements of that column. Thus every possible graph or network topology generates exactly one unique Markov transformation & conversely.

24. 24 Realization 2 Since any Markov transformation consists of columns which sum to unity, then they can be considered as probability distributions associated that the respective node. Thus we can compute an associated entropy (or generalized entropy) on this distribution which will give an entropy measure for that node: S i.

25. 25 Realization 3 Networks are generally asymmetric. The weights away from a node (row values) are different from the weights toward a node (column values). The same S i can be thus computed for rows as for columns. This gives a pair of metric values for each node thus reducing (n 2 – n) independent values down to 2n values.

26. 26 Realization 4 We do not really care which node is doing what but only “are the nodes on the whole doing about the same thing at one time as at another”. Thus we can sort the nodes by the value of the column entropy to obtain a non-increasing curve of the row entropy spectra. For each window of time that we capture a network image, we can recompute this spectra and see if it is ‘about the same’ and likewise for the row spectra with the same sort order.

27. 27 Realization 5 The spectral form can be studied by multiple means to see to what degree it deviates from the ‘normal profile’ for that time and for that network and type. Specifically, one can track the correlation coefficient, r 2, between the instantaneous row spectral curve and the ‘normal’ spectral curve. This reduces the entire network at a given time, to a correlation coefficient with the normal state of that network. One can also use wavelet theory and other means of expansion of the entropy spectra for analysis.

28. 28 We have filed a provisional patent on this procedure. For earlier and different work in 2001, Dr. Gudkov and I filed a patent based upon a different method based upon mutual entropies & generalized Renyi’ entropy. Another provisional patent was filed with Dr. Gudkov and Dr Nusinoff.

29. 29 Conclusion We can represent a network at a given instant by a pair of spectral curves representing the incoming and outgoing entropy (order/disorder) metrics. These metrics can be shown to have the interpretation the entropy rates of the dispersion of a conserved entity (fluid) over the equivalent network topology, with the flow rates specified. In this framework one can easily understand the meaning of the eigenvectors and eigenvalues of the connection matrix.

30. 30 Applications We intend to utilize these algorithms to monitor the behavior of complex networks. I have one patent application in progress with Dr Gudkov on related work, and another provisional application filed on the work you see here.

31. 31 3. Logical & Numerical Uncertainty – A more general mathematics "This special property of digital computers, that they can mimic any discrete machine, is described by saying that they are universal machines. The existence of machines with this property has the important consequence that... it is unnecessary to design various new machines."  A.M.Turing

32. 32 Concern & problem For many years I have been troubled by the use of the real number system to measure values such as length and time, when in fact the numbers are always truncated at the unknown digits. No one has EVER measured a real number. To do so requires an infinite effort and infinite information. Our means of combining these uncertain numerical values lack an exacting mathematical foundation that is at the core of the human reasoning process.

33. 33 A Radical Contention I contend that the real numbers are actually inappropriate for physical observations (length, time, mass,…). I contend that they should be replaced by a mathematical system that exhibits and tracks the associated uncertainties through the entirety of numerical processing. Such a system would be built upon the physical measurement process rather than the number system of pure logic (of Russell & Whitehead).

34. 34 Down to basics: Our most fundamental mathematical entities are the binary bits ‘1’ and the ‘0’ (or T/F). These are the ‘quanta’ of Shannon information. These represent the lowest level of decisions of ‘will the next increment of a binary power of that unit (time, length etc) ’ fit in the remaining space. But something at a still lower level exists which is the probability that the next increment would fit.

35. 35 Resulting Problem & Question The fundamental methodology of combining the 1 and 0 bits is Boolean logic, in the form of well defined truth tables giving closed forms with operations (AND, OR…). Is there a way to make the 1 and 0 ‘uncertain’ i.e. as probabilities, and articulate the structure into a cohesive mathematical system ? But ‘probabilities’ do not ‘close’.

36. 36 A Proposed New Fundamental ‘Bit’ Proposal: It occurred to me to use the two dimensional Markov monoid representation x = (x 1, x 0 ) to represent this fundamental bit with x 1 as the probability to be true (or ‘1’) and x 0 as the probability to be false (or ‘0’) Independent probabilities multiply. Thus we need to combine our fundamental ‘probability bit objects’ x and y to get a z in a way that generalizes Boolean logic.

37. 37 The Bittor: x = (x 1, x 0 ) This ‘Bit Vector’ is a (two dimensional) Markov group (monoid) representation space similar to the two-dimensional ‘Spinor’ representation space of the Unitary group. Thus I call this representation space a ‘Bittor’ Neither a ‘spinor’ nor a ‘bittor’ is a ‘vector’. A bittor is an ordered pair of non-negative reals that sum to unity: x 1 + x 0 =1 and x i >= 0

38. 38 Boolean Generalization Requirements Since probabilities multiply, and must give closure, we must have z i = c  ijk x j y k where the indices have the values 1,0 and we use the summation convention over repeated indices. The index  must range over all of the possible types of products that one can form (i.e. independent combinations for ‘AND’, ‘OR’, ‘NOR’, ‘NAND’ etc.) Furthermore, the c  ijk must give closure. Thus if x and y are Bittors (i.e. two dimensional Markov monoid representations with non-negative values) then z must be also be a bittor.

39. 39 Bittor Product Defined There are exactly 4 types of products generated: x 1 y 1, x 1 y 0, x 0 y 1, x 0 y 0 which exactly cover each of the four possible ‘outcomes’ as the probability of TT, TF, FT, and FF. It follows that the two components of the resulting bittor z must be the partitions of these four terms, each of which is positive and whose sum will then be unity as required.

40. 40 These can be indicated by the binary number (0 – 15) giving the presence of the four terms (or not) in z 1. Thus 1000 would mean that z 1 = x 1 y 1 while z 0 = x 1 y 0 + x 0 y 1 + x 0 y 0. Thus  = 1000 is the ‘AND’ operation. We have now defined 16 independent products by the 16 values of 

41. 41 Bittor Linear Combination We can additionally ask whether a linear combination of bittors z = a 1 x 1 + a 2 x 2 + … gives closure (i.e. a bittor z with non-negative values and a component sum to unity). The necessary and sufficient condition is that the a i linear coefficients themselves form an n-dimensional representation of the Markov monoid and thus represents a bittor of higher dimensionality. The interpretation of this is as a weighted average where the weights are non-negative and sum to unity.

42. 42 A New Mathematical Structure With bittors (Markov monoid representations) as the fundamental objects, we now have defined a new mathematical structure with 16 independent products, and a special type of linear combination. Additionally one can execute continuous Markov transformations on the bittor spaces. In regular Boolean logic, the NAND operation is sufficient to build all other products, but here one needs all 16 products.

43. 43 Bittor Based Numbers Just as the binary numbers are used to form the customary integers, rationales, and reals, we now propose a new bittor number that is the outer product of bittors: (1,0)(1,0). (0,1)(0.8,0.2)(0.5,0.5) …although we can abbreviate this as the upper part: 11.0(.8) Addition and multiplication follow the traditional rules for execution but with the generalized bittor logic replacing Boolean logic. The bittor numbers contain all past number systems, Boolean logic, and arithmetic as the limiting case of values where 1 = (1,0) and 0 = (0,1).

44. 44 New Computer Language Proposed One can define these two component bittors as ‘objects’ and then ‘overload’ the logic operations and the arithmetic operations in computer operations. But the situation is very complex as all branching commands become quite complex and allow for simultaneous threads to be spawned (and dissolved when minimal). One needs rather to redesign the base languages (CC++ or JAVA) to incorporate these new objects and the numbers (outer product representations) and the generalized arithmetic that they support.

45. 45 A True Paradigm Shift in Programming The foundation of logical decisions in computers is If x>y then A else B (or a similar expression). But now, there is a computable probability that x>y and also is not. This means that the program will neet to spawn two threads each carrying the associated probability for its truth. Threads with low probability will be set to automatically terminate below a threshold and the results will accumulate from multiple threads. These processes are not unlike the creation and annihilation of particles in physical processes.

46. 46 Information Content Both ‘1’ and ‘0’ represent one ‘bit’ of information in the definition of Shannon that I = log 2 (P). We can ask how much information is represented by the bittor (x 1, x 0 ) by imposing the additional boundary condition that the information of (0.5, 0.5) is zero. Since independent probabilities multiply, and the associated information is additive, we have I = log 2 a(x 1 b + x 0 b ) and need to determine ‘a’ and ‘b’. From the boundary conditions one easily determines that a = b = 2 thus I = log 2 2(x 1 2 + x 0 2 ). This is also known as Renyi’ entropy of the second order.

47. 47 General Information Equation For n dimensional bittors, one can show that one obtains: I = log 2 (n  x i 2 ). Furthermore one can show that this is a ‘smooth’ generalization of Shannon entropy down into the ‘fractional bit’ domain. One now has the tools to track the loss of information throughout all logical and arithmetic operations.

48. 48 Conclusions for this ‘New Math’ It is our proposal that the bittor based numbers be used as the foundational number system replacing the reals for observations and subsequent mathematics and computation. Note that this is a generalization of the Boolean and real number systems and all traditional logic, reals, rationals, integers, and mathematics are contained in this system.

49. 49 Utility of the Bittor System The bittor system will allow for the automated management of uncertainty and error in all scientific and engineering applications and computations. It furthermore can provide for multithreaded computation in a ‘Feynman path’ methodology that explores computational outcomes weighted with the associated probability. It provides a well defined method of tracking information and its loss in all operations. It thus provides a foundation for automated computation much closer to human reasoning. Last month (Feb. 7, 2006) I was awarded a patent for these concepts (US Patent 6996552)

50. 50 (image for quantum theory) Quantum Theory: Its all about information; what is knowable and when.

51. 51 3. Quantum Theory In one particle quantum theory we label the state vector with the eigenvectors of mutually commuting observables (position, energy, angular momentum..) e.g. with position: X |y..> = y |y..> where y is a real number. Yet we know that this is meaningless since we cannot measure the position exactly or even within the Compton wave length.

52. 52 Proposal: We propose that the real number eigenvalues be replaced by bittors. Thus eigenvalues are group (monoid) representations. Thus imagine that we make a series of measurements determining the position as a result of binary categories (boxes) as X|10.101(0.81)> = (1)(0).(1)(0)(1)(0.81 |10.101(0.81)> where each of these values is actually a bittor. Each subsequent binary digit that is measured results in more information for position space and an exact reduction in the information for the momentum space. This is superior to using , the standard deviation, as a measure of uncertainty because one has the same information if the particle is known to be within two boxes of a given size, no matter how far apart.

53. 53 The determination of a bittor value is to correspond to an action that links the quantum system to a classical ‘observation’ that is irreversible in a time of the age of the universe. The concept of having the representations of one Lie algebra (XPI) to be indexed by the representations of another Lie algebra (monoid) is complex and needs further study. In particular it would imply that the scalar product is mapped onto a group representation (a single bittor – representing the probability of the ‘overlap’ or equivalently that one state will be found in the other.) rather than the reals. We also note that the bittors are a way of ‘quantizing’ the real number system by saying that we can only know discrete ranges.

54. 54 Space and Time If space and time are now measured with bittor numbers, then space and time are a type of lattice without infinite divisibility. How might a theory of gravity be done in this environment?

55. 55 Thank You