A New Kind of NumberEncompassing Logical And Numerical UncertaintyWith Applications to Computer Systems and Quantum Theory

Joseph E. Johnson, PhDUSC Physics ColloquiumMarch 20, 2008

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Propose that probability is not a scalar. Propose a method of managing logical & numerical

uncertainty Generalize Boolean logic replacing ‘1’ and ‘0’ logical bits

with a Markov group representation with probabilities. Use this system to propose a new kind of number that

generalizes the existing numbers. Rebuild mathematics with these numbers Show that these new numbers encompass all probability

distributions & manage uncertainty Discuss computer and quantum theory applications.

Problems

Mathematics has no closed complete system for managing numerical uncertainty

Uncertain numbers are probability distributions and are no longer numbers

Computers do not do well with approximate data unless explicitly programmed

All observations and data are estimates, not real numbers Examples: pharmacy, medicine, finance, engineering –

even physics Mass, length, and time are never observed as real

numbers. - we impose the concept of infinite divisibility on them because we do not know what else to do.

Problem of how to begin

To incorporate numerical uncertainty is to incorporate probability

To have uncertain numbers suggests using uncertain logic since numbers are built upon Boolean logic.

But probability functions do not close nicely My work with Markov transformations suggested to me

to use the Markov representation space (x1, x2) to generalize 1 & 0 to continuous probabilities.

Thus probabilities are part of a group representation.

Postulate 1: Fundamental Logical Entity Postulate: (x1, x0) is the fundamental logical entity It is to be the probability to have a ‘1’ & ‘0’ respectively Thus x1+x0=1 & xi are non-negative These ‘bit vectors’ or ‘bittors’ are to be the

fundamental information entity. Representations of all integer dimensions exist but

are not needed yet. The technical group theory will be discussed later

Postulate 2: Definition of Product

The Boolean logical products (AND, OR..) must be defined If x, y, and z represent bittors then we must define z = x AND y etc.

as zi = cijk xj yk since probabilities multiply

For ‘AND’ we have z1 = x1y1 , z0 = x1y0 + x0y1 + x0y0

The other 15 Boolean products are all permutations of separating these 4 terms into the upper and lower components following the natural laws of probability

NOT reverses the components IT is easy to confirm that z is a bittor if x and y are bittors

Postulate 3: Linear combination

Having defined a ‘product’ of entities, we can ask if a sum can be defined.

A weighted linear combination can be defined as x = a1x1 + a2x2 +… anxn where xi

are different bittors and ais a higher dimensional bittor representation. (Thus the ai are nonnegative and sum to unity)

This new mathematics thus has 16 independent products and one linear ‘addition’.

Postulate 4: Bittor Based Numbers

Define a bittor number to be the outer product of several Markov monoid (usually two dimensional) representations (xj,yk) (xj,yk)() . ()()

This number is now a group representation space We only need to write the upper number of the two As error is always limited, we only need a few digits

to represent the probability as it is never known more accurately e.g. (1)(1)(0).(1)(011101) = 110.1(011101)()

Postulate 5: Bittor Arithmetic

Arithmetic is defined with Bittors in exactly the same schema as with binary numbers.

Adding two bits: 1+0 = 0+1 = 1 and 1+1 = 0+0 = 0 except carry 1 if 1+1 i.e. AND.

This is defined as XOR (exclusive OR) of the bittors: z1 = x1y0 + x0y1 & z0 = x1y1 + x0y0

The carry digit is computed as z1 = x1y1 for AND Multiplying uses the AND with z1 = x1y1

Postulate 6: Information Defined

Shannon’s definition of information I = 1 + Pi log2(Pi) gives us an information value of 1 when P is 0 or 1 and a value of 0 when P is ½ .

A more useful form is Renyi’s entropy which is already in the extended bittor logic as the operation ‘EQV’ of the bittor with itself:

I = log2 (2(x12 +x0

2)) is the log base 2 of self equivalence, is defined to be the information content of a bittor.

The information value in an entire bittor number is thus the sum of the information in each component bittor.

Shannon Information I = 1 – (Pi log2 Pi)

Renyi Information (2nd order) I = log2 (2 (P12 + P0

2))

Comparison of Shannon & Renyi Second Order Information

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Shannon

Renyi 2nd

Smooth Generalizations of Logic and Number. Bittor operations, and numbers, smoothly

reduce to the standard numbers, logic when the bittors are exact (1,0) or (0,1).

Specifically, the bittor structures include the full Boolean logic, binary values and existing number system (integer, rational, real, and complex numbers) with existing mathematical operations – all as a special case.

Summary:

The fundamental objects of information are proposed to be Markov Lie monoid representations

The generalized logic is defined by zi = cijk xj yk and x = a1x1 +

a2x2 +… anxn , provides an entirely new kind of mathematics among the bittor objects that consists of 16 different products plus bittor weighted linear combinations allowing addition

Bittor logic and bittor numbers generalize arithmetic and are thus capable of automated management of uncertainty

They constitute a new kind of mathematical structure that generalizes the existing number systems and contain them.

Examples

Consider the number 110.0(0.5)-> infinity The lower bound to this is 110.0000000… and

the upper bound is 110.0111111111.. which rounds to 110.0 and to 110.1 with a uniform distribution between – a square distribution.

Another example is 100(.5)10 which gives a bimodal distribution of 100010 and 100010 with equal probability.

Background Material on Markov Type Lie Groups and Monoids

Diffusion and Markov Transformations All diffusion can be represented by Markov

transformations on probability distributions Diffusion represents increasing entropy

and loss of information Thus a study of Markov transformations is

a study of probability, entropy, and information

Since diffusion (and thus Markov transformations) cannot be ‘undone’, they have no inverse

Thus no one had considered using group theory (discrete or continuous) to study them

We did. And showed a deep connection between Lie groups and Markov transformations as follows:

The General Linear Lie Group – GL(n,R) The continuous linear transformation

group can be represented as G() = eL

Where L = ij ijLij

And where there are n2 of the Lij with a 1 in the ij matrix position and 0 elsewhere

These Lij form a basis for the Lie Algebra that generate the group GL(n,R)

The Markov Lie Group M(n,R) and the Abelian Lie Group A(n,R) This set of Lmn can be decomposed into an

Abelian Lie Algebra A(n) : Lnnkl = n

k nl with a 1

in a single diagonal position and 0 elsewhere of n different diagonal elements.

And a Markov Lie Group M(n) with Lmnkl = m

k nl

- nk n

l giving a complete basis on n2-n off-diagonal elements.

The Markov Lie Group conserves the sum of the elements of a vector (not sums of squares)

Abelian Group

The Abelian group simply multiplies a particular axis or coordinate by e thus giving growth or contraction (if negative) by that factor.

Markov Group

The Markov transformations are always of the form M = 1 1-e-

0 e-

This transformation always preserves the sum of the components of a vector

Thus if x’ = M x then i xi’ = i xi

The Markov Monoid Removes the inverse from these Markov transformations:

If the xi are to be considered probabilities (or occupation numbers) then they must be non-negative and likewise x’ = M x must be non-negative.

The M will always give this if the are themselves non-negative. Thus the group looses its inverse, as is required for a diffusion process.

Such a group without its inverse is called a ‘monoid’

Markov Transformations Visualized

The Markov group moves one along the straight line x1 + x2 = 1

The Markov monoid moves any point in the positive quadrant to another point in the positive quadrant.

Thus the monoid never takes one from physical to unphysical states (like the unitary operator in quantum theory)

Lie Groups and Lie Algebras are now connected to Markov transformations

Now the power of continuous groups and algebras can be used with Markov theory

All continuous diffusion processes are actions of Markov monoid transformations generated by non-negative combinations of the Markov generators.

Infinitesimal Diffusion Generators

The infinitesimal generators (i.e. the Lie algebra for Markov transformations) are

0 1 1 0

0 1 1 0and

Properties:

One can easily show that [L12, L21] = L12 - L21 which is the lowest order nontrivial Lie algebra that exists (equivalent to [A,B]=A

Any power of any combination of the L basis is a member of the algebra (columns sum to zero).

Thus there are NO Casmir operators

Diffusion as a Markov Process

Consider a Markov diffusion transformation from left to right and also right to left:

12 21

21

1 1 0

0 1 1

sL sL

s s

s s

M s e M e

e e

e e

Markov one way diffusion

This transformation gradually transforms a fraction of the x2 value into the x1 value preserving x1 + x2 = constant.

One notes that the column sums are unity as is required for a Markov transformation.

Two way equal Markov diffusion

It is easy to see that this transformation conserves the probability and leads to an equal mixture in the distant future.

M tet L12 L21

1

21 e 2t 1 e 2t

1 e 2t 1 e 2t

Diffusion

This process is basically a “Rob Peter to Pay Paul” model.

One could equally begin with Peter having $1,000 and Paul $10. Every instant of time each gives a fraction of what they have to the other person.

Eventually we cannot discern who originally had the greater sum and information is lost as the ratio of the two sides approach unity.

Scaling Transformations

The general linear group contains two Markov transformations and two transformations that scale the axis proportionally (like an exponential birth or death rate on each side).

While the diffusion (Markov) transformation conserves the probability, the scaling transformations allow for both growth and decay.

The problem is identical to populations in two states that can migrate to each other conserving populations and also have a different birth and death rate.

Form of the Scaling Transformations

By combining this with diffusion one gets the most general linear transformation:

11 221 0 0 0

0 0 0 1

L and L

( )

tt

ch t sh t sh t

G t e e

sh t ch t sh t

Results and Future Work

Information Loss over Time

Since Bittor numbers are Markov group representations, this takes us closer to a true theory of measurement with information loss over time.

We suggest using the Markov transformation M=exp(t

L) where each value of acts upon the next highest bittor in the number with decreasing rates for diffusion L.

This allows the number to gradually diffuse and loose information beginning with the rightmost position

Path Dependence of Computation

If one takes just three bittors and forms z*(x+y) and compares this to z*x + z*y one can show that the second term has more information loss in both the carry bittor and the main bittor.

Information loss is path dependent (like friction with non-conservative forces).

We seek to always use the path of minimum loss. (What is it?)

A New World of Computing

The basic decision process in computing is IF { x >= y } THEN { do A } ELSE { do B }. But with numerical uncertainty or distributions, often there is a probability for both truth and false.

This means that each decision can spawn two threads, each with its own probability like cascading particle creation.

These in tern spawn more etc. We then terminate all processes whose probability falls below a stated threshold.

Application to Multiprocessors

This would allow the automated spreading of computation over multiple processors effectively utilizing perhaps 1024 processors for one computation.

Results would be collected back into bittors with appropriately weighted probabilities.

If databases of scientific and engineering values had uncertainty specifications, then the statistical analysis of building a bridge or finding a Higgs particle would be automatic.

Computer Management of Uncertainty I envision the direct representation of bittor logic and

numerical values in computer systems. First this will fully automate the management of

uncertainty and replace numerical rounding, Monte Carlo, and the more advanced uses of distributions.

Uncertainty will be put on a ‘solid’ foundation and computers will be able to make more informed ‘judgments’ and ‘decisions’ – begin to reason rather than to just compute.

Dr Ferdi Ponci in Electrical Engineering is overloading the operations in Mat Lab to manage bittor arithmetic and we have submitted a joint paper in Italy June 2008.

This New Math

A bittor number can be used to express any probability distribution, to any finite accuracy, by using higher order representations to manage correlations.

Since bittor numbers are isomorphic to the reals, and thus are a ‘number’ in the sense of Cantor, we are able to represent any probability distribution as a number!

Applications in Quantum Theory

We normally let linear operators act on vectors in Hilbert space or equivalently on creation and annihilation operators to get eigenvalues (numbers). The probability distributions have to be computed from (x)* (x)

Now the eigenvalues are bittors - group (monoid) representations themselves :

X | bittor_position> = bittor | bittor_position> where the bittor now gives the entire probability distribution for the particles position directly.

Thus a distribution can express the probabiltiy distribution for the wave function (x)

Some Conjectures

Quantum states are eigenvectors (a representation space) of the Lie algebra of observables with eigenvalues for position, momentum, angular momentum, energy, ….

Now these eigenvectors are a representation space for the Markov algebra with its connections to information theory and loss over time, and to measurement theory.

Can this new marriage lead to a deeper understanding of the collapse of entanglement and the connections between the quantum state and the classical world?

These new bittor numbers allow greater transparency of the information from measurements.

We can now have an eigenstate with simultaneous knowledge of X and P displayed as two bittor ‘eigenvalues’ (with an accuracy still limited by the uncertainty principle).

This allows us to imagine a set of X & P operators that measure these values only within certain limits eg X() & P() and these new operators ‘commute’ [X(), P()]=0 where the value is the wavelength of the measuring photon.

This does not violate the uncertainty principle and is in line with what we do in everyday life for simultaneous knowledge.

The Hilbert Space (x) =<x|>

Traditionally, |is a representation space for the algebra of observables providing both the probability amplitude distributions and the state phases that give us interference.

Using a complex bittor might give the interference as well as the probability distribution.

The foundations have always rested on <|the scalar product.

One can define a scalar product of two bittor (numbers/distributions) that exactly matches (with logic) the operation of the folding of two functions in the scalar product (probability of if then ) !!

Finally

Space and time are limited by the accuracy of measurements making their infinite divisibility questionable.

Could this new mathematics, describing mass, length, and time with group representations, non-singular distributions, information theory, and other differences, provide new approaches and insights?

Thank You