noise-based logic: from boolean logic gates to brain...

Texas A&M University, Department of Electrical and Computer Engineering

Noise-based logic: from Boolean logic gates to brain circuitry

Laszlo Kish

Department of Electrical and Computer Engineering, Texas A&M University, College Station

"We can't solve problems by using the same kind of thinking we used when we created them." (Albert Einstein)

When noise dominates an information system, like in nano-electronic systems of the foreseeable future, a natural

question occurs: can we perhaps utilize the noise as information carrier? Another question is: Can a deterministic

logic scheme be constructed that may be the explanation how the brain can efficiently process information, with

random neural spike trains of less than 100 Hz frequency, and with similar number of neurons than the number of

transistors in a 16 GB Flash dive? The answers to these questions are yes. Related developments indicate reduced

power consumption with noise-based deterministic Boolean logic gates and the more powerful multivalued logic

versions with arbitrary number of logic values. Similar scheme (logic hyperspace) as the Hilbert space of quantum

informatics can also be constructed with noise-based logic without some of the limitations of quantum computers.

Noise-based string search algorithm with higher speed than Grover's quantum search algorithm is obtained with the

same hardware complexity class as the quantum engine. This hyperspace scheme has also been utilized to construct a

deterministic multivalued logic scheme for the brain and the relevant circuitry of neurons.

See more at: http://www.ece.tamu.edu/%7Enoise/research_files/noise_based_logic.htm

Seminar at University of Texas, Grad. School of Biomedical Sciences, Houston, March 24, 2009.

Seminar at Rice University, Dept. Mech. Eng. and Materials Sci., Houston, March 25, 2009.

Seminar at RIKEN Brain Institute, Tokyo, Japan, June 22, 2009.

Seminar at Kanagawa Industrial Technology Center, Japan, June 26, 2009.

Seminar at Arizona State University, Dept of Electrical Engineering, Tempe, July 20, 2009.


Present active collaborators in noise-based logic:(In chronological order of the first joint paper submission, or expected submission. Brown color: joint results in this talk.)

Sunil Khatri, (computer engineering faculty, TAMU): "quantum-mimics", memory, chip, complexity, etc.

Swaminathan Sethuraman (mathematician, fresh PhD, TAMU): "quantum mimics", etc.

Sergey Bezrukov (chief scientist, NIH): brain: information processing/routing, circuitry, efficiency, etc.

Ferdinand Peper (senior computer scientist, Kobe Research Center, Japan): "quantum mimics", tokens, etc.

Zoltan Gingl (physics faculty, Univ. of Szeged, Hungary); modeling for circuit realization, etc.

Kamran Entesari (electrical engineering faculty,TAMU): noise generators for chip realization, etc.

Khalyan Bollapalli (computer engineering PhD student, TAMU): chip realization

Zoltan Bacskai (physics PhD student, Univ. of Szeged, Hungary): DSP circuit realization

Gabor Schmera (mathematician, US Navy, SPAWAR): Languevin equations and numeric solutions, etc.


Our related papers (brown: subject of this talk):

• L.B. Kish, "Thermal noise driven computing", Appl. Phys. Lett. 89 (2006) 144104;

http://arxiv.org/abs/physics/0607007

• L.B. Kish, "Noise-based logic: binary, multi-valued, or fuzzy, with optional superposition of

logic states.", Physics Letters A 373 (2009) 911-918; http://arxiv.org/abs/0808.3162

• L.B. Kish, S. Khatri, S. Sethuraman, "Noise-based logic hyperspace with the superposition of

2^N states in a single wire", Physics Letters A 373 (2009) 1928-1934,

http://arxiv.org/abs/0901.3947

• S. Bezrukov, L.B. Kish, "Deterministic multivalued logic scheme for information processing

and routing in the brain", Physics Letters A 373 (2009) 2338-2342,

http://arxiv.org/abs/0902.2033


Content

1. The device size-speed-error-energy issue in classical digital and single electron logic.

2. Quick comparison of the brain and a computer.

3. Continuum-noise-based logic, binary and multivalued logic.

4. Utilizing the logic hyperspace: 2N bits [2^(2^N) logic values] in a single wire, like in a

quantum computer.

5. Implementation of the hyperspace for neurons and their stochastic spike trains. Deterministic,

multivalued brain logic and routing the information in the brain.


It is fashionable to cite old, historical objections against the potentials of science and

then point it out how much science and technology has been outperforming even the

most courageous expectations.

For example the citation in Popular Mechanics (1949), forecasting the perspectives

of science:

Computers in the future may weigh no more than 1.5 tons.


However, we scientists want to be original so we should never go with the fashion...

Thus let's go against the fashion, while staying with computers ...:-)


In the "Blade Runner" movie (made in 1982) in Los Angeles, at 2019...


In the "Blade Runner" movie (made in 1982) in Los Angeles, at 2019,

the Nexus-6 robots are more intelligent than average humans.


However, 2019 is only 10 years from now and nowadays we have been observing

the slowdown of the evolution of computer chip performance.

We are simply nowhere compared a Nexus-6.


Isaac Asimov (1950's): The Three Laws of Robotics


1. A robot may not injure a human being, or, through inaction, allow a human to come to harm.

2. A robot must obey orders given to him by human beings except where such orders would

conflict with the First Law.

3. A robot must protect its own existence as long as such protection does not conflict with the

First or Second Law.

Isaac Asimov (1950's): The Three Laws of Robotics:


1. A robot may not injure a human being, or, through inaction, allow a human to come to harm.

2. A robot must obey orders given to him by human beings except where such orders would

conflict with the First Law.

3. A robot must protect its own existence as long as such protection does not conflict with the

First or Second Law.

Isaac Asimov (1950's): The Three Laws of Robotics:

However, not even the best supercomputer systems are able to address such refined

perception of situations!

We have great problems even with the most elementary necessities, such as

recognition of natural speech of arbitrary people or speech in background noise.


What is closing the slowdown and losing of hopes?

The Speed-Error-Energy triangle of microelectronics: the key is the noise

(Miniaturization) Bandwidth Error rate Increasing power need

• Claims about high performance without error rate and energy efficiency aspects are interesting but meaningless for practical

developments.

• Claims about high energy efficiency without error rate and performance drop aspects are interesting but meaningless for practical

developments.

• Claims about efficient error correction without energy requirement and performance drop aspects are interesting but meaningless for

practical developments.

• These performance-error-energy implications must be addressed at the system level otherwise they are meaningless for practical

developments. Maybe we won at the single gate level, which is interesting but unimportant, but lost at the system level.


Only two logic values are utilized on a single wire in today's digital circuitry

Usignal(t)

UH

UL

Time

U0 (power supply voltage)

0

Clock generator events

0 0 0 0

1 111


Model-picture of speed and dissipation versus miniaturization (LK, PLA, 2002)

U0

2

R

1C

CMOS gate

capacitance

CMOS drivers'

channel resistance

C s2

C s

f0(RC)

1

P1

f0E1(RC)

1CU

0

2 U0

2

R

PN

NU0

2/R NU

0

2U0

2/s2

Maximal clock frequency

Dissipation by a single unit

Total dissipation by the chip

number of units N1

s2

A switch is a potential barrier which

exists (off position) or not (on position).

To control/build the potential barrier we need energy.


R C

(T)

u(t)

1

2C u

c

2= E

C=kT

2 (one thermodynamical degree of freedom)

uc

2=kT

C

Nice, but how about the noise and errors? Unavoidable noise: Thermal noise (Johnson noise)

Su( f ) = 4kTR misleading!

Energy equipartition theorem;

only the capacitance matters!


For band-limited white noise, frequency band (0, fc) :

(Uth ) =2

3exp

Uth

2

2Un

2

fc Un = S(0) fcwhere

Same as the thermal activation formula, however, here we know the mean attempt frequency more accurately.

time

Amplitude

False bit flips. Gaussian noise can reach an arbitrarily great amplitude during

a long-enough period of time.

Usignal(t)

UH

UL

Time

U0 (power supply voltage)

0

Clock generator events

0 0 0 0

1 111


10-15

10-12

10-9

10-6

10-3

100

103

106

109

1012

8 9 10 11 12

Fre

qu

en

cy o

f b

it f

lip e

rro

rs

(1

/ye

ar)

Uth / U

n

Clock frequency: 2GHz

1 transistor

108 transistors

1010

transistors

109 transistors

Clock frequency: 20GHz

Conclusion:

11*Un noise margin is not

safe for future progress.

12 Un noise margin is very

safe.

Minimal energy need

The breakdown is extremely progressive.

20% change of the thermal noise or the

threshold yields a change by factor of 109


Thermal death of Moore's law (Kish, Physics Letters A, 2002)

See more: L.B. Kish, "Moore's Law and the Energy Requirement of Computing versus Performance", IEE Proc. - Circ. Dev. Syst., 2004.

Actual noise

margin, old

Actual noise

margin, new

Required noise

margin, old

Required noise

margin, new

0.1

1

10 100

Nois

e m

arg

in,

V

Size, nm

Clock frequency has not been increased since then!

2003

20022002

2003

• Optimistic estimation:

• No hot electron noise

• No 1/f noise

• No cross-talk noise

• No variability errors


Max radius of quantum dot in the single electron transistor to reach the required error rate,J.U. Kim, L.B. Kish, “Can single electronic microprocessors ever work at room temperature?”, Phys.Lett. A 2004.

To reach the required error rate in single electron transistor based microprocessosr,

the characteristic lithography size should be 1 nanometer or less with silicon!

1 10 1000.1

1

10

100

slope:-1/2

slope: -1

Vth=V

n

Classical for 109

a SETT

109 SETTs

Max

imu

m R

adiu

s o

f Q

uan

tum

Do

t (n

m)

Temperature (K)

DC working condition

Quantum confinement

Coulomb blockade

Coulomb blockade, 1

Coulomb

blockade 109

Kish’s condition for Moore’s Law

Room temperature

109


November 2002 January 2003

Conclusion was (2002): if the miniaturization is continuing

below 30-40 nm, then the clock frequency cannot be increased.

No increase since 2003 ! Prophecy fulfilled earlier!


Can we gain the energy back? Criticism of reversible computing approaches.

Wolfgang Porod, David Ferry, and coworkers. W. Porod, Appl. Phys. Lett. 52, 2191 (1988); and references

therein; W. Porod, R.O. Grondin, D.K. Ferry, Phys. Rev. Lett. 52, 232-235, (1984); W. Porod, R.O. Grondin, D.K. Ferry,

G. Porod, Phys. Rev. Lett. 52, 1206, (1984); and references therein. Their most important general argument:

Logical reversibility has nothing to do with physical reversibility.

How about the errors??? Cavin, et al, FNL 2005: If we want to do reversible computing with

the original error rate then we end up at more energy dissipation.


• Claims about high performance without error rate and energy efficiency aspects are interesting but

meaningless for practical developments.

• Claims about high energy efficiency without error rate and performance drop aspects are interesting

but meaningless for practical developments.

• Claims about efficient error correction without energy requirement and performance drop aspects

are interesting but meaningless for practical developments.

• These performance-error-energy implications must be addressed at the system level otherwise they

are meaningless for practical developments. Maybe we won at the single gate level, which is

interesting but unimportant, but lost at the system level.

Perhaps, the most important conclusion:


How does biology do it??? Comparison:

This Laptop Human Brain

Processor dissipation: 40 W Brain dissipation: 12-20 W

Deterministic digital signal Stochastic signal: analog/digital?

Very high bandwidth (GHz range) Low bandwidth (<100 Hz)

Sensitive for errors (freezing) Error robust

Deterministic binary logic Unknown logic

Potential-well based memory Unknown memory mechanism

Addressed memory access Associative memory access (?)


Noise as Information Carrier?

Neural signals (stochastic spike trains)

Noise driven informatics?


Quantum telecloning to 2 network Units, Fidelity 60%, at Furusawa's Lab (Tokyo)http://aph.t.u-tokyo.ac.jp/~furusawa/t_Lab_Setup.jpg Kirchhoff-Johnson network element tested

Fidelity 99.98%

Future Kirchhoff-Johnson network element

How about the energy dissipation? Just have a look at these pictures!


Noise as information carrier:

Noise-based logic

Continuum noise ? Random spike trains ?

Concerns:

Stochastic logic? Slow; repeated operations ?

Deterministic logic? Averaging (statistics) slowdown?

Speed?

Number of logic values?

Energy need; power dissipation/performance?

Devices and logic gates?

Error probability?


Vi(t)Vj(t) = i, j

X( t) = aiVi(t)

i=1

N

Generally, a logic state vector is the weighted superposition of logic base vectors:

N-dimensional logic space with orthogonal logic base vectors:

L2(t) = 1 H

2(t) = 1 H ( t)L( t) = 0

For example, a binary logic base is:

H

L

aLL + a

HH

aL

2+ a

H

2=1

fuzzy

(Binary L)

(Binary H)

Continuum-noise-based logic: Binary, multi-valued, or fuzzy, with optional superposition of logic states L.B. Kish, Physics Letters A 373 (2009) 911-918, ( http://arxiv.org/abs/0808.3162 )

Noises: independent realizations of a stochastic process (electronic noise) with zero mean.

Examples: thermal noises of different resistors or current noises of different transistors: Vk (t)

Multidimensional logic hyperspace was also introduced by multiplying the base noises, see later.


Basic structure of noise-based logic with continuum noises:

Input stage:

Correlator

Logic units

DC (fast errors)

Output stage:

Analog switches

Reference (base) noises Reference (base) noises

DC DC

Input signal

(noise)

Output signal

(noise)

These two units can together be realized by a system of analog switches

Note: analog circuitry but digital accuracy due to the saturation operation represented by the switches!


Analog Multiplier

X (Output)

(Inputs)

X1(t)

X2(t)

Y(t) = X1(t) X2(t)

If X>UH then switch is closed

If X<UL then switch is open

Analog switch, follower

(Input)

X

UL,UH

Analog switch, inverter

(Input)

If X>UH then switch is open

If X<UL then switch is closed

X

UL,UH

Time average

RC

(Output)(Input)

X(t) Y = X(t) where = RC

The basic building elements of noise based-logic (out of the noise generators which can be simply resistors or transistors)

are the same as that of analog computers: linear amplifiers; analog multipliers; adders; linear filters,

especially time average units which are low-pass filters; analog switches; etc.

Note: analog circuitry but digital accuracy due to the saturation operation represented by the switches!


Example: Binary, noise-based INVERTER gate

Y ( t) = X(t)H (t) L(t)+ X( t)H ( t) H ( t)

X(t) Y (t)

H(t) L(t)

L(t) H(t)


Y ( t) = X1( t)H ( t) + X

2( t)H ( t)[ ]H (t)+ X

1(t)H (t) + X

2(t)H (t)[ ] L( t)Binary OR :

Y ( t) = X1( t)H ( t) X

2(t)H (t)[ ]H ( t)+ X

1( t)H ( t) X

2(t)H (t)[ ]L( t)Binary AND:

Y ( t) = X1(t)X

2(t) H (t)+ X

1( t)X

2( t) L( t)Binary (or arbitrary input value) XOR with binary output:

X1(t) X

2(t) Y (t)

H(t) H(t) H(t)

L(t) H(t) L(t)

H(t) L(t) L(t)

L(t) L(t) L(t)

X1(t) X

2(t) Y (t)

H(t) H(t) H(t)

L(t) H(t) H(t)

H(t) L(t) H(t)

L(t) L(t) L(t)

X1(t) X

2(t) Y (t)

H(t) H(t) L(t)

L(t) H(t) H(t)

H(t) L(t) H(t)

L(t) L(t) L(t)


X1(t)

X2(t)

Analog Multiplier

X

Time average

RC(Inputs)

Analog switch, follower

Analog switch, inverter

H(t) "True"

L(t) "False"

(Output)

Y(t)

UL,UH

UL,UH

Example: XOR gate comparing two logic vectors in a space of arbitrary dimensions (binary, multi-

value, etc), with binary output giving "True" value only when the two input vectors are orthogonal.Even though the equation contains four multiplications, two saturation nonlinearities, one inverter, and two time averaging, the

hardware realization is much simpler. It requires only one multiplier, one averaging unit and two analog switches. Realizations of the

other gates also turns out to me simpler than their mathematical equations.

Y ( t) = X1(t)X

2(t) H (t)+ X

1( t)X

2( t) L( t)


If i k and Hi,k (t) Vi (t)Vk (t) then for all n =1...N , Hi,k (t)Vn (t) = 0

Logic hyperspace by multiplying the base noises:

The hyperspace can be grown further by multiplying hyperspace vectors made with

different base elements.

H

L

aLL + a

HH

aL

2+ a

H

2=1

fuzzy

(Binary L)

(Binary H)

Multidimensional (2N-1 dimensions with N noises)


V1

0V2

0V3

0= 0,0,0 , V

1

1V2

0V3

0= 1,0,0 , V

1

0V2

1V3

0= 0,1,0 , V

1

0V2

0V3

1= 0,0,1 ,

V1

1V2

1V3

0= 1,1,0 , V

1

0V2

1V3

1= 0,1,1 , V

1

1V2

0V3

1= 1,0,1 , V

1

1V2

1V3

1= 1,1,1

no signal = 0,0,0 , V1 = 1,0,0 , V2 = 0,1,0 , V3 = 0,0,1

V1V2 = 1,1,0 , V1V3 = 1,0,1 , V2V3 = 0,1,1 , V1V2V3 = 1,1,1

2N orthogonal hyperspace vectors from N base noises, noise-bits, used in binary (on/off) mode

2N-1 orthogonal hyperspace vectors from N base noises, noise-bits, used in binary (on/off) mode

The same type of Hilbert space as in quantum computing where the 0 and 1 means the state of different

spins: the qubit

Introducing the noise-bit (Kish, Khari, Sethuraman, Physics Letters A 373 (2009) 1928-1934;

http://arxiv.org/abs/0901.3947)

A string search algorithm outperforming Grovers quantum search is proposed with the same hardware

complexity as the quantum engine when it is used for real data.

2^(2^N-1)-1 different logic values can be represented if we use the hyperspace elements

in a binary (on/off superposition). For N=3, it is 127; for N=5, it is 2.1 billion values.


N=3 noise-bit hyperspaces (similar to quantum)

V1

0V2

0V3

0= 0,0,0

V1

1V2

0V3

0= 1,0,0

V1

0V2

1V3

0= 0,1,0

V1

0V2

0V3

1= 0,0,1

V1

1V2

1V3

0= 1,1,0

V1

0V2

1V3

1= 0,1,1

V1

1V2

0V3

1= 1,0,1

V1

1V2

1V3

1= 1,1,1

no signal = 0,0,0

V1= 1,0,0

V2= 0,1,0

V3= 0,0,1

V1V2= 1,1,0

V1V3= 1,0,1

V2V3= 0,1,1

V1V2V3= 1,1,1

1 2 3 1 2 32N noises and

2N dimensions

N noises and

at least 2N-1 dimensions

At least 255 logic values At least 127 logic valuesKish, Khatri, Sethuraman, Physics Letters A 373 (2009) 1928-1934


Conclusion of continuum noise-based logic Kish, Physics Letters A 373 (2009) 911-918

Kish, Khatri, Sethuraman, Physics Letters A 373 (2009) 1928-1934

Advantages:

i. Arbitrary number of logic values in a single wire. Utilization of quantum-like hyperspace is possible.

ii. Due to the zero mean of the stochastic processes, the logic values are AC signals and AC coupling can

make it sure that the variability-related vulnerabilities are strongly reduced.

iii. Robust against noises and interference. The different basic logic values are orthogonal not only to each

other but also to any transients/spikes or any background noise including thermal noise or circuit noise,

such as 1/f, shot, gr, etc, processes. Moreover, the usual binary switching errors do not propagate and

accumulate.

iv. Due to the orthogonality and AC aspects (points ii and iii), the logic signal on the data bus can have much

less effective value than the power supply voltage of the chip. This property and the robustness against

switching errors have a potential to reduce the energy consumption.

Disadvantages:

a. The (continuum noise based) noise-based logic is slower due to the need of averaging.

b. May need more complex hardware but that is fine with the multivalued logic abilities.


Conclusion for continuum-noise-based logic

Relevance for nanoelectronics:

1. Nanoelectronics: smaller device size, thus higher (small-signal) bandwidth. Relevant.

2. Nanoelectronics: smaller device size, thus more transistors on the chip. Relevant.

3. Nanoelectronics: deafening noise. Relevant.

Comparison with quantum computing:

4. Binary or multivalued, with optional superposition of states, like quantum.

5. Entanglement can be made in the superposition, like quantum.

6. However, collapse of the wavefuntion does not exist.

7. All the superposition components are accessible at all times: Good for general purpose.

8. A string search algorithm outperforming Grovers quantum search is proposed with the same

hardware complexity as the quantum engine when it is used for real data.

( Kish, Khatri, Sethuraman, Physics Letters A 373 (2009) 1928-1934 )


How does biology do it??? Comparison:

This Laptop Human Brain

Processor dissipation: 40 W Brain dissipation: 12-20 W

Deterministic digital signal Stochastic signal: analog/digital?

Very high bandwidth (GHz range) Low bandwidth (<100 Hz)

Sensitive for errors (freezing) Error robust

Deterministic binary logic Unknown logic

Potential-well based memory Unknown memory mechanism

Addressed memory access Associative memory access (?)


Often a Poisson-like spike sequence.

The relative frequency-error scales as the reciprocal of the

square-root of the number of spikes.=1/ n

Supposing the maximal frequency, 100 Hz, of

spike trains, 1% error needs 100 seconds averaging!

Pianist playing with 10 Hz hit rate would have 30%

error in the rhythm at the point of brain control.

Parallel channels needed, at least 100 of them.(Note: controlling the actual muscles is also a problem of

negative feedback but we need an accurate reference signal).

Let's do the naive math: similar number of neurons and transistors, but 10 million times slower

clock; plus 100 times slowing down due to averaging needed by the stochastics.

The brain should perform about 1 billion times lower than a computer!


Noise-based logic suggests a possible step toward the answer:

• Deterministic

• Multivalued

• But how to do it with neurons???


Introducing the neuro-bit to build the logic hyperspace for the brain (Bezrukov, Kish, Physics Letters A 373 (2009) 2338-2342 , http://arxiv.org/abs/0902.2033 )

• Neural signals are stochastic unipolar, overlapping spikes trains: their product has non-zero

mean value.

• Multiplying with neurons seems to be difficult, anyway.

• Can we build an orthogonal noise-bit type multidimensional Hilbert space here?


Introducing the neuro-bit (Bezrukov, Kish, Physics Letters A 373 (2009) 2338-2342 )

AB, AB and AB are orthogonal, they do not have common part! The partially

overlapping spike trains can be use as neuro-bits in the same was as it was with the

noise bits.N neuro bits will make 2N-1 orthogonal elements, that is a 2N-1 dimensional hyperspace.

• The answer is yes. Using set-theoretical operations. The A and B sets below represent two partially

overlapping neural spike trains.

• AB is the overlapping part.

• AB is the spike train A minus the overlapping spikes.

• AB is the spike train B minus the overlapping spikes.

AB

AB

AB

AB

The very same multidimensonal hyperspace as it was obtained with the noise-bits. ( Kish, Khatri, Sethuraman, Physics Letters A 373 (2009) 1928-1934)


A

B

Bezrukov, Kish, Physics Letters A 373 (2009) 2338-2342


A

B

AB



A

B

AB

AB

AB



A

B

C

AB C A BC

A B C

ABC

AB C A BC ABC

With 3 neuro-bits (N=3). It makes 2N-1 = 7 hyperspace vectors. Using these vectors in a

binary (on/off) superposition, we can represent 127 different logic values in a synthesized

neural spike train in a single line.

1,0,0 = AB C ,

0,1,0 = A BC

0,0,1 = A B C

1,1,0 = ABC

1,0,1 = AB C

0,1,1 = A BC

1,1,1 = ABCThe very same hyperspace as it was obtained with the noise-bits. ( Kish, Khatri, Sethuraman, Physics Letters A 373 (2009) 1928-1934)

(Bezrukov, Kish, Physics Letters A 373 (2009) 2338-2342 )


A key question: can we make these set theoretical operations with neurons?

Yes: the key role of inhibitory input of neurons becomes clear then.

N-th order orthogonator: N inputs for partially overlapping spike trains and 2N-1 output

ports with orthogonal spike trains.

_

+

_

+

+

_

A(t)

B(t)

A(t)B(t)

A(t)B (t)

A (t)B(t)

The second-order orthogonator gate circuitry utilizing both excitatory (+) and inhibitory (-) synapses of

neurons. The input points at the left are symbolized by circles and the output points at the right by free-

ending arrows. The arrows in the lines show the direction of signal propagation.

A

B

AB

AB-1

A-1B

AB

AB

AB

A

B



_

+

_

+

A(t)

B(t)

A(t)B(t)

A(t)B (t)

+_

A

BAB = 1,0

AB = 1,1

The orthon building element and its symbol.



The second order orthogonator circuit with orthon

+_

A

BAB = 1,0

AB = 1,1

+

_

A B = 0,1

_

+

_

+

+

_

A(t)

B(t)

A(t)B(t)

A(t)B (t)

A (t)B(t)



1,1,1

1,1,0

0,0,1

+_

B

A BC

A B C

A

B+_

C

AB C

AB C

+_

C

ABC

ABC

+_

AB

AB

+

_

+

_

+

_

A BC

C

C

1,0,1

1,0,0

0,1,0

0,1,1

A B

A C

The third order orthogonator circuit built with 4 orthons and 3 neurons



+_

S = ai

i=1

2N

1

c1,i ,c2,i , ... ,cN ,i

Rk= c1,k ,c2,k , ... ,c

N ,kS

k = ai

i=1

i k

2N

1

c1,i ,c2,i , ... ,cN ,i

ak Rk = ak c1,k ,c2,k , ... ,cN ,k

The projection operation circuitry for addressing and associative memory tasks (similarly to string search in

noise-based logic) needs a s ingle orthogon. The upper output extracts the k-th basis vector from the

superposition, indeed if it exists in the superposition. No averaging is necessary, it is a coincidence type of

detection; the first output spike there proves that the basis vector exists in the superposition. The lower

output will not contain the k-th vector but only the rest of the input superposition.

• Associative memory essential: search for a certain info content.

• It needs only a single orthon.

• No time-average is necessary. Due to the orthogonality, the first coinciding spike with a

spike in a hyperspace vector proves the existence of this vector in the superposition!

Here comes the piano player and the hunter !



A

B

AB

AB

AB

Coincidence detector utilizing the reference (basis vector) signals.

Very fast. No statistics/correlations are needed.


S = ai

i=1

2N

1

Ri

R1

S1 = a

i

i=2

2N

1

Ri

+_

+_

a1R

1

R2

a2R

2

S1 = a

i

i=2

2N

1

Ri

To 2N-4 further orthogons

"Neural Fourier transformation" analogous to quantum Fourier transformation. 2N - 2 orthogons used as

projection operators to get the full spectrum of orthogonal base vectors in the input superposition. The last

orthogon's lower output will provide the detection of the last superposition vector. Note, the number of

necessary orthogons can be radically decreased if only specific superpositions are needed to be detected.

Neural Fourier transformation: full analysis of a superposition. (Similar to quantum

Fourier transformation). Again: no averaging is needed, the first coincidence is enough.

Extremely fast. The hunter recognizes the situation in a fraction of a moment.

(Bezrukov, Kish, http://arxiv.org/abs/0902.2033 )


Essential open question: can these circuit elements be found in the brain???

... and a lot of open questions about details, operations, errors, etc, etc...


Noise-based logic: from Boolean logic gates to brain circuitry

Laszlo Kish

Department of Electrical and Computer Engineering, Texas A&M University, College Station

"We can't solve problems by using the same kind of thinking we used when we created them." (Albert Einstein)

When noise dominates an information system, like in nano-electronic systems of the foreseeable future, a natural

question occurs: can we perhaps utilize the noise as information carrier? Another question is: Can a deterministic

logic scheme be constructed that may be the explanation how the brain can efficiently process information, with

random neural spike trains of less than 100 Hz frequency, and with similar number of neurons than the number of

transistors in a 16 GB Flash dive? The answers to these questions are yes. Related developments indicate reduced

power consumption with noise-based deterministic Boolean logic gates and the more powerful multivalued logic

versions with arbitrary number of logic values. Similar scheme (logic hyperspace) as the Hilbert space of quantum

informatics can also be constructed with noise-based logic without some of the limitations of quantum computers.

Noise-based string search algorithm with higher speed than Grover's quantum search algorithm is obtained with the

same hardware complexity class as the quantum engine. This hyperspace scheme has also been utilized to construct a

deterministic multivalued logic scheme for the brain and the relevant circuitry of neurons.

See more at: http://www.ece.tamu.edu/%7Enoise/research_files/noise_based_logic.htm

Seminar at University of Texas, Grad. School of Biomedical Sciences, Houston, March 24, 2009.

Seminar at Rice University, Dept. Mech. Eng. and Materials Sci., Houston, March 25, 2009.

Seminar at RIKEN Brain Institute, Tokyo, Japan, June 22, 2009.

Seminar at Kanagawa Industrial Technology Center, Japan, June 26, 2009.

Seminar at Arizona State University, Dept of Electrical Engineering, Tempe, July 20, 2009.

noise-based logic: from boolean logic gates to brain...

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