Noise-based logic
Encyclopedia
Noise-based logic is a new class of multivalued deterministic logic schemes where the logic values and bit
s are represented by different realizations of a stochastic process
. The scheme is inspired by the brain
where the neural signals are stochastic spike sequences.
s (orthogonal functions
) and their superposition
, where the orthogonal
basis vectors
are independent noises. By the proper combination (products or set-theoretical products
) of basis-noises, which are called noise-bit, a logic hyperspace can be constructed with number of dimension
s, where N is the number of noise-bits. Thus N noise-bits in a single wire correspond to a system of classical bits that can express different logic values. Independent realizations of a stochastic process of zero mean have zero cross-correlation
with each other and with other stochastic processes of zero mean. Thus the basis noise vectors are orthogonal not only to each other but they and all the noise-based logic states (superpositions) are orthogonal also to any background noises in the hardware. Therefore, the noise-based logic concept is robust against background noises, which is a property that can potentially offer a high energy-efficiency.
and superpositions as logic signals in the basic logic space and their products and superpositions of their products in the logic hyperspace (see also. In the subsequent brain logic scheme, the logic signals were (similarly to neural signals) unipolar spike sequences generated by a Poisson process
, and set-theoretical unifications (superpositions) and intersections (products) of different spike sequences. Later, in the instantaneous noise-based logic schemes and computation works, random telegraph waves (periodic time, bipolar, with fixed absolute value of amplitude) were also utilized as one of the simplest stochastic processes available for NBL. With choosing unit amplitude and symmetric probabilities, the resulting random-telegraph wave has 0.5 probability to be in the +1 or in the -1 state which is held over the whole clock period.
can be classified according to the method the input identifies the logic value at the input. The first gates analyzed the statistical correlations between the input signal and the reference noises. The advantage of these is the robustness against background noise. The disadvantage is the slow speed and higher hardware complexity. The instantaneous logic gates are fast, they have low complexity but they are not robust against background noises. With either neural spike type signals or with bipolar
random-telegraph waves of unity absolute amplitude, and randomness only in the sign of the amplitude offer very simple instantaneous logic gates. Then linear or analog devices unnecessary and the scheme can operate in the digital domain. However, whenever instantaneous logic must be interfaced with classical logic schemes, the interface must use correlator
-based logic gates for an error-free signal.
and the AND
gates to prove universality, because having both of them is a satisfactory condition for the universality of a Boolean logic.
. The scheme is based on random telegraph waves and it is mentioned in the paper that the authors intuitively conclude that the intelligence
of the brain is using similar operations to make a reasonably good decision based on a limited amount of information. It is also interesting to note that the superposition of the first integer numbers can be produced with only 2N operations, which the authors call "Achilles ankle operation" in the paper.
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...
s are represented by different realizations of a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
. The scheme is inspired by the brain
Brain
The brain is the center of the nervous system in all vertebrate and most invertebrate animals—only a few primitive invertebrates such as sponges, jellyfish, sea squirts and starfishes do not have one. It is located in the head, usually close to primary sensory apparatus such as vision, hearing,...
where the neural signals are stochastic spike sequences.
The noise-based logic space and hyperspace
The logic values are represented by multi-dimensional vectorVector
Vector, a Latin word meaning "carrier", may refer in English to:-In computer science:*A one-dimensional array**Vector , a data type in the C++ Standard Template Library...
s (orthogonal functions
Orthogonal functions
In mathematics, two functions f and g are called orthogonal if their inner product \langle f,g\rangle is zero for f ≠ g. Whether or not two particular functions are orthogonal depends on how their inner product has been defined. A typical definition of an inner product for functions is...
) and their superposition
Superposition principle
In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...
, where the orthogonal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...
basis vectors
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
are independent noises. By the proper combination (products or set-theoretical products
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
) of basis-noises, which are called noise-bit, a logic hyperspace can be constructed with number of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
s, where N is the number of noise-bits. Thus N noise-bits in a single wire correspond to a system of classical bits that can express different logic values. Independent realizations of a stochastic process of zero mean have zero cross-correlation
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
with each other and with other stochastic processes of zero mean. Thus the basis noise vectors are orthogonal not only to each other but they and all the noise-based logic states (superpositions) are orthogonal also to any background noises in the hardware. Therefore, the noise-based logic concept is robust against background noises, which is a property that can potentially offer a high energy-efficiency.
The types of signals used in noise-based logic
In the paper, where noise-based logic was first introduced, generic stochastic-processes with zero mean were proposed and a system of orthogonal sinusoidal signals were also proposed as a deterministic-signal version of the logic system. The mathematical analysis about statistical errors and signal energy was limited to the cases of Gaussian noisesGaussian process
In probability theory and statistics, a Gaussian process is a stochastic process whose realisations consist of random values associated with every point in a range of times such that each such random variable has a normal distribution...
and superpositions as logic signals in the basic logic space and their products and superpositions of their products in the logic hyperspace (see also. In the subsequent brain logic scheme, the logic signals were (similarly to neural signals) unipolar spike sequences generated by a Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
, and set-theoretical unifications (superpositions) and intersections (products) of different spike sequences. Later, in the instantaneous noise-based logic schemes and computation works, random telegraph waves (periodic time, bipolar, with fixed absolute value of amplitude) were also utilized as one of the simplest stochastic processes available for NBL. With choosing unit amplitude and symmetric probabilities, the resulting random-telegraph wave has 0.5 probability to be in the +1 or in the -1 state which is held over the whole clock period.
The noise-based logic gates
Noise-based logic gatesLogic gate
A logic gate is an idealized or physical device implementing a Boolean function, that is, it performs a logical operation on one or more logic inputs and produces a single logic output. Depending on the context, the term may refer to an ideal logic gate, one that has for instance zero rise time and...
can be classified according to the method the input identifies the logic value at the input. The first gates analyzed the statistical correlations between the input signal and the reference noises. The advantage of these is the robustness against background noise. The disadvantage is the slow speed and higher hardware complexity. The instantaneous logic gates are fast, they have low complexity but they are not robust against background noises. With either neural spike type signals or with bipolar
Bipolar signal
In telecommunication, a bipolar signal is a signal that may assume either of two polarities, neither of which is zero.A bipolar signal may have a two-state non-return-to-zero or a three-state return-to-zero binary coding scheme....
random-telegraph waves of unity absolute amplitude, and randomness only in the sign of the amplitude offer very simple instantaneous logic gates. Then linear or analog devices unnecessary and the scheme can operate in the digital domain. However, whenever instantaneous logic must be interfaced with classical logic schemes, the interface must use correlator
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
-based logic gates for an error-free signal.
Universality of noise-based logic
All the noise-based logic schemes listed above are proven universal. The papers typically produce the NOTInverter (logic gate)
In digital logic, an inverter or NOT gate is a logic gate which implements logical negation. The truth table is shown on the right.This represents perfect switching behavior, which is the defining assumption in Digital electronics. In practice, actual devices have electrical characteristics that...
and the AND
AND gate
The AND gate is a basic digital logic gate that implements logical conjunction - it behaves according to the truth table to the right. A HIGH output results only if both the inputs to the AND gate are HIGH . If neither or only one input to the AND gate is HIGH, a LOW output results...
gates to prove universality, because having both of them is a satisfactory condition for the universality of a Boolean logic.
Computation by noise-based logic
The string verification work over a slow communication channel shows a powerful computing application where the methods is inherently based on calculating the hash functionHash function
A hash function is any algorithm or subroutine that maps large data sets to smaller data sets, called keys. For example, a single integer can serve as an index to an array...
. The scheme is based on random telegraph waves and it is mentioned in the paper that the authors intuitively conclude that the intelligence
Intelligence
Intelligence has been defined in different ways, including the abilities for abstract thought, understanding, communication, reasoning, learning, planning, emotional intelligence and problem solving....
of the brain is using similar operations to make a reasonably good decision based on a limited amount of information. It is also interesting to note that the superposition of the first integer numbers can be produced with only 2N operations, which the authors call "Achilles ankle operation" in the paper.