Representation of Multiple-Valued Logic Functions

Stanković, Radomir S.; Astola, Jaakko; Moraga, Claudio

doi:10.2200/s00420ed1v01y201205dcs037

Cited by 19 publications

(21 citation statements)

References 142 publications

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“…In general, for logic functions of Boolean variables, x 1 ,…, x n , computing all the outputs for all possible inputs, that is to build its decision tree, is equivalent to writing the function in terms of its MIN terms . The MIN term expansion of a Boolean function is a Sum Of Products (SOP) form, that is, a sum over the 2 n possible sets of n inputs

{(i_{1}, . . ., i_{n})}_{j}

[Eq.…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

“…The SOP form for a Boolean function, Equation (1), can also be viewed as a spectral decomposition of a logic function . It can be generalized to a logic function of n variables of radix r in terms of sum of (Hadamard) products, see ref.…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

“…It can be generalized to a logic function of n variables of radix r in terms of sum of (Hadamard) products, see ref. section 2.3. SOP decompositions of discrete functions of multivalued discrete variables are the analog to integral transforms of functions of continuous variables.…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

“…To decompose one of these nine functions as an SOP, we need to define the characteristic functions J i ( x 1 ) and J i ( x 2 ), which are defined as [Eq. ]:

true J_{i} ((), x_{j}) = ({, \begin{matrix} 1 if x_{j} = i \\ 0 otherwise \end{matrix})

…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

“…A generalization of decision trees is that they implement the spectral decomposition of a logic function, in the sense of the MIN term expansions of multivariable Boolean functions . For multivalued variables, the spectral decomposition is called a Sum of Product (SOP) decomposition . Multivalued variables allow encoding and processing more information in one cycle of the device.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Implementation of Multivariable Logic Functions in Parallel by Electrically Addressing a Molecule of Three Dopants in Silicon

et al. 2017

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To realize low‐power, compact logic circuits, one can explore parallel operation on single nanoscale devices. An added incentive is to use multivalued (as distinct from Boolean) logic. Here, we theoretically demonstrate that the computation of all the possible outputs of a multivariate, multivalued logic function can be implemented in parallel by electrical addressing of a molecule made up of three interacting dopant atoms embedded in Si. The electronic states of the dopant molecule are addressed by pulsing a gate voltage. By simulating the time evolution of the non stationary electronic density built by the gate voltage, we show that one can implement a molecular decision tree that provides in parallel all the outputs for all the inputs of the multivariate, multivalued logic function. The outputs are encoded in the populations and in the bond orders of the dopant molecule, which can be measured using an STM tip. We show that the implementation of the molecular logic tree is equivalent to a spectral function decomposition. The function that is evaluated can be field‐programmed by changing the time profile of the pulsed gate voltage.

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{(i_{1}, . . ., i_{n})}_{j}

[Eq.…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

“…To decompose one of these nine functions as an SOP, we need to define the characteristic functions J i ( x 1 ) and J i ( x 2 ), which are defined as [Eq. ]:

true J_{i} ((), x_{j}) = ({, \begin{matrix} 1 if x_{j} = i \\ 0 otherwise \end{matrix})

…”

Section: Decision Trees and Spectral Decomposition Of Logic Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Implementation of Multivariable Logic Functions in Parallel by Electrically Addressing a Molecule of Three Dopants in Silicon

et al. 2017

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The Reed-Muller-Fourier Transform—Computing Methods and Factorizations

Stanković

2016

Claudio Moraga: A Passion for Multi-Valued Logic and Soft Computing

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The power of qutrits for non-adaptive measurement-based quantum computing

et al. 2023

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Non-locality is not only one of the most prominent quantum features but can also serve as a resource for various information-theoretical tasks. Analysing it from an information-theoretical perspective has linked it to applications such as non-adaptive measurement-based quantum computing (NMQC). In this type of quantum computing the goal is to output a multivariate function. The success of such a computation can be related to the violation of a generalised Bell inequality. So far, the investigation of binary NMQC with qubits has shown that quantum correlations can compute all Boolean functions using at most $2^n-1$ qubits, whereas local hidden variables (LHVs) are restricted to linear functions. Here, we extend these results to NMQC with qutrits and prove that quantum correlations enable the computation of all three-valued logic functions using the generalised qutrit Greenberger-Horne-Zeilinger (GHZ) state as a resource and at most $3^n-1$ qutrits. This yields a corresponding generalised GHZ type paradox for any three-valued logic function that LHVs cannot compute. We give an example for an $n$-variate function that can be computed with only $n+1$ qutrits, which leads to convenient generalised qutrit Bell inequalities whose quantum bound is maximal. Finally, we prove that not all functions can be computed efficiently with qutrit NMQC by presenting a counterexample.

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Representation of Multiple-Valued Logic Functions

Cited by 19 publications

References 142 publications

Implementation of Multivariable Logic Functions in Parallel by Electrically Addressing a Molecule of Three Dopants in Silicon

Implementation of Multivariable Logic Functions in Parallel by Electrically Addressing a Molecule of Three Dopants in Silicon

The Reed-Muller-Fourier Transform—Computing Methods and Factorizations

The power of qutrits for non-adaptive measurement-based quantum computing

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