New XOR & XNOR Operations in Instantaneous Noise-Based Logic

We report the implementation of the XOR and XNOR logical operations using an electro-optic circuit, which is fabricated by CMOS-compatible process in the silicon-on-insulator (SOI) platform. The circuit consists of two cascaded add-drop microring resonators (MRRs), which are modulated through electric-field-induced carrier depletion in reverse biased pn junctions embedded in the ring waveguides. The resonance wavelength mismatch between the two nominally identical MRRs caused by fabrication errors is compensated by thermal tuning. Simultaneous bitwise XOR and XNOR operations of the two electrical modulating signals at the speed of 12.5 Gb/s are demonstrated. And 20 Gb/s XOR operation at one output port of the circuit is achieved. We explain the phenomena that one half of the resonance regions of the device are much more sensitive to the round-trip phase shift in the ring waveguides than the other half resonance regions. Characteristic graphs with logarithmic phase coordinate are proposed to analyze the sensitivity of the demonstrated circuit, as well as several typical integrated optical structures. It is found that our circuit with arbitrary chosen parameters has similar sensitivity to MRRs under the critical coupling.

We report the simultaneous implementation of the XOR and XNOR operations at two ports of a directed logic circuit based on two cascaded microring resonators (MRRs), which are both modulated through thermo-optic effect. Two electrical modulating signals applied to the MRRs represent the two operands of each logic operation. Simultaneous bitwise XOR and XNOR operations at 10 kbit/s are demonstrated in two different operating modes. We show that such a circuit can be readily realized using the plasma dispersion effect or the electric field effects, indicating its potential for high-speed operation. We further employ the scattering matrix method to analyze the spectral characteristics of the fabricated circuit, which can be regarded as a Mach-Zehnder interferometer (MZI) in whole. The two MRRs in the circuit act as wavelength-dependent splitting and combining units of the MZI. The degradation of the spectra observed in the experiment is found to be related to the length difference between the MZI's two arms. The evolution of the spectra with this length difference is presented.

In this article, we study a generalized system of mixed ordered variational inequalities problems with various operations in a real ordered product Banach space and discuss the existence of the solution of our considered problem. Further, we discuss the convergence analysis of the proposed iterative algorithm using XNOR and XOR operations techniques. Most of the variational inequalities solved by the projection operator technique but we solved our considered problem without the projection technique. The results of this paper are more general and new than others in this direction. Finally, we give a numerical example to illustrate and show the convergence of the proposed algorithm in support of our main result has been formulated by using MATLAB programming. 2010 AMS Subject Classification: 47H09; 49J40.

We propose and demonstrate a directed optical logic circuit, which can perform the XOR and XNOR logical operations based on U-to-U waveguides and two microring resonators. Compared to the previous structure, no waveguide crossings exist in the proposed circuit, which is beneficial to reduce the insertion loss and crosstalk of the device. Two electrical signals representing the two operands of the logical operation are employed to modulate two microring resonators through the thermo-optic effect, respectively. The operation result is represented by the output optical signal. For proof of principle, XOR and XNOR operations at 10 kb/s are demonstrated.

Metasurface‐based logic devices have pioneered an innovative paradigm for advancing computing technology owing to customizable, flexible, and controllable characteristics. However, further improvements in performance typically come at the cost of escalated system complexity, constraining computational response speeds and parallel processing capability. Here, synergizing the picosecond (ps) response of silicon with ultrafast broadband terahertz (THz) modulation, a 3‐bit all‐logic parallel operation strategy is proposed based on light‐driven metadevice. Leveraging 3D input channels including optical pump fluence, polarization‐decoupled response, and ultrafast relaxation dynamics, the 3‐bit NAND, XNOR, OR, XOR, NOR, and AND logic operations can operate in parallel within 8 ps at a fractional bandwidth of 179%. Besides, the light‐driven logic metadevice operates without complex external stimulation while delivering outstanding behaviors with single‐silicon integration. It is believed the high‐efficiency and multi‐path logic parallel computing ability of the proposed metadevice holds significant potential for large‐scale data processing, photonic quantum computers, and optoelectronic encryption devices.

The Schnorr-Stimm dichotomy theorem (Schnorr and Stimm, 1972) concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. The theorem asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S. (2) If S is normal, then any finite-state gambler loses money at an exponential rate betting on S. In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||α) of a probability measure α on Σ from a sequence S over Σ and the upper asymptotic divergence Div(S||α) of α from S in such a way that a sequence S is α-normal (meaning that every string w has asymptotic frequency α(w) in S) if and only if Div(S||α)=0. We also use the Kullback-Leibler divergence to quantify the total risk Risk <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (w) that a finite-state gambler G takes when betting along a prefix w of S. Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to α-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S. ( $1~'$ ) The infinitely-often exponential rate of winning in 1 is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Div(S||α)|w|</sup> . ( $2~'$ ) The exponential rate of loss in 2 is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">- Risk_G(w)</sup> . We also use (1 $'$ ) to show that 1- Div(S||α)/c, where c = log(1/ min <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a ∈ Σ</sub> α(a)), is an upper bound on the finite-state α-dimension of S and prove the dual fact that 1- div(S||α)/c is an upper bound on the finite-state strong α-dimension of S.

Hamming distance for conjugates

In information theory, the Hamming distance between two strings of equal length is the number of positions at which the corresponding symbols are different. It measures the minimum number of substitutions required to change one string into the other, or the number of errors that transformed one string into the other. The process of detecting errors in a sequence of bits is determined by comparing this sequence to a dictionary of encodings, if found then the pattern is correct, otherwise an error has occurred. In order to correct the errors, the original pattern is substituted with the closest pattern in the dictionary, i.e. the pattern with the smallest hamming distance from the original. A code having a minimum Hamming distance d, can typically detect up to d - 1 and correct up to (d - 1)/2 errors in a code word. In this paper we are going to study the possibility to design an encoding scheme by using random patterns in the dictionary that are created by a secure encryption algorithm, and we are going to test the error correction capabilities of our code. For our study, we used a model in which the code words are represented by whole numbers and an error of size r will change the message number by ± r. So, for example, a single error in the message x=7 will change the value of x to 8 or 6, and if x=7 is received as y=10 then an error of size 3 has occurred. The study will be as follows: if we have K numbers selected randomly from {1, 2, 3 ... N} where K<;N, what is the expected minimum distance between the numbers that we will get? To reach our objective, we designed a tool that will find the smallest distance between some K selected numbers out of N, and we were able to show theoretically and practically that for any N, there exist K<;N such that the minimum distance between those K selected numbers is definitely going to be 1, and therefore we cannot detect or correct any error.

Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries

The Schnorr-Stimm dichotomy theorem (Schnorr and Stimm, 1972) concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet $\Sigma $ . The theorem asserts that, for any such sequence $S$ , the following two things are true. (1) If $S$ is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in $S$ ), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on $S$ . (2) If $S$ is normal, then any finite-state gambler loses money at an exponential rate betting on $S$ . In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence ${\mathrm {div}}(S||\alpha)$ of a probability measure $\alpha $ on $\Sigma $ from a sequence $S$ over $\Sigma $ and the upper asymptotic divergence ${\mathrm {Div}}(S||\alpha)$ of $\alpha $ from $S$ in such a way that a sequence $S$ is $\alpha $ -normal (meaning that every string $w$ has asymptotic frequency $\alpha (w)$ in $S$ ) if and only if ${\mathrm {Div}}(S||\alpha)=0$ . We also use the Kullback-Leibler divergence to quantify the total risk ${\mathrm {Risk}}_{G}(w)$ that a finite-state gambler $G$ takes when betting along a prefix $w$ of $S$ . Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to $\alpha $ -normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes $w$ of $S$ . ( $1~'$ ) The infinitely-often exponential rate of winning in 1 is $2^{{\mathrm {Div}}(S||\alpha)|w|}$ . ( $2~'$ ) The exponential rate of loss in 2 is $2^{- {\mathrm {Risk}}_{G}(w)}$ . We also use (1 $'$ ) to show that $1- {\mathrm {Div}}(S||\alpha)/c$ , where $c= \log (1/ \min _{a\in \Sigma }\alpha (a))$ , is an upper bound on the finite-state $\alpha $ -dimension of $S$ and prove the dual fact that $1- {\mathrm {div}}(S||\alpha)/c$ is an upper bound on the finite-state strong $\alpha $ -dimension of $S$ .

Two strings of equal length are called order-isomorphic if their relative orders are identical at every position. The classical order-preserving pattern matching (OPPM) problem finds all substrings in a text T that are order-isomorphic to a pattern P. However, measurement errors can cause data loss or inaccuracies, making exact pattern detection difficult and motivating the active study of approximate OPPM variants, such as 2-partition OPPM. In this paper, we extend the existing partition-based relaxation of order-isomorphism and define the 3-partition OPPM problem. The 3-partition OPPM problem is to find all substrings in a text T that can be divided into three segments such that each partitioned segment is order-isomorphic to the corresponding segment of a pattern P. We propose an efficient algorithm to solve the problem in O(nm+m2logm) time, where n=|T| and m=|P|. We conduct experiments on various time series datasets, comparing the number of occurrences and the runtime efficiency among the OPPM, 2-partition OPPM, and proposed 3-partition OPPM algorithms. Our experimental evaluation shows that the proposed algorithm becomes increasingly cost-effective for longer patterns.

This paper investigates the approximability of the Longest Common Subsequence (LCS) problem. The fastest algorithm for solving the LCS problem exactly runs in essentially quadratic time in the length of the input, and it is known that under the Strong Exponential Time Hypothesis the quadratic running time cannot be beaten. There are no such limitations for the approximate computation of the LCS however, except in some limited scenarios. There is also a scarcity of approximation algorithms. When the two given strings are over an alphabet of size k, returning the subsequence formed by the most frequent symbol occurring in both strings achieves a 1/k approximation for the LCS. It is an open problem whether a better than 1/k approximation can be achieved in truly subquadratic time (O(n^{2-δ}) time for constant δ > 0). A recent result [Rubinstein and Song SODA'2020] showed that a 1/2+e approximation for the LCS over a binary alphabet is possible in truly subquadratic time, provided the input strings have the same length. In this paper we show that if a 1/2+e approximation (for e > 0) is achievable for binary LCS in truly subquadratic time when the input strings can be unequal, then for every constant k, there is a truly subquadratic time algorithm that achieves a 1/k+δ approximation for k-ary alphabet LCS for some δ > 0. Thus the binary case is the hardest. We also show that for every constant k, if one is given two strings of equal length over a k-ary alphabet, one can obtain a 1/k+e approximation for some constant e > 0 in truly subquadratic time, thus extending the Rubinstein and Song result to all alphabets of constant size.

We have designed and fabricated a directed logic architecture consisting of two silicon microring resonators that can perform XOR and XNOR operations. The microring resonators are modulated through thermo-optic effect. Two electrical modulating signals applied to the microring resonators represent the two operands of the logical operation. The logical function is evaluated through the directed propagation of light in the device, and the result is represented by the output optical signal. Both XOR and XNOR operations at 20 kbits are demonstrated.

A directed logic circuit consisting of two silicon microring resonators which can perform XOR and XNOR operations has been proposed. These two operations are evaluated as the optical signal is directed along the circuit, and the results appear at some specific output ports of the circuit. A potential advantage of this new logical paradigm is that it has markedly less state delay than traditional logic circuits. Devices based on the above proposal are fabricated on an 8-inch silicon-on-insulator wafer. The microring resonators in the circuit are modulated through thermo-optic effect. Two electrical modulating signals applied to the microring resonators represent the two operands of the logical operation respectively. Bitwise XOR and XNOR operations at 20 kbit/s are demonstrated. The scattering matrix method is adopted to analyse the transmission spectra of the circuit. The symmetrical characteristics of the spectra are well described by this model. The potential applications of this device are given.

A framework for designing space-efficient dictionaries for parameterized and order-preserving matching