Proof Of Universality Research Articles

Universality in the 2d Quasi-periodic Ising Model and Harris–Luck Irrelevance

We prove that in the 2D Ising model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case; that is, the critical exponents for the specific heat and energy-energy correlations are identical, and no logarithmic corrections are present. The disorder produces a quasi-periodic modulation of the amplitude of the correlations and a renormalization of the velocities, that is, the coefficients of the rescaling of positions, and of the critical temperature. The result establishes the validity of the prediction based on the Harris–Luck criterion, and it provides the first rigorous proof of universality in the Ising model in the presence of quasi-periodic disorder in both directions and for any angle. Small divisors are controlled assuming a Diophantine condition on the frequencies, and the convergence of the series is proved by Renormalization Group analysis.

Research on aquatic biological signal recognition based on mode decomposition and deep learning

Abstract As an important research content in the field of marine biology and ecology, aquatic biological signal (ABS) recognition is of great significance for understanding marine ecosystems, protecting biodiversity, and monitoring environmental changes. To improve the recognition accuracy of ABS, this paper proposes a new ABS recognition method based on mode decomposition and deep learning. First, real ABS data were obtained from the public website, and some data were selected for the simulation experiment. Secondly, the signal is decomposed using improved variational mode decomposition by human evolutionary optimization algorithm (HEOA-VMD), and the decomposed intrinsic mode function (IMF) set is classified into low complexity and high complexity IMF using improved Lempel-Ziv complexity (ILZC) and reverse permutation entropy (RPE). Then, mutual information (MI) is used to select double eigenvectors from low and high complexity IMF, respectively, and recognition is performed based on the double eigenvectors using weighted-convolutional neural network (CNN)-bidirectional gate recurrent unit (BIGRU)-Attention model. Finally, the proposed ABS recognition method is applied to both chaotic and real signals, and additional proof of universality is performed using real signals. The result of the study shows that the accuracy of the proposed method for the recognition of chaotic and real signals is as high as 97.3% and 98.0%. In conclusion, the research on ABS recognition in this paper is successful and has a broad application prospect.

Universal proof theory: Semi-analytic rules and Craig interpolation

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including UL−, MTL, R, Łn (for n⩾3), Gn (for n⩾4), and almost all extensions of IMTL, Ł, BL, RMe, IPC, S4, and Grz (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.

A Machine Proof System of Point Geometry Based on Coq

An important development in geometric algebra in recent years is the new system known as point geometry, which treats points as direct objects of operations and considerably simplifies the process of geometric reasoning. In this paper, we provide a complete formal description of the point geometry theory architecture and give a rigorous and reliable formal verification of the point geometry theory based on the theorem prover Coq. Simultaneously, a series of tactics are also designed to assist in the proof of geometric propositions. Based on the theoretical architecture and proof tactics, a universal and scalable interactive point geometry machine proof system, PointGeo, is built. In this system, any arbitrary point-geometry-solvable geometric statement may be proven, along with readable information about the solution’s procedure. Additionally, users may augment the rule base by adding trustworthy rules as needed for certain issues. The implementation of the system expands the library of Coq resources on geometric algebra, which will become a significant research foundation for the fields of geometric algebra, computer science, mathematics education, and other related fields.

The Investigation of the Discrete Universality of L-Functions of Elliptic Curves

In the paper, we prove the discrete universality theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the L-functions of elliptic curves. We consider an approximation of analytic functions by translations LE (s+imh) , where h > 0 is a fixed number, the translations of the imaginary part of the complex variable take values from some discrete set such as arithmetical progression. We suppose that the number h > 0 is chosen so that exp{2πk/h } is an rational number for some non-zero integer. The proof of discrete universality of the derivatives of L-functions of elliptic curves is based on a limit theorem in the sense of weak convergence of probability measures in the space of analytic functions.

Implementation and Optimization of Zero-Knowledge Proof Circuit Based on Hash Function SM3

With the increasing demand for privacy protection in the blockchain, the universal zero-knowledge proof protocol has been developed and widely used. Because hash function is an important cryptographic primitive in a blockchain, the zero-knowledge proof of hash preimage has a wide range of application scenarios. However, it is hard to implement it due to the transformation of efficiency and execution complexity. Currently, there are only zero-knowledge proof circuits of some widely used hash functions that have been implemented, such as SHA256. SM3 is a Chinese hash function standard published by the Chinese Commercial Cryptography Administration Office for the use of electronic authentication service systems, and hence might be used in several cryptographic applications in China. As the national cryptographic hash function standard, the zero-knowledge proof circuit of SM3 (Chinese Commercial Cryptography) has not been implemented. Therefore, this paper analyzed the SM3 algorithm process, designed a new layered circuit structure, and implemented the SM3 hash preimage zero-knowledge proof circuit with a circuit size reduced by half compared to the automatic generator. Moreover, we proposed several extended practical protocols based on the SM3 zero-knowledge proof circuit, which is widely used in blockchain.

General Conditions for Universality of Quantum Hamiltonians

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.

Proof of universality in one-dimensional few-body systems including anisotropic interactions

We provide an analytical proof of universality for bound states in one-dimensional systems of two and three particles, valid for short-range interactions with negative or vanishing integral over space. The proof is performed in the limit of weak pair-interactions and covers both binding energies and wave functions. Moreover, in this limit the results are formally shown to converge to the respective ones found in the case of the zero-range contact interaction.

Black hole zeroth law in the Horndeski gravity

The four laws of black hole mechanics have been put forward for a long time. However, the zeroth law, which states that the surface gravity of a stationary black hole is a constant on the event horizon, still lacks universal proof in various modified gravitational theories. In this paper, we study the zeroth law {in a special Horndeski gravity}, which is an interesting gravitational theory with a nonminimally coupled scalar field. After assuming that {the nonminimally coupled scalar field has the same symmetries with the spacetime,} the minimally coupled matter fields satisfy the dominant energy condition and the Horndeski gravity has a smooth limit to Einstein gravity when the coupling constant approaches zero, we prove the zeroth law based on the gravitational equation in Horndeski gravity without any assumption to the spacetime symmetries.

Extreme gaps between eigenvalues of Wigner matrices

This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the entries. The proof relies on the Erdős–Schlein–Yau dynamic approach. We exhibit a new observable that satisfies a stochastic advection equation and reduces local relaxation of the Dyson Brownian motion to a maximum principle. This observable also provides a simple and unified proof of gap universality in the bulk and the edge, which is quantitative. To illustrate this, we give the first explicit rate of convergence to the Tracy–Widom distribution for generalized Wigner matrices.

A study of universal zero-knowledge proof circuit-based virtual machines that validate general operations & reduce transaction validation

Recently, blockchain technology accumulates and stores all transactions. Therefore, in order to verify the contents of all transactions, the data itself is compressed, but the scalability is limited. In addition, since a separate verification algorithm is used for each type of transaction, the verification burden increases as the size of the transaction increases. Existing blockchain cannot participate in the network because it does not become a block sink by using a server with a low specification. Due to this problem, as the time passes, the data size of the blockchain network becomes larger and it becomes impossible to participate in the network except for users with abundant resources. Therefore, in this paper, we studied the zero knowledge proof algorithm for general operation verification. In this system, the design of zero-knowledge circuit generator capable of general operation verification and optimization of verifier and prover were also conducted. Also, we developed an algorithm for optimizing key generation. Based on all of these, the zero-knowledge proof algorithm was applied to and tested on the virtual machine so that it can be used universally on all blockchains.

Elipsinių kreivių L funkcijų išvestinės diskretusis universalumas

Straipsnyje įrodyta elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo teorema silpnojo tikimybinių matų konvergavimo prasme analizinių funkcijų erdvėje. Nagrinėjamas analizinės funkcijos aproksimavimas postūmiais L‘E (s + imh), čia kompleksinio kintamojo menamosios dalies postūmiai įgyja reikšmes iš diskrečiosios aibės, pavyzdžiui, aritmetinės progresijos. Fiksuotas skaičius h > 0 pasirenkamas taip, kad exp{2πk/h} būtų iracionalusis skaičius visiems k ∈ Z \{0} . Elipsinių kreivių L funkcijų išvestinės diskrečiojo universalumo įrodymas remiasi diskrečiąja ribine teorema tikimybinių matų silpnojo konvergavimo prasme analizinių funkcijų erdvėje.

Fundamental formulation of light-matter interactions revisited

The basic physics disciplines of Maxwell's electrodynamics and Newton's mechanics have been thoroughly tested in the laboratory, but they can nevertheless also support nonphysical solutions. The unphysical nature of some dynamical predictions is demonstrated by the violation of symmetry principles. Symmetries are fundamental in physics since they establish conservation principles. The procedures explored here involve gauge transformations that alter basic symmetries, and these alterations are possible because gauge transformations are not necessarily unitary despite the widespread assumption that they are. That gauge transformations can change the fundamental physical meaning of a problem despite the preservation of electric and magnetic fields is a universal proof that potentials are more basic than fields. These conclusions go to the heart of physics. Problems are not evident when fields are perturbatively weak, but the properties demonstrated here can be critical in strong-field physics where the electromagnetic potential becomes the dominant influence in interactions with matter.

First-Order Automated Reasoning with Theories: When Deduction Modulo Theory Meets Practice

We discuss the practical results obtained by the first generation of automated theorem provers based on Deduction modulo theory. In particular, we demonstrate the concrete improvements such a framework can bring to first-order theorem provers with the introduction of a rewrite feature. Deduction modulo theory is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We introduce two automated reasoning systems that have been built to extend other provers with Deduction modulo theory. The first one is Zenon Modulo, a tableau-based tool able to deal with polymorphic first-order logic with equality, while the second one is iProverModulo, a resolution-based system dealing with first-order logic with equality. We also provide some experimental results run on benchmarks that show the beneficial impact of the extension on these two tools and their underlying proof search methods. Finally, we describe the two backends of these systems to the Dedukti universal proof checker, which also relies on Deduction modulo theory, and which allows us to verify the proofs produced by these tools.

Spiking neural P systems with structural plasticity and anti-spikes

Spiking neural P systems (in short, SNP systems) are a class of distributed parallel computing devices, abstracted from the way neurons communicate by means of spikes. This paper discusses spiking neural P systems with structural plasticity and anti-spikes (in short, SNP-SPA systems), a new variant of SNP systems with two interesting features: structural plasticity and anti-spike. By means of plasticity rules in neurons, SNP-SPA systems can provide a dynamic directed graph structure. Turing universality of SNP-SPA systems is discussed. It is proven that SNP-SPA systems as number generating/accepting devices are Turing universal, and a small example with 56 neurons that computes a universal function is constructed. The introduction of anti-spikes allows to reduce the modules in the proof of universality.

Universal MBQC with generalised parity-phase interactions and Pauli measurements

We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form exp⁡(−iπ2nZ⊗Z). When n=2, these are equivalent, up to local Clifford unitaries, to graph states. However, when n>2, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli Z and X measurements and feed-forward. Even though, for n>2, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every n>2 this family is capable of producing all Clifford gates and all diagonal gates in the n-th level of the Clifford hierarchy.

Measurement-Device-Independent Twin-Field Quantum Key Distribution

The ultimate aim of quantum key distribution (QKD) is improving the transmission distance and key generation speed. Unfortunately, it is believed to be limited by the secret-key capacity of quantum channel without quantum repeater. Recently, a novel twin-field QKD (TF-QKD) is proposed to break through the limit, where the key rate is proportional to the square-root of channel transmittance. Here, by using the vacuum and one-photon state as a qubit, we show that the TF-QKD can be regarded as a measurement-device-independent QKD (MDI-QKD) with single-photon Bell state measurement. Therefore, the MDI property of TF-QKD can be understood clearly. Importantly, the universal security proof theories can be directly used for TF-QKD, such as BB84 encoding, six-state encoding and reference-frame-independent scheme. Furthermore, we propose a feasible experimental scheme for the proof-of-principle experimental demonstration.

Random Band Matrices in the Delocalized Phase, II: Generalized Resolvent Estimates

This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\ll W\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\left(z 1_{1\leq i \leq W}+\widetilde z 1_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:{\rm Im} z>0\}$ and $\widetilde z\in \mathbb C_+\cup \mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in \cite{PartI}. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in \cite{PartIII}.

Universality at Weak and Strong Non-Hermiticity Beyond the Elliptic Ginibre Ensemble

We consider non-Gaussian extensions of the elliptic Ginibre ensemble of complex non-Hermitian random matrices by fixing the trace $\operatorname{Tr}(XX^*)$ of the matrix $X$ with a hard or soft constraint. These ensembles have correlated matrix entries and non-determinantal joint densities of the complex eigenvalues. We study global and local bulk statistics in these ensembles, in particular in the limit of weak non-Hermiticity introduced by Fyodorov, Khoruzhenko and Sommers. Here, the support of the limiting measure collapses to the real line. This limit was motivated by physics applications and interpolates between the celebrated sine and Ginibre kernel. Our results constitute a first proof of universality of the interpolating kernel. Furthermore, in the limit of strong non-Hermiticity, where the support of the limiting measure remains an ellipse, we obtain local Ginibre statistics in the bulk of the spectrum.

Two types of universal proof systems for all variants of many-valued logics and some properties of them

Two types of propositional proof systems are described in this paper such that proof system for each variant of propositional many-valued logic can be presented in both of the described forms. The first of introduced systems is a Gentzen-like system, the second one is based on the generalization of the notion of determinative disjunctive normal form, formerly defined by first coauthor. Some generalization of Kalmar’s proof of deducibility for two-valued tautologies in the classical propositional logic gives us a possibility to suggest the easy method of proving the completeness for first type of described systems. The completeness of the second type one is received from its construction automatically. The introduced proof systems are “weak” ones with a “simple strategist” of proof search and we have investigated the quantitative properties, related to proof complexity characteristics in them as well. In particular, for some class of many-valued tautologies simultaneously optimal bounds (asymptotically the same upper and lower bounds for each proof complexity characteristic) are obtained in the systems, considered for some versions of many-valued logic.