Infinite State Research Articles

Confinement in the Transverse Field Ising Model on the Heavy Hex Lattice.

Inspired by a recent quantum computing experiment [Y. Kim etal., Nature (London), 618, 500-5 (2023)NATUAS0028-083610.1038/s41586-023-06096-3], we study the emergence of confinement in the transverse field Ising model on a decorated hexagonal lattice. Using an infinite tensor network state optimized with belief propagation we show how a quench from a broken symmetry state leads to striking nonthermal behavior underpinned by persistent oscillations and saturation of the entanglement entropy. We explain this phenomenon by constructing a minimal model based on the confinement of elementary excitations. Our model is in excellent agreement with our numerical results. For quenches to larger values of the transverse field and/or from nonsymmetry broken states, our numerical results display the expected signatures of thermalization: a linear growth of entanglement entropy in time, propagation of correlations, and the saturation of observables to their thermal averages. These results provide a physical explanation for the unexpected classical simulability of the quantum dynamics.

First order Büchi automata and their application to verification of LTL specifications

Büchi automata have applications in formal verification, e.g., in deciding whether a system satisfies given properties. We provide a definition of Büchi automata based on first order logics for representing infinite state systems, and investigate rules for proving emptiness and non-emptiness of such automata. We then apply these rules to solve the problem of verifying correctness of concurrent transition systems, leading to a relatively complete approach for proving and disproving LTL (Linear Temporal Logic) specifications. This approach overcomes weaknesses of existing work based on well-founded sets in the sense that the relative completeness does not depend on additional specification for ensuring progress of non-stuttering transitions. On the practical aspect, we provide a set of examples with an experimental verification condition generation tool to demonstrate the potential applicability of the approach for the verification of concurrent systems.

The maximum principle for lumped-distributed control systems.

This paper concerns the optimal control of lumped-distributed systems, an examplar of which is a mechanical system with rotating components connected by flexible rods. The underlying mathematical model is a controlled semilinear evolution equation, in which nonlinear terms involve a projection of the full state onto a finite dimensional subspace. We derive necessary conditions of optimality in the form of a maximum principle, for a problem formulation which involves pathwise and end-point constraints on the lumped components of the state variable. Perturbational methods are employed in the derivation, which allow for non-smooth data. A key step in the analysis is the reduction of the optimization problem with an infinite dimensional state space, to one with finite dimensional aspects. The computational implications of this reduction will be explored in future work. Necessary conditions for problems with state constraints and end-point constraints can only be achieved under rather strict compatibility hypotheses on the way the dynamics interact with the state constraint and end-point constraint sets, due the infinite dimensionality of the state space. This paper identifies a class of infinite dimensional problems, of engineering significance, for which necessary conditions of optimality can be derived, in the absence of any additional compatibility hypotheses.

Reinforcement Learning for Jointly Optimal Coding and Control Policies for a Markovian System Controlled over a Communication Channel

This paper develops approximation and optimality results for the optimal control of a networked system. In this system, a Markovian process is managed over a finite-rate, noiseless communication channel. Solving this problem involves determining joint optimal coding and control policies that minimize cost over time. While theoretical results on optimal coding and control structures exist, practical implementation has remained largely infeasible for non-linear systems due to the computational complexities and uncountable state spaces. This research introduces an approach that combines structural results with reinforcement learning (RL) techniques. The method uses regularity properties of the system to approximate the uncountable state space with a countable one, which allows reinforcement learning algorithms to achieve near-optimal solutions. Specifically, we establish that finite model approximations (where infinite state spaces are quantized to finite ones) and sliding finite window approximations (where a finite memory "window" of past control actions is maintained at each time step) can be employed to develop near-optimality. These approximations allow the system to be reformulated as a Markov Decision Problem (MDP) with a finite state space, making RL algorithms implementable. The convergence of the reinforcement learning algorithm to a near-optimal policy (under both the previous approximations) is supported by theoretical analysis and performance simulations. We ensure that the solutions obtained are not only computationally feasible but also nearly optimal with respect to the original problem. The applications of this work extend to many networked control systems, especially those with zero-delay coding and partially observable Markov decision processes (POMDPs). By integrating structural results with learning algorithms, this paper provides a practical framework for implementing optimal control in finite-rate environments.

A novel comprehensive method for weak signal blind detection based on adaptive quasi-periodic potential stochastic resonance and its application in bearing fault detection

In order to solve the common output saturation of stochastic resonance systems and the limitation of classical index SNR for blind detection, a novel adaptive quasi-periodic potential stochastic resonance blind detection method is proposed. First, a model of quasi-periodic potential stochastic resonance (QPPSR) possessing infinite steady state is constructed and analyzed for its structure change pattern. The superior performance of the model is verified by using the fourth-order Runge–Kutta algorithm. Secondly, the mechanism of QPPSR is analyzed using the probability flow method, which reveals the relationship between system parameters and performance. Again, a novel comprehensive blind detection index (CBDI) is exquisitely constructed to make up for the shortcomings of each indicator. Finally, CBDI and QPPSR are constructed into an adaptive blind detection system and applied to bearing fault detection. The results analyzed by experiments verify the good engineering application prospect of CBDI-QPPSR.

Robust Effective Ground State in a Nonintegrable Floquet Quantum Circuit.

An external periodic (Floquet) drive is believed to bring any initial state to the featureless infinite temperature state in generic nonintegrable isolated quantum many-body systems in the thermodynamic limit, irrespective of the driving frequency Ω. However, numerical or analytical evidence either proving or disproving this hypothesis is very limited and the issue has remained unsettled. Here, we study the initial state dependence of Floquet heating in a nonintegrable kicked Ising chain of length up to L=30 with an efficient quantum circuit simulator, showing a possible counterexample: the ground state of the effective Floquet Hamiltonian is exceptionally robust against heating, and could stay at finite energy density even after infinitely many Floquet cycles, if the driving period is shorter than a threshold value. This sharp energy localization transition or crossover does not happen for generic excited states. The exceptional robustness of the ground state is interpreted by (i)its isolation in the energy spectrum and (ii)the fact that those states with L-independent ℏΩ energy above the ground state energy of any generic local Hamiltonian, like the approximate Floquet Hamiltonian, are atypical and viewed as a collection of noninteracting quasiparticles. Our finding paves the way for engineering Floquet protocols with finite driving periods realizing long-lived, or possibly even perpetual, Floquet phases by initial state design.

Likelihood calculation in a multistate model of vocabulary evolution for linguistic dating

Abstract Computational methods of language dating make inferences about the divergence times of protolanguages by evaluating the patterns of inheritance in the vocabulary of modern languages, given the specification of a model of vocabulary evolution. We consider a model that describes vocabulary evolution as the replacement of traits by new traits from an infinite state space along a tree. This model has been introduced in previous literature but so far it has not been used in many applications. We give a general recursive algorithm for calculating likelihoods and argue that the model gives a more realistic representation of vocabulary evolution over time, compared to existing models like the Stochastic Dollo model. We also provide a case study demonstrating the model’s potential applications.

Efficient analysis of belief properties in process algebra

Protocols are typically specified in an operational manner by specifying the communication patterns among the different involved principals. However, many properties are of epistemic nature, e.g., what each principal believes after having seen a run of the protocol. We elaborate on a unified algebraic framework suitable for epistemic reasoning about operational protocols. This reasoning framework is based on a logic of beliefs and allows for the operational specification of untruthful communications. The information recorded in the semantic models to support reasoning about the interaction between the operational and epistemic aspects intensifies the state-space explosion. We propose an efficient on-the-fly reduction for such a unifying framework by providing a set of operational rules. These operational rules automatically generate efficient reduced semantics for a class of epistemic properties, specified in a rich extension of modal μ-calculus with past and belief modality, and can potentially reduce an infinite state space into a finite one. We reformulate and prove criteria that guarantee belief consistency for credulous agents, i.e., agents that are ready to believe what is told unless it is logically inconsistent. We adjust our reduction so that the belief consistency of an original model is preserved. We prove the soundness and completeness result for the specified class of properties.

Fractional quantum Hall states with variational projected entangled-pair states: A study of the bosonic Harper-Hofstadter model

An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum. Published by the American Physical Society 2024

Dynamic minimisation of the commute time for a one-dimensional diffusion

AbstractMotivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to 1 and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion’s drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form $$[0,\ell )\cup (i,1]$$ [ 0 , ℓ ) ∪ ( i , 1 ] . This leads to a type of stochastic control problem with infinite dimensional state.

Wall modes and the transition to bulk convection in rotating Rayleigh-Bénard convection

We investigate states of rapidly rotating Rayleigh-Bénard convection in a cylindrical cell over a range of Rayleigh numbers 3×105≤Ra≤5×109 and Ekman numbers 10−6≤Ek≤10−4 for Prandtl number Pr=0.8 and aspect ratios 1/5≤Γ≤5 using direct numerical simulations. We characterize, for perfectly insulating sidewall boundary conditions, the first transition to convection via wall mode instability and the nonlinear growth and instability of the resulting wall mode states, including a secondary transition to time dependence. We show how the radial structure of the vertical velocity uz and the temperature T is captured well by the linear eigenfunctions of the wall mode instability where the radial width of uz is δuz∼Ek1/3r/H whereas δT∼e−kr (k is the wave number of a laterally infinite wall mode state). The disparity in spatial scales for Ek=10−6 means that the heat transport is dominated by the radial structure of uz since T varies slowly over the radial scale δuz. We further describe how the transition to a state of bulk convection is influenced by the presence of the wall mode states. We use temporal and spatial scales as measures of the local state of convection and the Nusselt number Nu as representative of global transport. Our results elucidate the evolution of the wall state of rotating convection and confirm that wall modes are strongly linked with the boundary zonal flow being the robust remnant of nonlinear wall mode states. We also show how the heat transport (Nu) contributions of wall modes and bulk modes are related and discuss approaches to disentangling their relative contributions. Published by the American Physical Society 2024

On comparisons of information structures with infinite states

Blackwell's theorem on the comparison of information structures is by now sufficiently well-understood for a finite state space, but important gaps remain in the infinite case. While the equivalence of (i) sufficiency and (ii) more-informativeness is known, we present a comprehensive theory that establishes equivalences between these two orders (in both their original and almost all versions) and three additional prior-dependent criteria on general (Polish) state spaces. We consider (iii) Bayesian preference, (iv) convex dominance, and (v) mean-preserving-spread (dilation) for all priors as well as for a given full-support prior. We provide counterexamples to underscore the necessity of the assumptions underlying some of our findings, and offer a generalization of the Hirschleifer-Schlee theorem as an application.

Phase diagram of generalized totally asymmetric simple exclusion process on an open chain: Liggett-like boundary conditions.

The totally asymmetric simple exclusion process with generalized update is a version of the discrete time totally asymmetric exclusion process with an additional interparticle interaction that controls the degree of particle clustering. Though the model was shown to be integrable on the ring and on the infinite lattice, on the open chain it was studied mainly numerically, while no analytic results existed even for its phase diagram. In this paper, we introduce boundary conditions associated with the infinite translation invariant stationary states of the model, which allow us to obtain the exact phase diagram analytically. We discuss the phase diagram in detail and confirm the analytic predictions by extensive numerical simulations.

A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

AbstractThis paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin–Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite-horizon considerations. However, there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin–Wentzell quasipotential is indeed the rate function.

Stability of Stochastic Functional Differential Equations With Past-Dependent Random Switching Involving Countably Infinite States

Stability of Stochastic Functional Differential Equations With Past-Dependent Random Switching Involving Countably Infinite States

Linking infinite bond-dimension matrix product states with frustration-free Hamiltonians

Linking infinite bond-dimension matrix product states with frustration-free Hamiltonians

White-box validation of quantitative product lines by statistical model checking and process mining

We propose a novel methodology to validate software product line (PL) models by integrating Statistical Model Checking (SMC) with Process Mining (PM). We consider the feature-oriented language QFLan from the PL engineering domain. QFLan allows to model PL equipped with rich cross-tree and quantitative constraints, as well as aspects of dynamic PLs such as the staged configurations. This richness allows us to easily obtain models with infinite state-space, calling for simulation-based analysis techniques, like SMC. For example, we use a running example with infinite state space. SMC is a family of analysis techniques based on the generation of samples of the dynamics of a system. SMC aims at estimating properties of a system like the probability of a given event (e.g., installing a feature), or the expected value of quantities in it (e.g., the average price of products from the studied family). Instead, PM is a family of data-driven techniques that uses logs collected on the execution of an information system to identify and reason about its underlying execution process. This often regards identifying and reasoning about process patterns, bottlenecks, and possibilities for improvement. In this paper, to the best of our knowledge, we propose, for the first time, the application of Process Mining (PM) techniques to the byproducts of Statistical Model Checking (SMC) simulations. This aims to enhance the utility of SMC analyses. Typically, if SMC gives unexpected results, the modeler has to discover whether these come from actual characteristics of the system, or from bugs in the model. This is done in a black-box manner, only based on the obtained numerical values. We improve on this by using PM to get a white-box perspective on the dynamics of the system observed by SMC. Roughly speaking, we feed the samples generated by SMC to PM tools, obtaining a compact graphical representation of the observed dynamics. This mined PM model is then transformed into a mined QFLan model, making it accessible to PL engineers. Using two well-known PL models, we show that our methodology is effective (helps in pinpointing issues in models, and in suggesting fixes), and that it scales to complex models. We also show that it is general, by applying it to the security domain.

Probabilistic Reachability Prediction of Unbounded Petri Nets: A Machine Learning Method

Unbounded Petri nets (UPNs) can describe and analyze discrete event systems with infinite states (DESIS). Due to the infinite state space and the combination explosion problem, the reachability analysis of UPNs is an NP-Hard problem. The existing reachability analysis methods cannot achieve an accurate result at reasonable costs (computational time and space) due to the finite reachability tree with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega$</tex-math> </inline-formula> -numbers. Based on the idea of approximating infinite space with finite states, given some limited reachable markings of a UPN, we propose a method that can quantitatively solve the UPN’s reachability problem with machine learning. Firstly, we define the probabilistic reachability of markings and transform the UPN’s reachability problem into the prediction problem of markings. The proposed method based on positive and unlabeled learning (PUL) and bagging trains a classifier to predict the probabilistic reachability of unknown markings. Finally, to predict the markings outside the positive sample set and unlabeled sample set, an iterative strategy is designed to update the classifier. Based on seven general UPNs, the results of the experiments show that the proposed method has a good performance in the accuracy and time consumption for the UPN’s reachability problem. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —In discrete event systems, the reachability problem mainly studies reachable states of the system and the relationship between states, which is the basis of the system’s states, behaviors, attributes and performance analysis. For discrete event systems with infinite states, it is hard to analyze the reachable relationship between states within a finite time due to the infinite state space and the combination explosion problem. The main motivation of the paper is to propose a method that can predict the reachable relationship between the states with a probability value within a finite time. By machine learning algorithms, the method learns the feature information of the known reachable states. The reachability of unknown states in the infinite state space can be predicted approximately. The proposed approximation method can be applied to analyze the reachability properties of general discrete event systems with infinite states, such as checking whether a fault occurs in operating systems, whether a message is delivered in communication and so on.

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 2

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.

The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 1

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.