Set Of Target States Research Articles

Optimization of Time-Ordered Processes in the Finite and Asymptotic Regimes

Many problems in quantum information theory can be formulated as optimizations over the sequential outcomes of dynamical systems subject to unpredictable external influences. Such problems include many-body entanglement detection through adaptive measurements, computing the maximum average score of a preparation game over a continuous set of target states, and limiting the behavior of a (quantum) finite-state automaton. In this work, we introduce tractable relaxations of this class of optimization problems. To illustrate their performance, we use them to: (a) compute the probability that a finite-state automaton outputs a given sequence of bits; (b) develop a new many-body entanglement-detection protocol; and (c) let the computer an adaptive protocol for magic state detection. As we further show, the maximum score of a sequential problem in the limit of infinitely many time steps is in general incomputable. Nonetheless, we provide general heuristics to bound this quantity and show that they provide useful estimates in relevant scenarios. Published by the American Physical Society 2024

In Search of Dispersed Memories: Generative Diffusion Models Are Associative Memory Networks.

Uncovering the mechanisms behind long-term memory is one of the most fascinating open problems in neuroscience and artificial intelligence. Artificial associative memory networks have been used to formalize important aspects of biological memory. Generative diffusion models are a type of generative machine learning techniques that have shown great performance in many tasks. Similar to associative memory systems, these networks define a dynamical system that converges to a set of target states. In this work, we show that generative diffusion models can be interpreted as energy-based models and that, when trained on discrete patterns, their energy function is (asymptotically) identical to that of modern Hopfield networks. This equivalence allows us to interpret the supervised training of diffusion models as a synaptic learning process that encodes the associative dynamics of a modern Hopfield network in the weight structure of a deep neural network. Leveraging this connection, we formulate a generalized framework for understanding the formation of long-term memory, where creative generation and memory recall can be seen as parts of a unified continuum.

Reach-Avoid Verification Based on Convex Optimization

In this paper we propose novel sufficient conditions for verifying reach-avoid properties of continuous-time systems modelled by ordinary differential equations (ODEs). Given a system, an initial set, a safe set and a target set of states, we say that the reach-avoid property holds, if for all initial conditions in the initial set, any trajectory of the system starting at them will eventually, i.e. in unbounded yet finite time, enter the target set while remaining inside the safe set until that first target hit (that is, if the system starting from the initial set can reach the target set safely). Based on a discount value function, two sets of quantified constraints are derived for verifying the reach-avoid property via the computation of exponential/asymptotic guidance-barrier functions (they form a barrier escorting the system to the target set safely at an exponential or asymptotic rate). It is interesting to find that one set of constraints whose solution is termed exponential guidance-barrier functions is just a simplified version of the existing one derived from the moment based method, while the other one whose solution is termed asymptotic guidance-barrier functions is completely new. Furthermore, built upon this new set of constraints, we derive a set of more expressive constraints, which includes the aforementioned two sets of constraints as special instances, providing more chances for verifying the reach-avoid property successfully. Finally, several examples demonstrate the theoretical developments and performance of proposed sufficient conditions using semi-definite programming methods.

Stochastic Games with Synchronization Objectives

We consider two-player stochastic games played on a finite graph for infinitely many rounds. Stochastic games generalize both Markov decision processes (MDP) by adding an adversary player, and two-player deterministic games by adding stochasticity. The outcome of the game is a sequence of distributions over the graph states, representing the evolution of a population consisting of a continuum number of identical copies of a process modeled by the game graph. We consider synchronization objectives, which require the probability mass to accumulate in a set of target states, either always, once, infinitely often, or always after some point in the outcome sequence; and the winning modes of sure winning (if the accumulated probability is equal to 1) and almost-sure winning (if the accumulated probability is arbitrarily close to 1). We present algorithms to compute the set of winning distributions for each of these synchronization modes, showing that the corresponding decision problem is PSPACE-complete for synchronizing once and infinitely often and PTIME-complete for synchronizing always and always after some point. These bounds are remarkably in line with the special case of MDPs, while the algorithmic solution and proof technique are considerably more involved, even for deterministic games. This is because those games have a flavor of imperfect information, in particular they are not determined and randomized strategies need to be considered, even if there is no stochastic choice in the game graph. Moreover, in combination with stochasticity in the game graph, finite-memory strategies are not sufficient in general.

On a Notion of Resilience for Markov Decision Processes with Reachability Objectives

We propose and study a notion of resilience for Markov decision processes (MDPs) with the almost-sure reachability objective to action losses. Given an MDP with an initial state and a set of target states, we define the resilience degree of the MDP as the minimum number of actions that must be removed so that the target states cannot be reached almost surely from the initial state. This notion measures the level of tolerance of an MDP to action losses under the reachability objective. We first preprocess the MDP to remove irrelevant states and actions and construct a reduced transition diagram. Then, we show that computing the resilience degree is an NP-hard problem and provide an exact solution based on the mixed-integer linear programming.

Semi-Device-Independent Framework Based on Restricted Distrust in Prepare-and-Measure Experiments.

A semi-device-independent framework for prepare-and-measure experiments is introduced in which an experimenter can tune the degree of distrust in the performance of the quantum devices. In this framework, a receiver operates an uncharacterized measurement device and a sender operates a preparation device that emits states with a bounded fidelity with respect to a set of target states. No assumption on Hilbert space dimension is required. The set of quantum correlations is investigated and bounded from both the interior and the exterior. Furthermore, the optimal performance of quantum state discrimination with bounded distrust is derived and applied to certification of detection efficiency. Quantum-over-classical advantages are demonstrated and the magnitude of distrust compatible with such advantages is explored. Finally, efficient schemes for semi-device-independent random number generation are developed.

Lackadaisical quantum walk for spatial search

Lackadaisical quantum walk (LQW) has been an efficient technique in searching for a target state in a database which is distributed in a two-dimensional lattice. We numerically study the quantum search algorithm based on the lackadaisical quantum walk in one and two dimensions. It is observed that specific values of the self-loop weight at each vertex of the graph is responsible for such a speedup of the algorithm. Searching for a target state in one-dimensional lattice with periodic boundary conditions is possible using lackadaisical quantum walk, which can find a target state with [Formula: see text] success probability after [Formula: see text] time steps. In two dimensions, our numerical simulation up to [Formula: see text] for specific sets of target states suggests that the lackadaisical quantum walk can search one of the [Formula: see text] target states in [Formula: see text] time steps.

Optimal common resource in majorization-based resource theories

We address the problem of finding the optimal common resource for an arbitrary family of target states in quantum resource theories based on majorization, that is, theories whose conversion law between resources is determined by a majorization relationship, such as it happens with entanglement, coherence or purity. We provide a conclusive answer to this problem by appealing to the completeness property of the majorization lattice. We give a proof of this property that relies heavily on the more geometric construction provided by the Lorenz curves, which allows to explicitly obtain the corresponding infimum and supremum. Our framework includes the case of possibly non-denumerable sets of target states (i.e. targets sets described by continuous parameters). In addition, we show that a notion of approximate majorization, which has recently found application in quantum thermodynamics, is in close relation with the completeness of this lattice. Finally, we provide some examples of optimal common resources within the resource theory of quantum coherence.

Uncertainties on atomic data. A case study: N iv

We consider three recent large-scale calculations for the radiative and electron-impact excitation data of N IV, carried out with different methods and codes. The scattering calculations employed the relativistic Dirac $R$-matrix (DARC) method, the intermediate coupling frame transformation (ICFT) $R$-matrix method, and the B-spline $R$-matrix (BSR) method. These are all large-scale scattering calculations with well-tested and sophisticated codes, which use the same set of target states. One concern raised in previous literature is related to the increasingly large discrepancies in the effective collision strengths between the three sets of calculations for increasingly weak and/or high-lying transitions. We have built three model ions and calculated the intensities of all the main spectral lines in this ion. We have found that, despite such large differences, excellent agreement (to within $\pm$~20\%) exists between all the spectroscopically-relevant line intensities. This provides confidence in the reliability of the calculations for plasma diagnostics. We have used the differences in the radiative and excitation rates amongst the three sets of calculations to obtain a measure of the uncertainty in each rate. Using a Monte Carlo approach, we have shown how these uncertainties affect the main theoretical ratios which are used to measure electron densities and temperatures.

An improved probability hypothesis density filter for multi-target tracking

For the problem that the standard probability hypothesis density is unable to correctly estimate the states of targets and their number when tracking multiple targets in possible missed detection environments, an improved probability hypothesis density filter based multi-target tracking algorithm is proposed under the linear Gaussian conditions. Two assisted parameters, namely label and probability of existence, are introduced to expand the standard target state in the proposed algorithm which includes three robust schemes compared with the Gaussian mixture probability hypothesis density filter. Firstly, the extended component set of target states representing the target intensity can be correctly updated in the proposed target intensity update scheme. Secondly, by optimizing the component set that approximates the target posterior intensity, the excess and invalid components are effectively reduced in the improved component fusion scheme. Lastly, a new target state extraction scheme is able to accurately estimate the states of targets by comprehensively utilizing both the weight and existent probability of the target. Simulation results show that the proposed algorithm not only provides relatively accurate multi-target estimates, but also has a relatively low computational burden.

Indefinite-Horizon Reachability in Goal-DEC-POMDPs

DEC-POMDPs extend POMDPs to a multi-agent setting, where several agents operate in an uncertain environment independently to achieve a joint objective. DEC-POMDPs have been studied with finite-horizon and infinite-horizon discounted-sum objectives, and there exist solvers both for exact and approximate solutions. In this work we consider Goal-DEC-POMDPs, where given a set of target states, the objective is to ensure that the target set is reached with minimal cost.We consider the indefinite-horizon (infinite-horizon with either discounted-sum, or undiscounted-sum, where absorbing goal states have zero-cost) problem. We present a new and novel method to solve the problem that extends methods for finite-horizon DEC-POMDPs and the RTDP-Bel approach for POMDPs. We present experimental results on several examples, and show that our approach presents promising results.

A Symbolic SAT-Based Algorithm for Almost-Sure Reachability with Small Strategies in POMDPs

POMDPs are standard models for probabilistic planning problems, where an agent interacts with an uncertain environment. We study the problem of almost-sure reachability, where given a set of target states, the question is to decide whether there is a policy to ensure that the target set is reached with probability 1 (almost-surely). While in general the problem is EXPTIME-complete, in many practical cases policies with a small amount of memory suffice. Moreover, the existing solution to the problem is explicit, which first requires to construct explicitly an exponential reduction to a belief-support MDP. In this work, we first study the existence of observation-stationary strategies, which is NP-complete, and then small-memory strategies. We present a symbolic algorithm by an efficient encoding to SAT and using a SAT solver for the problem. We report experimental results demonstrating the scalability of our symbolic (SAT-based) approach.

Optimal cost almost-sure reachability in POMDPs

We consider partially observable Markov decision processes (POMDPs) with a set of target states and an integer cost associated with every transition. The optimization objective we study asks to minimize the expected total cost of reaching a state in the target set, while ensuring that the target set is reached almost surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost, both double exponential in the POMDP state space size; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we also present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest.

Optimal Cost Almost-Sure Reachability in POMDPs

We consider partially observable Markov decision processes (POMDPs) with a set of target states and every transition is associated with an integer cost. The optimization objective we study asks to minimize the expected total cost till the target set is reached, while ensuring that the target set is reached almost-surely (with probability 1). We show that for integer costs approximating the optimal cost is undecidable. For positive costs, our results are as follows: (i) we establish matching lower and upper bounds for the optimal cost and the bound is double exponential; (ii) we show that the problem of approximating the optimal cost is decidable and present approximation algorithms developing on the existing algorithms for POMDPs with finite-horizon objectives. While the worst-case running time of our algorithm is double exponential, we present efficient stopping criteria for the algorithm and show experimentally that it performs well in many examples of interest.

Enhancement of a Fault Measure for AMSs Using Probabilistic Timed Automata

A novel approach for probabilistic timed structure that is based on combining the formalisms of timed automata and probabilistic automata representation of the system is proposed. Due to their real-valued clocks can measure the passage of time and transitions can be probabilistic such that it can be expressed as a discrete probability distribution on the set of target states. The usage of clock variables and the specification of state space are illustrated with real value time applications. The transitions between states are probabilistic by events which describe either the occurrence of faults or normal working conditions. Additionally, the passage of discrete time and transitions can be probabilistic by mean of the theory of expectation sets to obtain a unified measure reasoning strategy.

Networks of automata with read arcs: a tool for distributed planning

AbstractA planning problem consists in driving a system from its current state to a set of target states. Problems are generally expressed by a collection of state variables, and actions that change the value of a subset of these variables. Such problems can be modeled as networks of automata, one per state variable, that partially synchronize on some actions. Distributed planning (or factored planning) consists in driving each of these automata to its goal state(s), while preserving the coherence of their interactions. Real planning problems, however, need to model actions that can only be performed in one component when another component is in a specific state. This paper proposes a mechanism to capture this phenomenon, under the form of automata with read arcs, reproducing what already exists for Petri nets. It is shown that a previous approach to distributed planning, based on automata computations, can be extended to this new setting.

Reachability analysis based transient stability design in power systems

This paper provides a systematic framework to determine switching control strategies to stabilize the system after a fault if the stabilization is possible. A method to compute the stability region of a stable equilibrium point with the purpose of power system stability analysis is proposed and the validity of discrete controls in transient stability design is studied. First, a Hamilton–Jacobi–Isaas (HJI) partial differential equation (PDE) is constructed to describe the set of backward reachable states as a function of time starting from a target set of states. The backward reachable set of a stable equilibrium point is computed by numerically solving the HJI PDE backwardly in time using level set methods. This backward reachable set yields the stability region of the equilibrium point. Based on such reachability analysis, a transient stability design method is presented. The validity of a discrete control is determined by examining the stability region of the power system with the said control on. If a post-fault initial state is in the stability region of the system with a control on, the control is valid. A control strategy is provided based on the validity of controls. Finally, this method is illustrated by applying to a single machine infinite bus system with the compensation of shunt and series capacitors.

Relativistic convergent close-coupling method: Calculations of electron scattering from cesium

We present a detailed formulation of the relativistic convergent close-coupling (RCCC) method which is based on the solution of the Dirac equation describing electron scattering from quasi-one-electron atoms. A square-integrable Dirac $L$ spinor basis has been used to obtain a set of target states representing both the bound and continuum spectra of the target. A set of momentum-space coupled Lippmann-Schwinger equations for the $T$ matrix is then formulated and solved. We use spin asymmetries, particularly those that are identically zero in a nonrelativistic formulation, to check the accuracy of the RCCC method and find good agreement with experiment on a broad energy range.

Design of Logic Feedback Controllers for Discretely Controlled Continuous Systems

This contribution presents an approach to synthesize logic feedback controllers for Discretely Controlled Continuous Systems such that given specifications are fulfilled. This class of systems consists of a continuous plant with nonlinear dynamics which is controlled by a logic controller that processes discrete measurements. The control problem is to optimally steer the system from a set of initial states to a set of target states without reaching unsafe states. For given thresholds, the algorithm first computes the optimal discrete inputs for each boundary section to obtain optimal trajectories from each boundary section to the target by a dynamic programming approach. The controller then has to be verified to guarantee that the unsafe states are not reached for arbitrary trajectories. The approach is demonstrated for a two-tank benchmark.

Model Checking Probabilistic Timed Automata with One or Two Clocks

Probabilistic timed automata are an extension of timed automata with discrete probability distributions. We consider model-checking algorithms for the subclasses of probabilistic timed automata which have one or two clocks. Firstly, we show that PCTL probabilistic model-checking problems (such as determining whether a set of target states can be reached with probability at least 0.99 regardless of how nondeterminism is resolved) are PTIME-complete for one-clock probabilistic timed automata, and are EXPTIME-complete for probabilistic timed automata with two clocks. Secondly, we show that, for one-clock probabilistic timed automata, the model-checking problem for the probabilistic timed temporal logic PCTL is EXPTIME-complete. However, the model-checking problem for the subclass of PCTL which does not permit both punctual timing bounds, which require the occurrence of an event at an exact time point, and comparisons with probability bounds other than 0 or 1, is PTIME-complete for one-clock probabilistic timed automata.