Hierarchical attentive multiple models for execution and recognition of actions According to the motor theories of perception, the motor systems of an observer are actively involved in the perception of actions when these are performed by a demonstrator. In this paper we review our computational architecture, HAMMER (Hierarchical Attentive Multiple Models for Execution and Recognition), where the motor control systems of a robot are organised in a hierarchical, distributed manner, and can be used in the dual role of (a) competitively selecting and executing an action, and (b) perceiving it when perfomed by a demonstrator. We subsequently demonstrate that such arrangement can provide a principled method for the top-down control of attention during action perception, resulting in significant performance gains. We assess these performance gains under a variety of resource allocation strategies. Following the increased interest in mechanisms that will endow robots with the capability to imitate human action, several computational architectures have been proposed to match visual information from observing a demonstrator to motor plans that would achieve the corresponding action for the observer [1] [2] [3] . Internal models of the motor systems of the observer and the demonstrator and their capabilities have been frequently suggested as useful tools aiding in this matching (e.g. [4] , for a review, see [5] ). The particular method we will employ in this paper relies on the concept of motor simulation. Mental simulation theories of cognitive function [6] , of which motor simulation is an instance, advocate the use of the observer's cognitive and motor structures in a dual role: on-line, for the purposes of perceiving and acting overtly, and off-line, for simulating and imagining actions and their consequences [7] . With the discovery of the mirror system, first in monkeys [8] and subsequently in humans [9] , the idea that motor systems can be used for the perception of others' actions became increasingly popular [10] . In our previous work [2] , we have performed a computational investigation of the mirror system using robotic devices as models, which resulted in plausible explanations and testable predictions regarding its behaviour, and in particular its sensitivity to the velocity profile of the demonstrator (for a review see [11] ).. In this paper, we describe HAMMER, our computational architecture for recognizing and executing actions, utilising hierarchical, attentive, multiple models. After a review of related work, we will review the operation of the building blocks of the architecture. Subsequently we will perform a number of experiments, (a) illustrating that such architecture can provide a principled method for the top-down control of attention during the perception of actions of others, and (b) demonstrating that it results in significant computational performance increases. We reviewed our approach to the development of architectures that incorporate distributed, hierarchical networks of inverse and forward models, and described how HAMMER can be used to perceive a demonstrated action. We performed computational experiments demonstrating how different resource allocation strategies in the control of attention can influence the performance of HAMMER.. The novelty of our approach lies in the idea that the features that the observer will choose to attend are the ones that (s)he would have to control if (s)he was in the same situation and would have to perform the same action. This is compatible with recent biological evidence [32] on the use of action plans in action recognition. This stems naturally from the simulation approach to action perception [11] , which has been the main inspiration underlying our work, and provides a timely opportunity to study the interplay between the two important topics of attention and action perception. The key contribution of this paper is the introduction of a novel way of exerting top-down control to the attention mechanism, using a competitive, parallel implementation of the simulation theory of mind. Our experiments demonstrate that the addition of an attention mechanism which arbitrates among the state requests of multiple inverse models results in a significant amount of computational savings, without significant loss of separability between the inverse models. This is particularly true with a combination of a conservative approach (initially equal-share, RR) switched to a more aggressive one (HCAW) when a strong candidate is found. As expected, taking into consideration the environmental context (condition RR-HCAW-A) further improves the results by removing inapplicable inverse models. We expect that the combination of our system with a joint attention mechanism [31] which extracts the gaze of the demonstrator, and through it, attempts to infer the focus of attention of the demonstrator will further reduce the number of applicable inverse models. We intend to pursue this in future work.. The savings in our current implementation, although significant, are only computational given that saccades within our system take place within the image and do not involve camera movements. However, should the situation require camera movements to locate the state information requested, these savings would increase given the additional cost associated with each camera movement. The fact that at the motor level, resources will always need to be shared, makes our architecture useful even for implementation on parallel machines that could afford to process, from a computational perspective, more than one inverse model at a time.. One point worth discussing is the "resource-slice" that the winning inverse model is given at each iteration by the attention mechanism. Although in this particular set of experiments we have used one single frame, and we perform one iteration of the arbitration mechanism at each frame, that does not imply it is the only solution. We have found that for some of the experimental conditions, it has the undesirable effect of linking the numbers of inverse models with the frame rate: for example, if the number of inverse models in-creases and the attention system is working in the RR condition, the rate that each inverse model will be processed (and thus, the accuracy of its predictions) decreases. On the positive note, using the applicability initialization (RR-HCAW-A condition), the number of inverse models that are involved is reduced, reducing this problem, but it would be interesting to experiment with different resource-slices, and methods of allocating resources. Our current implementation utilizes an arbitration mechanism that takes into consideration the requests blindly, without considering what they represent; for example, more than one inverse model can be asking for the same information, the state of the arm. We are currently considering a further optimization method that allocates resources according to the current popularity of state requests across the inverse models. A "working memory" mechanism can also maintain a number of recent states and immediately return them when they are required. Furthermore, path planning algorithms can also be utilized so future requests are optimized topographically (for example, if three inverse models are requesting the state of the head, arm and leg of a human, these will be served in that order, irrespective as to whether the order that the requests were made, or their priorities) to minimize lengthy saccades. In these experiments, we captured eight video sequences of a demonstrator performing different tasks in front of the robot to ensure fair comparison between the different conditions, and performed 40 experiments on these videos under different combinations of scheduling algorithms. Figures 6 to 9 give representative results with respect to the performance of the architecture under five different experimental conditions (figure 5 serves as a legend for the other figures, and shows the assigned numbers (B1 to B8) for the different inverse models):. • No attention mechanism -at each frame all inverse models receive the state information they require, and all compute and update their confidences. This condition is the same used in previous experiments [2, 4] , and will serve as the reference condition for comparing the performance of the attention subsystem.. • Round-Robin implementation: a circular linked list is created and all requests are served in a first-in first-out fashion (one frame per request).. • HCAW (Highest Confidence Always Wins) condition, in which the inverse model with the current highest confidence is always given the state it is requesting.. • RR-HCAW condition, which combines conditions two and three above. Initially all inverse models are given equal treatment (RR), but when one of them takes a clear lead (its confidence becomes higher than 50% of the average confidence of all inverse models with positive confidences), the arbitration mechanism switches to HCAW.. • RR-HCAW-A condition, which repeats the last condition but performs an initialization step that removes from the candidate inverse models the ones that are not applicable -for example if an object is not present, the corresponding inverse models for handling it are not activated. Figure 6 gives the evolution of the confidences of all the inverse models when no attention mechanism is employed. The demonstrator is picking up an orange and the corresponding inverse model (B4) correctly receives the highest confidence score, with the "picking up the can" action (B1) receiving the second highest given its similarity to B4 for the initial part of the trajectory. Figure 7 gives the evolution of the confidences of all the inverse models when attention is given to each of the inverse models in an equal-share manner. The corresponding inverse model (B4) again correctly receives the highest confidence score, but the separability of B4 from B1 is now significantly smaller since the inverse models are not as frequently updated as in the first experimental condition. The corresponding computational savings however are significant (assuming equal computational resources are required for each inverse model, this is of the order of n−1 n (87.5% in this case), with n being the number of inverse models -the arbitration mechanism's overheads are negligible), since under this condition (and the following three), only one of the eight inverse models gets processed at each frame. This point is further elaborated in the discussion section. Figure 8 gives the evolution of the confidences of all the Fig. 7 . Evolution of the confidence values with a Round Robin implementation inverse models when attention is given only to the inverse model with the highest confidence. The corresponding inverse model (B4) still correctly receives the highest confidence score, and the separability of B4 from B1 (and all the rest) is now clearer. However this condition is more susceptible to noise with respect to the RR condition and the overall confidence portait of the inverse models is noisier than before. Figure 9 gives the evolution of the confidences of all the inverse models when attention is given to an inverse model using a combination of the two previous conditions. Initally all inverse models are receiving equal treatment (RR), but when the confidence of one of them in-creases to more than 50% of the average of all inverse models with a positive score, the selection algorithm switches to one that favours the one with the highest confidence. This condition combines the benefits of the two previous conditions: it displays clear separability between the winner inverse model and the rest, while not displaying the initial noisy profile of the HCAW condition. This condition essentially amounts to an attention strategy of "playing it safe" at first, and then focusing the system's resources to the winner once a strong candidate is found. Fig. 9 . Evolution of confidence values using the RR-HCAW condition Finally, we repeat the last experiment, removing the soda can from the setup. Figure 10 shows the evolution of confidences of the inverse models with the initialisation step; the inverse models that have to deal with the soda can are not considered in the arbitration process, and (as a matter of convention) are constantly receiving negative confidence reinforcement of -1. This boosts the separability even further since there are fewer inverse models competing for attention. Figure 11 gives a summary of the performance of the attention Fig. 10 . Evolution of the confidence values using the RR-HCAW-A condition mechanism for each of the behaviours. In all conditions, all behaviours were recognised correctly. Results of the final confidence value of the demonstrated action are shown for all experimental conditions. Percentages are calculated by making a relative comparison between each confidence value and the corresponding one from the plain version of the architecture without the attention mechanism. Equipping robots with the ability to imitate enables them to learn to perform tasks by observing a human demonstrator [12] . In the center of this ability lies a mechanism that matches demonstrated actions with motor actions available to the robot [2, 1, 5] . Several architectures have been proposed for implementing this mechanism (for reviews see [13, 12, 5, 14] ), including a number of proposals utilizing a shared substrate between execution, planning, and recognition of actions [2, 5, 15] . This methodological shift, compared to other successful approaches to learning by demonstration [16] was inspired by the discovery of the mirror system [17, 18] , which indicated that, at least in primates, there is indeed a shared neural substrate between the mechanisms of action execution and those of action recognition. Apart from being compatible with the motor theories of perception, from an engineering perspective this approach is also attractive since it allows reuse of subsystems in multiple roles.. An important issue that remains unresolved is where the attention of the observer should be focused when a demonstrator performs an action. Whilst there is little agreement on an universal definition of attention [19] , from an engineering point of view it can be defined as a mechanism for allocating the limited perceptual and motor resources of an agent to the most relevant sensory stimuli. The control inputs to the attention mechanism can be divided into two categories: stimulus-driven (or bottom-up) and goal-directed (or topdown). Stimulus-driven attention models work by attaching levels of saliency to low-level features of the visual scence, e.g. colour, texture or movement, and then deriving the corresponding saliency maps, fusing them, and applying a winner-take-all strategy to direct the attention to the most salient part of the scene [20] . However, it is well known from human psychophysical experiments that top-down information can also influence bottom-up processing (e.g. [21, 22] ). Wolfe [21] put forward the hypothesis that the goal-directed element of attention selects bottom-up features that are relevant to the current task (in what is termed visual search) by varying the weighting of the feature maps. However, the fundamental question of what determines the features that are relevant to the task has not been answered in a principled way. This is the case particularly when the tasks are performed by someone else without the observer having access to the internal motivational systems of the demonstrator. The task demonstrated is not even known in advance, therefore the top-down selection of features to attend to is not obvious, and an online method for selecting features dynamically is needed. We will utilize our previous work implementing the motor theory of perception [2] for deriving a principled mechanism for the goal-directed control of attention during action perception, which we consider the key contribution of this paper. We implemented and tested this architecture on an experimental setup involving an ActivMedia Peoplebot; in these experiments the on-board camera was used as the only sensor. A human demonstrator performed an object oriented action (an example is shown in figure 3) , and the robot, using HAMMER, attempted to match the action demonstrated (an example is shown in figure 4) , with the equivalent in its repertoire. In the experiments reported here, the robot captured the demonstration at a rate of 30Hz, with an image resolution of 160×120, and the demonstrations lasted an average of 2 seconds. In the following sections, we will describe the implementation of the architecture, along with its performance with and without several implementations of the attention mechanism. We will compare its performance against previous implementations of this approach [2, 4] to demonstrate the performance improvements that the attention subsystem of HAMMER brings. Fig. 2 . The complete architecture, incorporating the attention systems; forward models have been omitted (but are shown in figure 1 ); B1 to Bn are the available inverse models, and the arbitration block has to decide which of their requests to satisfy.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Real-Time, Load-Adaptive Processing of Continuous Queries Over Data Streams We introduce a new type of query, called a real-time continuous query (RCQ), that captures the real-time requirements of processing data streams. We develop techniques to efficiently process the RCQs in the presence of fluctuating query load and data load. We show that Rate-Monotonic scheduling is applicable to this problem domain, and show how to make this method adaptive to varying load conditions. When a set of queries becomes unschedulable due to load variations, we perform controlled input load shedding by dropping tuples using a novel feedback-based approach to decide which tuples to drop. Our work shows how to provide response time guarantees for processing RCQs, and enables making the appropriate trade-off between penalty due to missed deadlines and result accuracy. Our experiments show that our approach works very well and is usable in practice. A data stream is a continuous, unbounded, and time-varying sequence of data elements [7] . Streams arise naturally in numerous applications. Important examples include data transmitted by sensors, stock market data, and network monitoring data. It is common for users to issue continuous queries (CQ) over data streams [19] . A Data Stream Management System (DSMS) holds the query and executes it whenever new data arrives, returning the result to the user. Continuous queries are also called standing queries.. There has been considerable recent interest in building Data Stream Management Systems [11, 10, 20, 12, 23, 18, 21, 1] . Most such work addresses the issues of efficiency or memory utilization. However, very little attention has been paid in the literature to the processing of continuous queries in real-time environments. We have proposed a novel approach to schedule real-time queries over data streams. We have focussed on the problem of overrunning queries in a weakly hard real-time systems, where a small degree of missed deadlines and imprecision can be tolerated. Our work is novel because, we make no assumptions about data arrival rates, query load or the data distribution. Meeting query deadline in such setting is challenging. Our approach of meeting the query deadline by reduction the execution time of queries is very practical, and is based on well-known properties of rate-monotonic scheduling. Our load adaptive rate-monotonic (LRM) scheduling policy is dynamic and quickly adapts to the changing load conditions. LRM is able to minimize the total error caused to reduction in the processing load. Moreover, experimental results show that LRM works for various workloads and consistently out-performs the simple ratemonotonic policy and a non preemptive priority schedule. Since most continuous queries are long-running, rescheduling may be required. For dynamic settings, permissive rescheduling seems more suitable. For a workload where the arrival rate or query load can be estimated a priori, periodic rescheduling is sufficient.. As part of future work, we wish to explore how our scheduling can be applied in Aurora/Borealis [10] . Work exists in the area of real-time database systems [16] . However, all work in this area assumes that the data is static and hence, most of the data-related properties, such as the cardinalities of the relations, data distribution etc are known a priori. Thus, we believe that the techniques studied for relation-data are not directly applicable to streaming data.. Recently there has been numerous works on optimizing continuous queries over data streams. However, the bulk of the work is related to minimizing the memory utilization [20, 8, 9, 6] during query execution. A novel technique to maximize the output rate is proposed in [25] . However, very little has been done in the context of real-time application of these techniques.. The load shedding [25] approach has some overlap with our work, but the authors do not study this in the context of real-time queries. Moreover, the quality of service metric used in the semantic approach does not consider performance measures for real-time systems, such as miss ratios or latency of query execution.. Imprecise computation of real-time tasks was first proposed in [13] . This and other extensions to this work [22] , however, assume that the task can be logically divided into 2 parts, namely the mandatory and the optional part. Moreover, they study the performance of schedule under static load conditions. We make no such assumptions and we are interested in quantifying the imprecision in the context of data streams.. The authors in [15] address how to handle over-running tasks and propose a techniques to run the optional part of the task on an aperiodic server. We make no such assumptions about the query, and treat each query as an indivisible task that cannot be run separately. [26] is the only work we have come across in the context of realtime queries over data streams. However, this work is still preliminary and the paper does not provide any theoretical results of how to tune the query performance. The paper only outlines an approach to drop tuples when the arrival rates are high. The parameters that define the tuple drop rate are chosen experimentally, which is largely impractical for dynamic data stream environment. Moreover, the scheduling scheme is not discussed in detail. We address the problem of meeting deadline in a much broader context, by considering the fluctuations in the query load, data load, as well as selectivity of the queries.. A straight-forward approach to our problem is to run a overrunning query until it passes its deadline, and then stop its execution. This approach is similar to the one studied in [15] , and is not suitable in our context because of various reasons. First, a query with very high frequency (small period), but a high execution time (≥ the period) will hog the processor at all times and hence starve other queries in the system. Moreover, in a hard real-time systems the penalty due to unprocessed data is likely to become unacceptable.. Our goal is the address the problem in the context of a weakly hard DSMS, the goal being to meet all the deadlines of the queries -at the cost of some imprecision in the computation. When the queries are have a long life-time, this advantage of meeting deadlines easily outweighs the approximation error.

O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Ambiguous chance constrained problems and robust optimization In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We focus primarily on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : ρ p (Q, Q 0 ) ≤ β}, where ρ p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q 0 . The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. We also show that the robust sampled problem can be solved efficiently both in theory and in practice. A large class of decision problems in finance and engineering can be formulated as optimization problems of the form min c T x s.t. f (x, h) = max 1≤i≤l {f i (x, h)} ≤ 0,. where x ∈ X is the decision vector, X ⊆ R n is a bounded, closed, convex set, h ∈ R m are problem parameters and each of the functions f i (x, h) : X × H → R are convex in x for a fixed h. We assume, without loss of generality, that the objective is linear and independent of h. Boundedness of the set X is needed for theoretical convenience and is not a restrictive assumption for practical purposes.. The deterministic optimization approach to solving optimization problems computes a solution to (1) assuming that the problem parameters h are known and fixed. In practice, however, the parameters h are the result of some measurement or estimation process, and are, therefore, never known for certain. This uncertainty is of serious concern in applications because solutions of optimization problems are often very sensitive to fluctuations in the problem parameters. This phenomenon is well documented in several different application areas [3, 27] .. Recently robust optimization has emerged as an attractive optimization framework for reducing the sensitivity of the optimal solution to perturbations in the parameter values. In this framework, the uncertain parameters h are assumed to belong to a bounded uncertainty set H and the robust optimization problem corresponding to the nominal problem (1) is given by min c T x s.t. f (x, h) ≤ 0, ∀ h ∈ H, x ∈ X .. This framework was introduced in Ben-Tal and Nemirovski [4, 5, 6] . There is also work on robust formulations of optimization problems originating from the robust control literature [18, 20] . In many applications the uncertainty set H is given by the confidence region around the point estimates of the parameters allowing one to provide probabilistic guarantees on the performance of the optimal solution of the robust problem [27] . The robust problem (2) is solved by reformulating the semi-infinite constraints, f (x, h) ≤ 0, for all h ∈ H, as a finite collection of constraints. Such a reformulation is only possible when the uncertainty set H and the function f (x, h) satisfy some regularity conditions. See [4, 6, 7] for robust formulations that can be solved efficiently. Even when the reformulation is possible, the resulting problem is typically harder than the nominal problem (1) ( [7] proposes a new framework wherein the robust problem remains in the same complexity class as the nominal problem). In general, however, the robust problem is intractable. Another criticism of the robust framework is that it gives the same "weight" to all perturbations h ∈ H. Also, in certain applications one might have the flexibility of violating the constraints corresponding to a small fraction of the set H. An alternative optimization framework that mitigates this criticism to some extent is called chance-constrained optimization. In this framework, one assumes the parameters h are distributed according to a known distribution Q on H, and replaces the nominal problem (1) by the following chance-constrained problem min c T x s.t. x ∈ X (Q),. where. for some 0 < < 1. The parameter controls the probability that the optimal solution of (3) violates the constraints -as ↓ 0 the chance-constrained problem starts to resemble the robust problem (2) . Although chance-constrained problems have a long history dating back to at least the work of Charnes and Cooper [12] , they have not found wide applicability. This is primarily because computing the optimal solution for chance-constrained problems is extremely difficult. To begin with just evaluating Q(H : f (x, H) > 0) involves a multidimensional integral that becomes difficult as the number of parameters grows. Moreover, even if the function f (x, h) is convex (or even linear) in x the feasible set X (Q) of (3) is in general not convex. A detailed discussion of the chance-constrained programs and, more generally, stochastic programs can be found in [37] . Recently, Calafiore and Campi [10, 11] and de Farias and Van Roy [14] independently proposed tractable approximations to (3) based on constraint sampling and statistical learning techniques. In this approach, one approximates chance-constrained problem (3) by the following sampled problem. where H i , i = 1, . . . , N , are N independent, identically distributed (IID) samples from the distribution Q. de Farias and Van Roy [14] consider the special case where and f (x, h) = f (x, (h, h 0 )) = h T x + h 0 and use results from Computational Learning Theory [1, 31, 45] to show that for all N ≥ 4n ln 12 + 4 ln 2 δ , the feasible set of the sampled problem (5) is contained in X (Q) with probability at least 1 − δ. Thus, in this sampling based method there are two possible sources of errors: with probability δ, the feasible set of (5) (and consequently, the optimal solution of (5)) may not be contained in X (Q); and, in the event that this is not the case, the feasible points of (5) can still violate the constraint f (x, H) ≤ 0 with a probability . The analysis in [14] can be extended to general f (x, h) to yield the bound N ≥ 4d f ln 12 + 4 ln 2 δ , where d f denotes the Vapnik-Chervonenkis (VC) dimension of the class of function {f (x, ·) : x ∈ X } (see Section 3 for details). Calafiore and Campi [10, 11] consider general convex functions f (x, h) and show that for N ≥ 2n ln 12 + 2 ln 2 δ + 2n, the optimal solution of the sampled problem (5) is feasible for (3) with probability at least 1−δ. On the one hand, this bound is weak in the sense that it is only valid for the optimal solution, and not the entire feasible set. On the other hand, since the VC dimension d f of a class of functions can be orders of magnitude larger than the problem dimension n, the number of samples needed to ensure that the optimal solution is feasible for (3) with a high probability can be orders of magnitude smaller. The result in [10, 11] was proved using a fundamental fact that the optimal solution of a convex program is "supported" by at most n constraints. We will briefly review this work in Section 3.3. Recently, Nemirovski and Shapiro [34, 33] established logarithmically separated upper and lower bounds on the number of samples required to approximate a chance constrained problem when the measure Q has a well defined moment generating function.. Although the bounds on the sample size N are distribution-free, one must be able to sample from Q in order to construct the sampled problem (5). Also, there is an implicit assumption that the distribution Q of the random parameters H is fixed. A major criticism raised against chance constrained problems and, more generally, stochastic programs is that, in practice, the measure is never known exactly. Just as the point estimates for the parameters, the distribution Q is also estimated from data or measurements, and is, therefore, known only to within some error, i.e. the measure Q ∈ Q where Q is a set of measures. Since our primary interest in the chance constrained problem (3) was to use it as an approximation (or even a refinement) of the robust problem (2), the natural problem to consider when the measure Q is uncertain is given by. We will call (6) an ambiguous chance-constrained problem. A problem of the form (6) has two sources of uncertainty: The distribution Q of the parameter h is uncertain, and, given a measure Q, the particular realization of the parameter h is also uncertain. In the decision theory literature the uncertainty in the distribution is referred to as ambiguity, and hence the name for the problem. Modeling ambiguity and its consequence has been received attention in several different fields. The minimax formulation has a long history in stochastic programing [46, 8, 16, 17, 30, 42, 40, 41] . Ruszczynski and Shapiro [38] show the equivalence between minimax stochastic programming and minimizing a convex risk measure [2, 24] of the second-stage cost. [39] extends the minimax approach to a multiperiod setting. The study of ambiguity in Economics began with the work of Gilboa and Schmeidler [26] . This work was extended to a multiperiod setting by Hansen and Sargent [28] and Epstein and his co-authors [13, 21, 22] . Ambiguity in the context of Markov decision processes was independently investigated by Iyengar [29] and El Ghaoui and Nilim [19] .. The main contributions of this paper are as follows.. (a) We consider uncertainty sets Q of measures that are of the form Q = {Q : ρ(Q, Q 0 ) ≤ β} where ρ(·, ·) denotes a suitable metric between probability measures, i.e. the uncertainty sets are "balls" centered around a central measure Q 0 . We approximate the ambiguous chanceconstrained problem (6) by a robust sampled problem defined as follows. where H 0 i , i = 1, . . . , N , denote IID samples drawn according to the central measure Q 0 and the norm · on the H space is related to the probability metric ρ(·, ·) (details are given in Section 4). Results in [7] imply that for a large class of constraint functions f (x, h) and suitably defined norms · the robust sampled problem (8) is in the same complexity class as the nominal problem (1).. (b) We combine results from Computational Learning Theory with results for coupling of random variables [44] to compute upper bounds on the number of samples N required to ensure that the feasible set of the robust sampled problem (8) is contained in X (Q), for any Q ∈ Q, with a high probability. This bound depends on the Vapnik-Chervonenkis (VC) dimension of the function f (x, h).. (c) We use coupling to extend the results of Calafiore and Campi [10, 11] to the ambiguous chance constrained problems, i.e. we compute upper bounds on the number of samples required to ensure that the optimal solution of the robust sampled problem (8) is contained inX with a high probability. The bound in this case depends on the number of "support" constraints, and is independent of the VC dimension of f (x, h).. The issue of ambiguity of measures was also raised in [11] where the authors considered a finite uncertainty set Q. They proposed a solution strategy where one samples from all of these measures and showed that the samples from different measures "help" each other. In contrast, we consider the case where Q is uncountably infinite and we draw samples from only the central measure Q 0 . The rest of this paper is structured as follows. In Section 3 we briefly review the known results for chance constrained problems. The results in this section are not new -they have been included to set the context for our extensions. Section 4 introduces probability metrics, coupling and the Strassen-Dudley Representation Theorem. Section 5 uses this Representation Theorem to establish bounds for ambiguous chance constrained problems. In Section 6 we identify particular classes of functions f (·, ·) and norms · on the parameter space H that allow the robust sampled problem (8) to be solved efficiently. Section 7 has some concluding remarks. In this paper we study ambiguous chance constrained problem where the uncertainty set for the ambiguous measure is given by a ball defined in terms of the Prohorov metric. We approximate the ambiguous chance constrained problem by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the "central" distribution in the uncertainty set of measures. The main contribution of this paper is to show that all the sample complexity results known in the context of usual chance constrained problems extend to the ambiguous setting. Our extensions are based on the Strassen-Dudley Representation Theorem, which states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability. Coupling is just a construct needed to prove the results; it is never used in computing the solution to the robust sampled problem.. The results in this paper should be viewed as a first step towards solving ambiguous chance constrained problems. Several issues still remain unresolved. Foremost is the issue of selecting the constant β. This constant should reflect the "confidence" in estimating the distribution of the uncertain parameters H from finite data. It is not clear how this can be done.. We only consider uncertainty sets that are norm balls defined in terms of the Prohorov metric. One could consider "tiling" a more general uncertainty set by norm balls of a given radius and construct a robust sampled problem by drawing samples according to the centers of the balls (a simplified version of this idea appears in [11] ). Since the constant that controls the violation probability in the ambiguous chance constrained problem has to be greater than the radius β of the norm ball, such an approach is attractive even when the uncertainty is a norm ball. However, it is not clear how to efficiently select the centers of the balls to "tile" the uncertainty set.. In Section 4 we introduced several probability metrics and show that uncertainty set Q = {Q : ρ(Q, Q 0 ) ≤ γ} can be conservatively approximated by an uncertainty setQ = {Q : ρ p (Q, Q 0 ) ≤ β(γ)} defined in terms of the Prohorov metric. However, we have no way of measuring the "blowup" of the uncertainty set that occurs in changing the metrics. This issue can be resolved by either establishing tight lower bounds on the Prohorov metric or by constructing Representation results for the other metrics. Ideally one would like to get logarithmically separated upper and lower bounds as in [33, 34, 23] .. Finally, there is the issue of proving worst-case lower bounds on the number of samples required to learn the solution of an ambiguous chance constrained problem, i.e. a refinement of Lemma 3 that accounts for ambiguity in the measure.. Define the measureQ as follows .. Then it is easy to verify thatQ is a probability measure that achieves the upper bound in (48) and satisfies ρ tv (Q, Q 0 ) ≤ β. Thus, the inequality in (48) is, in fact, an equality. The bound (48) leads to the following conclusions.. (a) The ambiguous chance constraint sup{Q(B x ) : ρ tv (Q, Q 0 ) ≤ β} ≤ is feasible only if ≥ β + Q 0 (B x ) ≥ β. Since the Prohorov metric ρ p ≤ ρ tv , it follows that > β is necessary for the feasibility of the robust chance constraint sup{Q(B x ) : ρ p (Q, Q 0 ) ≤ β} ≤ .. (b) For ≥ β, the ambiguous chance constraint sup{Q(B x ) : ρ tv (Q, Q 0 ) ≤ β} ≤ is equivalent to the (nominal) chance constraint Q 0 (H : f (x, H) > 0) ≤ − β, i.e. a chance constraint with tighter requirement. One interpretation of this result is that by introducing ambiguity via ρ tv one does not protect against a set of measures; one only increases the probabilistic guarantee with respect to the central measure Q 0 .. We have not been able to establish a similar result for the Prohorov metric. Since ρ p ≤ ρ tv , a neighborhood defined in terms of the Prohorov metric is more conservative and is likely to protect against measures other than the central measure. At the very least the bound (48) alerts us that the power of the ambiguous chance constraint can be quite sensitive to the metric used to define the neighborhood of measures.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Shared quantum control via sharing operation on remote single qutrit Abstract Two qubit operation sharing schemes [J. Phys. B 44 (2011) 165508] are generalized to qutrit ones. Operations to be shared are classified into three different classes in terms of different probabilities (i.e, 1/3, 2/3 and 1). For the latter two classes, ten and three restricted sets of operations are found out, respectively. Moreover, the two generalized schemes are amply compared from four aspects, namely, quantum and classical resource consumption, necessaryoperation complexity, success probability and efficiency. It is found that the second scheme is overall more optimal than the first one as far as three restricted sets of operations are concerned.Keywords: quantum operation sharing, generalized Bell state, restricted sets of qutrit operations, success probability, efficiency.PACS numbers: 03.65.Ta, 03.67.-aEntanglement is admitted as a kind of important quantum resource nowadays. In the last two decades it has been extensively exploited and utilized in the many fields of quantum information science to fulfill various quantum tasks involving classical information processing in a quantum manner and quantum information (i.e., quantum state) processing , such as quantum key distribution, quantum state teleportation, remote state preparation, operation teleportation, state sharing, direct secure communication, quantum computing, and so on. Enlightened by the generalization of quantum state teleportation to quantum state sharing, in 2011 Zhang and Cheung [26] definitely presented quantum operation sharing with shared entanglements. They introduced the sharing idea into quantum operation teleportation and proposed two specific schemes with different groups of shared entanglements. The first one is a universal but nontrivial scheme for sharing any single-qubit operation with one group of shared entanglements, while the latter treats the sharing of two restricted sets of operations with less resource consumptions. Incidentally, these schemes are uniformly referred to as the ZC schemes later. As same as quantum operation teleportation, QOS can be viewed as a remote control (encryption, decryption or destruction) on quantum information in the future quantum network. If the target state as an important quantum information is initially encrypted with a given unitary operation, then the inverse operation on it is obviously the decryption on it. Moreover, it is natural to regard a random operation as the destruction on it. As a consequence, quantum operation sharing can be taken as a key to activate some important actions in the future lives by some sharer entities, such as missile emissions, quantum collective seal or unseam, remote joint destruction of quantum money, and so on. Therefore, recently this topic has already attracted some attentions [27] [28] [29] . In spite of this, it is worthy emphasizing that all these works treat only the sharing of single-qubit operations with shared qubit entanglements.In the intending quantum networks, various quantum states involving high-dimensional qudit state might be employed and distributed among different nodes due to some special demands, such as peculiar quantum tasks in some concrete quantum scenarios, some definite security requirements, and so on. Because of this, some researchers have been attracted by the issues related to high-dimensional qudit cases and started to explore them [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] . Surely, their works includes the employment of qutrit entanglements in addressing various relative quantum problems [38] . In this condition, it is intriguing to ask what will happen in the field of QOS if the accessible quantum channels are composed of shared qutrit entanglements? Specifically, what kind of qutrit states can be used to fulfill QOS? If quantum channels are determined, what is the maximal success probability for sharing a given operation? Can the operation be shared more economically with less operation complexity? Can operations be classified with different classes corresponding to different success sharing probabilities? How to characterize those Entanglement is admitted as a kind of important quantum resource nowadays. In the last two decades it has been extensively exploited and utilized in the many fields of quantum information science to fulfill various quantum tasks involving classical information processing in a quantum manner and quantum information (i.e., quantum state) processing , such as quantum key distribution, quantum state teleportation, remote state preparation, operation teleportation, state sharing, direct secure communication, quantum computing, and so on. Enlightened by the generalization of quantum state teleportation to quantum state sharing, in 2011 Zhang and Cheung [26] definitely presented quantum operation sharing with shared entanglements. They introduced the sharing idea into quantum operation teleportation and proposed two specific schemes with different groups of shared entanglements. The first one is a universal but nontrivial scheme for sharing any single-qubit operation with one group of shared entanglements, while the latter treats the sharing of two restricted sets of operations with less resource consumptions. Incidentally, these schemes are uniformly referred to as the ZC schemes later. As same as quantum operation teleportation, QOS can be viewed as a remote control (encryption, decryption or destruction) on quantum information in the future quantum network. If the target state as an important quantum information is initially encrypted with a given unitary operation, then the inverse operation on it is obviously the decryption on it. Moreover, it is natural to regard a random operation as the destruction on it. As a consequence, quantum operation sharing can be taken as a key to activate some important actions in the future lives by some sharer entities, such as missile emissions, quantum collective seal or unseam, remote joint destruction of quantum money, and so on. Therefore, recently this topic has already attracted some attentions [27] [28] [29] . In spite of this, it is worthy emphasizing that all these works treat only the sharing of single-qubit operations with shared qubit entanglements.. In the intending quantum networks, various quantum states involving high-dimensional qudit state might be employed and distributed among different nodes due to some special demands, such as peculiar quantum tasks in some concrete quantum scenarios, some definite security requirements, and so on. Because of this, some researchers have been attracted by the issues related to high-dimensional qudit cases and started to explore them [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] . Surely, their works includes the employment of qutrit entanglements in addressing various relative quantum problems [38] . In this condition, it is intriguing to ask what will happen in the field of QOS if the accessible quantum channels are composed of shared qutrit entanglements? Specifically, what kind of qutrit states can be used to fulfill QOS? If quantum channels are determined, what is the maximal success probability for sharing a given operation? Can the operation be shared more economically with less operation complexity? Can operations be classified with different classes corresponding to different success sharing probabilities? How to characterize those classes and find out them? and so on and so forth. To our best knowledge these mentioned issues have not been touched by far. Hence, it is of interest to consider the generalization of the ZC schemes [26] from the aspect of particle degree. Since there exist many open questions, in this paper we will consider a comparatively simple generalization, i.e., generalize the simplest QOS schemes (i.e., the ZC schemes) to the qutrit ones by employing the qutrit Bell and GHZ states as quantum channels. Such extensions is actually a little intricate but will lead to quite abundant results. More importantly, they indicate that QOS as remote control can be achieved with shared qutrit entanglement, too. We will show them later. Additionally, at present it is broadly recognized that resource consumption and operation complexity as hot topics in many fields are always concentrated. How to consume less amount of resources and how to degrade the difficulty and intensity of necessary operations are continuously attracting much attention and pursued by many researchers. Because of this, in this paper we will focus our attention on this issue during the generalizations, too.. The rest of this paper is organized as follows. In section 2, two three-party schemes for sharing single-qutrit operations on a remote qutrit in any state are preciously proposed with different quantum channels. In section 3, two schemes are amply compared and discussed from the four aspects of quantum and classical resource consumption, necessary-operation complexity, success probability and efficiency. Finally, a concise summary is made in section 4. To summarize, in this paper we have actually considered the shared remote control in the form of quantum operation sharing on a single-qutrit state. By integrating the ideas of quantum operation teleportation and quantum secret sharing, we have presented two possible schemes. The first scheme is universally applicable for any arbitrary single-qutrit unitary operation. The second scheme is generally a probabilistic one. However, after intensive investigations we have found that, if the operation U in question is known to belong to some restricted sets, both the scheme success probability and its efficiency can be doubled or even tripled. We have concretely compared the schemes in different cases from the four aspects of quantum and classical resource consumption, necessary-operation complexity, success probability and efficiency. In the tripled case the latter scheme becomes a deterministic one and is more optimal than the general scheme.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O B I I B I I B I I I I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O


Configuration prefetch for single context reconfigurable coprocessors Current reconjigurable systems sufferfrom a significaizt overhead due to the time it takes to reconfigure their hardware. In order to deal with this overhead, and increase the power of reconjigurable systems, it is important to develop hardware and software systems to reduce or eliminate this delay. In this paper we propose-one technique for significantly reducing the reconfiguration latency: the prefetching of configurations. By loading a configuration into the reconfigurable logic in advance of when it is needed, we can overlap the reconfiguration with useful computation.We demonstrate the power of this technique, and propose an algorithm for automatically adding prefetch operations into reconfigurable applications.This results in a significant &crease in the reconfiguration overheadfor these applications. When FPGAs were first introduced in the mid 1980s they were viewed as a technology for replacing standard gate arrays for some applications. In these first generation systems, a single configuration is created for the FPGA, and this configuration is the only one loaded into the FPGA. A second generation soon followed, with FpGAs that could use multiple configurations, but reconfiguration was done relatively infrequently [Hauck97a] . In such systems, the time to reconfigure the FPGA was of little concern Many of the most exciting applications being developed with FPGAs today involve run-time reconfiguration [Hauck97a] . In such systems the configuration of the FF'GAs may change multiple times in the course of a computation, reusing the silicon resources for several different parts of a computation. Such systems have the potential to make more effective use of the chip resources than even standard ASICs, where fixed hardware may only be used in a portion of the computation. However, the advantages of run-time reconfiguration do not come without a cost. By requiring multiple reconfigurations to complete a Permission to make d@taVhvd copies of all or part of this material for pemnal or classroom use is mted without fee provided that the copies are not made or distributed for profit or commercial advantage, tic copyn'& notice, the tide ofthe publication and its date appear, and not& is &'a that mpy@ht is by permission of the ACM, Inc. To copy otherv&e, to republish, to post on SOWS or to redistribute to lists, requires specific permission and/or fee FF'GA 98 Monterey CA USA Copyright 1998 ACM O-89791-978-5/98/ 01..$5.00 computation, the time it takes to reconfigure the FPGA becomes a significant concern. In most systems the FFGA must sit idlewhile it is being reconfigured, wasting cycles that could otherwise be performing useful work. For example, applications on the DISC and DISC II system have spent 25% [Withlin96] to 71% [wirthlin95] of their execution time performing reconfiguration.. It is obvious from these overhead numbers that reductions in the amount of cycles wasted to reconfiguration delays can have a significant impact on the performance of run-time reconfigured systems. For example, if an application spends 50% of its time in reconfiguration, and we were somehow able to reduce the overhead.per reconfiguration by a factor of 2, we would reduce the application's runtime by at least 25%. In fact, the performance improvement could be even higher than this. Specifically, consider the case of an FPGA used in conjunction with a host processor, with only the most time-critical portions of the code mapped into reconfigurable logic. An application developed for such a system with a given reconfiguration delay may be unable to take advantage of some optimizations because the speedups of the added functionality are outweighed by the additional reconfiguration delay required to load the functionality into the FPGA. However, if we can reduce the reconfiguration delay, more of the logic might profitably be mapped into the reconfigurable logic, providing an even greater performance improvement. For example, in the UCLA ATR work the system wastes more than 75% of its cycles in reconfiguration [Villasenor96, Villasenor971 . This overhead has limited the optimizations explored with the algorithm, since performance optimizations to the computation cycles will yield only limited improvement in the overall runtimes. This has kept the researchers from using higher performance FPGA families and other optimizations which can significantly reduce the computation cycles required.. Because of the potential for improving the performance of reconfigurable systems, developing techniques for reducing the reconfiguration delay is an important research area. In this paper we consider one method for reducing this overhead: the overlapping of computation with reconfiguration via the prefetching of FPGA configurations. In this paper we have introduced the concept of configuration prefetch for reconfigurable systems. By adding instructions into the code of an application, configurations for a reconfigurable coprocessor can be loaded in advance. This allows the overlapping of computation and reconfiguration, reducing the reconfiguration overhead of reconfigurable systems. We have also developed an algorithm which can automatically determine the placement of these prefetch operations, avoiding burdening the user with the potentially difficult task of placing these operations by hand. Finally, we have developed the Optimal Prefetch technique, which provides a bound on the potential improvement realizable via configuration prefetch. The results indicate that these techniques can reduce the reconfiguration overhead of reconfigurable systems by more than a factor of two, which will have a direct impact on the performance of reconfigurable systems.. We believe that such techniques will become even more critical for more advanced recontigurable systems. In order to investigate the impact of prefetching on the reconfiguration overhead in reconfigurable systems, we have tested prefetching on some standard software benchmarks from the SPEC benchmark suite [Spec95] . Note that these applications 67 have not been optimized for reconfigurable systems, and may not be as accurate in predicting exact performance as would real applications for reconfigurable systems. However, such real applications are not in general available for experimentation. Also, applications of reconfigurable systems are tailored to a specific system, and can be carefully optimized in reaction to a specific reconfiguration overhead. These applications may change significantly if they were mapped to a system with a much higher or lower reconfiguration delay, with different portions of the source code mapped to thereconfigurable logic. Thus, we feel that the only feasible way to investigate optimizations to the reconfiguration system is to use current, general-purpose applications, and make reasonable assumptions in order to mimic the structure of future reconfigurable system. In order to conduct these experiments, we must perform three steps. First, some method must be developed to choose which portions of the software algorithms should be mapped to the reconfigurable coprocessor. Second, a prefetch algorithm must be developed to automatically insert prefetch operations into the source code. Third, a simulator of the reconfigurable system must be employed to measure the performance of these applications. Each of these three steps will be described in paragraphs that follow.. The first step in the experiments is to choose which portions of the source code should be mapped to the reconfigurable coprocessor (these mappings will be referred to as RFUOPs here). As mentioned before, in this paper we will assume that arbitrary code sequences can be mapped to the reconfigurable logic as long as they have a single entry and a single exit point, and have no backward branches or jumps. This ensures that only combinational code sequences are considered. This is a somewhat conservative assumption, since in many reconfigurable systems it is possible to implement loops and other sequential control flow operations in the reconfigurable logic.. One complexity in deciding which portions of the source code should be mapped to the reconfigurable logic is to find that set of mappings that provide the best performance improvement in the face of a potentially substantial delay for each reconfiguration. In general this is a complex problem, and one we do not attempt to solve here. Our solution is to simply find all potential mappings to the reconfigurable logic, and then simulate the impact of including each candidate. This is done by repeatedly calling the reconfigurable system simulator, and assuming optimal prefetching (both of which are described later in this paper). Our algorithm then greedily chooses the candidate which provides the best performance improvement, and retests the remaining candidates. These retests examine the impact of including any one candidate in with the already chosen candidates. This repeats until the simulator determines that there is at most a potential 1% improvement available in the remaining candidates. In this way a reasonable set of RFUOPs can be developed which produces a significant performance improvement even when reconfiguration delay is taken into consideration. The result of this operation is to create a file that specifies which portion of the source executable should be mapped into RFUOPs, and which can be given to the simulator to compute the delays seen in the target reconfigurable system.. The simulator we have developed takes in an executable for a Sun SPARCstation. a specification of the location of RFUOPs and PREFETCH instructions in the executable, and a parameter that specifies the number of cycles it takes to reconfigure the coprocessor. This simulator is developed from the SHADE simulator [Cmelik93a]. This allows us to track the cycle-by-cycle operation of the system, and get exact cycle counts. Note that only one program can be executed at a time, and operating system calls are not instrumented, so context switch effects and the potential to overlap reconfiguration with cycles in the operating system are not considered. This simulator reports the reconfiguration time and overall performance for the application under both normal and optimal prefetching, as well as performance assuming no prefetching occurs at all. These numbers are used to measure the impact of the various prefetching techniques.. Note that for simplicity we model reconfiguration costs as a single delay constant. Issues such as latency verses bandwidth in the reconfiguration system, conflicts between configuration load and other memory accesses in systems which use a single memory port, and other concerns are ignored. Such effects can be considered to simply increase the average delay for each configuration load, and thus should not significantly impact the accuracy of the results. We consider a very wide range of reconfiguration overheads, from 10 cycles to 10,000 per reconfiguration. This delay range should cover most systems that are likely to be constructed, including the very long delays found in current systems, as well as very short delays that might be achieved by future highly cached architectures.. The remaining component of the experimental setup is the prefetch insertion program. This algorithm decides where prefetch instructions should be inserted into the executable given the set of RFUOPs determined by the RFU picker. The specific algorithm used will be described in a later section. The prefetch insertion program takes in the specification of RFUOPs from the RFU picker, as well as a control flow graph for the executable, and produces a file for the simulator that specifies the PREFETCH locations. Note that in a production system both the RFU picker and the prefetch insertion program would directly modify the executable. However, in order to allow us to use a standard processor simulator to simulate the reconfigurable system this information is maintained in a separate file.

O O O O B O O O O B O O O O O O O O O O O O O B O B O O O O O O O O B O O B O O O O O B O B B O O O O O B O O B O O O O O O O O B O B O O O O O O O O O B O O O O O O O O O O O O O O O B O B O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O B O O O O O O O B O O O O O O O O O O O B O O O O B O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O B O O O O O B O O O O O O B O O O B O O O B O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O B O O B O O O B O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O B O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O