Natural person-following behavior for social robots We are developing robots with socially appropriate spatial skills not only to travel around or near people, but also to accompany people side-by-side. As a step toward this goal, we are investigating the social perceptions of a robot's movement as it follows behind a person. This paper discusses our laser-based person-tracking method and two different approaches to person-following: direction-following and path-following. While both algorithms have similar characteristics in terms of tracking performance and following distances, participants in a pilot study rated the direction-following behavior as significantly more humanlike and natural than the path-following behavior. We argue that the path-following method may still be more appropriate in some situations, and we propose that the ideal personfollowing behavior may be a hybrid approach, with the robot automatically selecting which method to use. Robots that operate around people in the real world need to move in coherent, easily-understood ways, so that they will not startle or harm the people around them. In particular, for robots that operate in hospitals or in nursing homes [5, 11] , we argue that such robots need to move in ways that are socially acceptable to people-particularly to Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. HRI'07, March 8-11, 2007 , Arlington, Virginia, USA. people who are not roboticists. Moving in easily understood and predictable ways will both improve people's trust in and comfort with the robot as well as help insure the safety of people moving near the robot.. Our overall goal is to develop robots with spatial social skills-robots that navigate in ways that people understand and expect. We are interested in developing robots that can operate as social assistants, such as encouraging nursing home residents to participate in social activities and accompanying them to common areas. Such a robot cannot simply drive toward a goal and expect the person to follow it; rather, it needs to engage the person and accompany him as a human would, traveling side-by-side with the person. Many social conventions exist for side-by-side accompaniment, and a robot must be able to obey these conventions. When two people walk next to each other, they may use cues such as utterances ("let's turn here"), gestures, and gross body movement into and away from each other's personal space. A robot also needs to offer and respond to these social cues, both verbal and non-verbal. In addition, when traveling next to a person, a robot must behave appropriately at bottlenecks, such as doorways or crowded hallways, when the robot may need to move ahead of, or follow be-hind, the person. Furthermore, the robot's behavior may need to be significantly different depending on the social situation, as side-by-side travel may take the form of purely social travel or of escorting with the robot either leading or following the person.. As a first step toward developing robots that can accompany people in socially acceptable ways, we are investigating social perceptions of a robot's movement as it follows behind a person (Figure 1) , as a social assistant robot might when passing through doorways or navigating around obstacles. We have designed and tested two modes of person-following to determine which is more natural and socially acceptable. Participants in a pilot study agreed that the robot's behavior was more human-like when the robot always drove in the direction of the person, rather than when it followed the person's exact path. Thus, the "direction-following" method may be the more socially acceptable behavior for the robot to use. However, one can imagine situations when the path-following behavior may be more appropriate for the robot, such as in cluttered or unsafe environments where the robot's sensors may not be as reliable. We therefore propose a hybrid person-following model, and suggest that the robot may be able to learn the appropriate situations in which to use each method. Finally, we provide a discussion of what steps must be taken to move from a person-following robot to a robot that is capable of socially accompanying people side-by-side. In the work presented here, the robot has little to no understanding of the social cues that a person might use when leading a robot. That is, the robot responds only to the person's speed and location, ignoring aspects such as the person's identity or personality, spoken or gestured commands from the person, how and when to engage the person, appropriateness of behavior for the task (such as which person-following behavior to use when), or how to respond to other people in the vicinity.. This lack of understanding will only be amplified in a robot that accompanies a person side-by-side, as we are developing. Unfortunately, all of these aspects are currently open problems in the field of human-robot interaction. At a minimum, we intend to address the following in our future work:. • Better person-tracking. A sideways-facing laser rangefinder will allow the person-tracker described above to track people-shaped objects next to the robot, and we are currently investigating various laser configurations. To identify people, however, the tracker will likely need to be combined with a vision-based system for recognizing a person's face or clothing or by requiring the person to wear a visual fiducial or electronic tag.. • Generate appropriate social cues. We intend to study how people walk in pairs in order to gain a better understanding of what conventions people use when walking together, including interpersonal distance, use of gestures, and handling of obstacles and bottlenecks.. • Respond appropriately to unexpected behavior. If the person stops suddenly or turns unexpectedly, the robot should be able to react appropriately. This may require the robot to replan its path to follow the person or to interact with the person to determine the cause of his or her behavior.. Developing such social competencies will be a necessary step in allowing our person-following robot to travel sideby-side with a person in a socially acceptable manner. We have presented a laser-based person-tracker and have described two different methods for a robot to follow a person: direction-following and path-following. We analyzed both behaviors and found that, while their quantitative performances are very similar, they are perceived differently by people. Specifically, observers of the robot's behaviors felt that the direction-following behavior appeared more natural and human-like, and more closely matched their expectations.. We believe that this study is a first step toward socially assistive robots that are capable of traveling with people in socially acceptable ways. We have shown that people's expectations for the robot were that it would move in a humanlike manner, following human social conventions. We intend to expand this research by studying further both how people walk in pairs and how people expect robots to navigate with people. Specifically, we plan to perform observational studies of how people walk together, develop both the hybrid person-following model described above and a model for side-by-side accompaniment, and perform extensive field testing with the robot. By developing robots that are capable of following human social conventions, we believe that we are making great strides toward robots that can function as social assistants for the elderly. Quantitatively, the two methods of person-following are equivalent; the behaviors do not differ in laser tracking performance, and both allow the robot to follow smoothly behind a person. The primary difference between the two behaviors occurs at corners, when the direction-following behavior causes the robot to curve much more gently than the person, given sufficient space to do so. If the following occurs in narrow corridors, the differences between the behaviors lessens, as obstacle avoidance constrains the robot's movement.. Qualitatively, however, people indicate that the directionfollowing behavior is significantly more human-like and more closely matches their expectations than when the robot follows the person's path. Several participants commented that, when performing path-following, the robot did not appear to react "quickly enough" to the person's turns (since the robot turned at the location where the person turned, rather than at the same time as the person), which may help explain this finding.. To date, we have performed only the small pilot study as described above. Obviously, many caveats apply to this sort of study, as the participants were all familiar with robots and most likely had very different expectations of robotic behavior than non-roboticists. However, while the participant population may have influenced the exact values of the survey questions, we expect that the relative differences between the different robot behaviors would be similar across other populations, as well. A further shortcoming of this study is that it analyzed only people's third-person observations of the robot's behaviors, and thus the results may not capture the full spectrum of people's preferences. However, while people's in situ experiences of a robot's behavior are clearly valuable, such testing is difficult to perform and evaluate when the robot's behavior occurs strictly behind-and thus out of sight of-the person. Four trials with each approach were run at relatively high speeds. Each approach was run for a total of about 30 minutes (5-10 minutes per trial) and covered a total traversal of over 1 kilometer, with average speeds of close to the maximum allowed speed of the robot, 70cm/s. On average, the robot was able to track the person over a distance of about 30 meters (1 minute) before an error in tracking occurred. Both approaches had a highest distance between tracking errors of over 160 meters (over 3.5 minutes). Using an analysis of variance (ANOVA), no significant difference was found between the two approaches in terms of the distance or time between tracking errors (distance F [1, 65] = 0.79, p = 0.3; time F [1, 65] = 0.27, p = 0.6). In all cases of tracker failure, the robot was able to reacquire the person within moments of the person re-approaching the robot.. The tracking performs considerably better when the robot moves more slowly. When the person traveled at speeds of about 45cm/s, the robot was able to follow for over 320 meters (over 10 minutes) before tracker failure, using each of the methods. During these slower traversals, the robot was able to remain on average approximately 1.3m (standard deviation 25cm) from the person. Again, both personfollowing behaviors performed equivalently.. Due to the within-subjects nature of this study, we analyzed the survey responses using paired t-tests across trials. Average responses and t-values for each question are given in Table 2 .. Although the two behaviors are very similar, participants noticed their differences. Participants were asked to rate the robot's behavior according to whether it met their expectations ("not at all" (1)-"very much" (7)) and how natural the behavior was ("not at all" (1)-"human-like" (7)). As shown in Table 2 , participants rated the robot's behavior as significantly more natural and human-like in the direction-following condition. In addition, participants felt that the direction-following robot behaved more according to their expectations. Notably, none of the participants rated the path-following behavior as better than the direction-following behavior on either of these questions. Furthermore, the answers to these two questions were highly correlated (r = 0.80, p < 0.0001), indicating that participants expected the robot's behavior to be human-like, despite the robot's non-anthropomorphic physical shape.. Participants were also asked whether the robot followed and stopped at appropriate distances from the experimenter. They rated the distances on a scale of 1 ("too far away") to 7 ("too close"). Overall, participants felt that the robot stayed a little too far away from the experimenter while moving (overall mean 2.95, SD 1.10), but stopped at an appropriate distance (overall mean 5.15, SD 1.46). There were no significant differences in participants answers across the two person-following behaviors. Person-tracking is currently an active area of research in robotics, and many person-trackers are demonstrated by having a robot follow the person being tracked. One of the more common methods of tracking people is to use a camera for face or color blob detection or for contour tracking [16] . Any such camera-based method, however, is prone to suffer from varying background colors and illumination conditions as a robot moves through various environments. Montemerlo and colleagues developed a laser-based persontracker for mobile robots [12] ; however, their tracker requires the robot to have an a priori map of the environment, and will thus not work in a new or highly dynamic environment. Kluge et al demonstrated a method of object-tracking with a laser range-finder using network optimization techniques, but this method fails if any occlusions occur [9] . Several researchers have investigated combining camera-based and laser-based tracking methods, typically using a laser to find legs and a camera either to detect faces [8] or to track other visual regions of interest [10] . Using face detection, however, requires that the person always face the robot, and is thus awkward for the robot to follow behind or even walk next to a person. A final method of person-tracking requires attaching a tracking device to the person [2] . However, for such sensors to provide accurate location information, they are often large and therefore too awkward and intrusive to be worn. Our tracking approach is most similar to that of Topp et al [20] , who use a single laser rangefinder and sample-based joint probabilistic data association filters to identify and track people [17] .. In contrast with the above-mentioned literature, which focuses primarily on the technology of sensing and tracking a person, we are interested in people's perceptions of the robot's person-following behavior. Our research seeks to combine the technological aspects with social psychol- ogy and interaction design, to develop robots that can accompany people in socially acceptable ways. Even though some work has been done toward a robot that can navigate safely side-by-side with a person [15] , such work also accounts only for the algorithms involved and does not consider human perception of the robot's behavior. In contrast, work that has focused on the social nature of navigation includes passing people in a hallway [14] and standing in line [13] in socially acceptable ways. Each of these studies has provided insight into people's perceptions of robotic behavior. We provide similar insight into people's perceptions of the specific behavior of person-following, and we hope to expand this knowledge into the general domain of spatial social human-robot interaction.

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 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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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


Regularized locality preserving indexing via spectral regression We consider the problem of document indexing and representation. Recently, Locality Preserving Indexing (LPI) was proposed for learning a compact document subspace. Different from Latent Semantic Indexing (LSI) which is optimal in the sense of global Euclidean structure, LPI is optimal in the sense of local manifold structure. However, LPI is not efficient in time and memory which makes it difficult to be applied to very large data set. Specifically, the computation of LPI involves eigen-decompositions of two dense matrices which is expensive. In this paper, we propose a new algorithm called Regularized Locality Preserving Indexing (RLPI). Benefit from recent progresses on spectral graph analysis, we cast the original LPI algorithm into a regression framework which enable us to avoid eigen-decomposition of dense matrices. Also, with the regression based framework, different kinds of regularizers can be naturally incorporated into our algorithm which makes it more flexible. Extensive experimental results show that RLPI obtains similar or better results comparing to LPI and it is significantly faster, which makes it an efficient and effective data preprocessing method for large scale text clustering, classification and retrieval. Document representation and indexing have been a key problem for document analysis and processing, such as clustering, classification and retrieval [8] , [12] , [23] . If we denote by document space the set of all the documents, different indexing algorithms see different structures of the document space. The Vector Space Model (VSM) might be one of the most popular model for document representation. Each document is represented as a bag of words. Correspondingly, the document space is associated with a Euclidean structure and the inner product (or, cosine similarity) is used as the standard similarity measure for documents. Unfortunately, VSM suffers from some problems such as synonymy and polysemy.. To deal with these problems, Latent Semantic Indexing (LSI) was proposed [8] . Given a term-document matrix X, LSI applies Singular Value Decomposition (SVD) to project the document vectors into a subspace so that cosine similarity can accurately represent semantic similarity. LSI aims to find the best subspace approximation to the original document space in the sense of minimizing the global reconstruction error. The basis functions obtained by LSI are the eigenvectors of the matrix XX T . It would be important to note that LSI is different from Principal Component Analysis (PCA) We have proposed a new algorithm for document indexing and representation, called Regularized Locality Preserving Indexing. RLPI shares the same locality preserving character as LPI while avoids the expensive computation. Specifically, RLPI can be computed by a sparse matrix eigen-decomposition followed with a regularized least squares. Moreover, with a specific graph design in supervised case, the eigen-decomposition becomes trivial and RLPI only needs to solve a set of regularized least squares problems. Such property makes RLPI can be efficiently computed even for a large scale data set. Extensive experiments on TDT2 and 20News-groups datasets demonstrated the effectiveness and efficiency of RLPI.. There is a parameter α which controls the smoothness of the basis functions of RLPI. Our theoretical analysis shows that RLPI gives the same solution of LPI when α decreases to zero. For supervised learning, when the training set is small, LPI tends to over-fit the training data. In such case, a smoother (with small norm) basis function is usually preferred and RLPI with α > 0 can have better performance than LPI. However, it remains unclear that how to automatically estimate the best α. Besides clustering the whole data set into 30 clusters, the evaluations were also conducted with different number of clusters, ranging from 2 to 10. For each given cluster number c, 50 tests were conducted on different randomly chosen categories, and the average performance was computed over these 50 tests (except the 30 cluster case). For each test, K-means algorithm was applied 10 times with different start points and the best result in terms of the objective function of K-means was recorded. After LSI, LPI, or RLPI, how to determine the dimensions of the subspace is still an open problem. In this experiment, we keep c dimensions for all the three algorithms as suggested by previous study [4] . Table 2 and Figure 1 show the average accuracy and average mutual information of the six algorithms. LSI seems not promising in dimension reduction for clustering because the K-means on the LSI subspace is even worse than K-means on the original document space. One may iterate all the possible dimensions for better performance of LSI as suggested in [4] . However, it may not possible to do so in a real case. Clustering using PLSI is even worse. NMF-NCW method achieves better performance than Baseline which is consistent with previous study [23] , [4] . Both LPI and RLPI achieve significant improvements over other four algorithms. The reason is that LPI and RLPI try to reveal the local geometric structure of document space. More detailed analysis and experiments of document clustering using LPI are provided in [4] . Table 3 and Figure 2 show the processing time of the six algorithms. The processing time of LSI, LPI and RLPI include two parts: dimensionality reduction time and time of K-means on the reduced subspace. The processing time of Baseline and NMF-NCW methods are simply the time of clustering approaches (Kmeans and nonnegative matrix factorization). PLSI estimate the probability of each document belongs to each cluster, which can be directly used to infer the clustering result. Thus, the processing time of PLSI is only the dimensionality reduction (model estimation) time. After dimensionality reduction of LSI (LPI and RLPI), K-means is performed in a very low dimensional subspace thus is much more efficient then K-means in the original document space. The results here further show the advantage of dimensionality reduction for clustering. Clustering based on LSI is the most efficient approach. However, the low clustering accuracy makes LSI approach less attractable. Although the NMF-NCW method achieves better performance than Baseline method, the high computational cost (NMF-NCW spent more than 4 hours for clustering 9,394 documents into 30 classes!) makes it not applicable on large document set. The same shortcoming exists for LPI approach. It can not be applied with 9,394 documents due to the memory limit. Consider both accuracy and efficiency, RLPI is obviously the best among the six compared algorithms for document clustering.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 I 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 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


Interference alignment-based sum capacity bounds for random dense Gaussian interference networks We consider a dense K user Gaussian interference network formed by paired transmitters and receivers placed independently at random in a fixed spatial region. Under natural conditions on the node position distributions and signal attenuation, we prove convergence in probability of the average per-user capacity C Σ /K to 1 2 E log(1+2SNR). The achievability result follows directly from results based on an interference alignment scheme presented in recent work of Nazer et al. Our main contribution comes through an upper bound, motivated by ideas of 'bottleneck capacity' developed in recent work of Jafar. By controlling the physical location of transmitter-receiver pairs, we can match a large proportion of these pairs to form so-called ǫ-bottleneck links, with consequent control of the sum capacity. In this paper, we have deduced sharp bounds for the sum capacity of a Gaussian interference network. Our main contribution comes through the upper bound, which uses arguments based on controlling the position of pairs of vertices to match them into bottleneck states. Although our main result is proved under the assumption of deterministic fading, with signal strength decaying as a function of distance, in Section 4.2 we describe an extension to a model with random fading.

O O B I 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 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 B I 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 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


Hyperboloidal layers for hyperbolic equations on unbounded domains We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques. Hyperbolic equations typically admit wavelike solutions that oscillate infinitely many times in an unbounded domain. Take a plane wave in one spatial dimension with frequency ω and wave number k, u(x, t) = e 2πi(kx−ωt) .. Any mapping of such an oscillatory solution from an infinite domain to a finite domain results in infinitely many oscillations near the domain boundary, which can not be resolved numerically. We refer to this phenomenon as the compactification problem [1] . It is commonly stated that hyperbolic partial differential equations are not compatible with compactification, and therefore can not be solved on unbounded domains accurately. A suitable transformation of the time coordinate, however, leads to a finite number of oscillations in an infinite spatial domain. Introduce. where C is a finite, positive constant. The plane wave (1) becomes u(x, τ) = e −2πi(kC/(1+x)+ωτ) .. This representation of the plane wave has only k C cycles along a constant time hypersurface in the unbounded space x ∈ [0, ∞), and is therefore compatible with compactification.. The simple idea just described has far reaching consequences. In numerical calculations of hyperbolic equations one typically truncates the unbounded solution domain by introducing an artificial outer boundary that is not part of the original problem. Boundary conditions, called transparent, absorbing, radiative, or nonreflecting, are constructed to simulate transparency of this artificial outer boundary. There has been significant developments in the treatment of artificial outer boundaries since the 70s, but there is no consensus on an optimal method [2, 3] . Especially the construction of boundary conditions for nonlinear problems is difficult [4] . A successful technique for numerical calculations on unbounded domains resolves this problem for suitable hyperbolic equations and provides direct quantitative access to asymptotic properties of solutions.. Furthermore, the numerical construction of oscillatory solutions as (3) can be very efficient. Numerical accuracy requirements for hyperbolic equations are typically given in terms of numbers of grid points per wavelength. In the example presented above, the free parameter C determines the number of cycles to be resolved, which can be chosen small. This suggests that high order numerical discretizations requiring a few points per wavelength can be very efficient in combination with time transformations of the type (2) .. The rest of the paper is devoted to the discussion of time transformation and compactification for hyperbolic equations. The theoretical part of the paper (sections 2 and 3) includes a detailed description of the method. We discuss the compactification problem (section 2.1) and its resolution (section 2.2) for the advection equation in one dimension. In section 2.3 we discuss the wave equation with incoming and outgoing characteristics. We show that the method works also for systems of equations (section 2.4). Hyperboloidal layers are introduced in section 2.5 in analogy to absorbing layers. In multiple spatial dimensions, compactification is performed in the outgoing direction in combination with rescaling to take care of the asymptotic behavior (sections 3.1 and 3.2). The layer strategy in multiple dimensions allows us to employ arbitrary coordinates in an inner domain, where sources or scatterers with irregular shapes may be present (section 3.3). We finish the theoretical part discussing possible generalizations of the method to nonspherical coordinate systems (section 3.4). Section 4 includes numerical experiments in one and three spatial dimensions. In one dimension, we solve the Maxwell equations using finite difference methods (section 4.1). A stringent test of the method is the evolution of off-centered initial data for the wave equation in three spatial dimensions with and without nonlinear source terms (section 4.2). We conclude with a discussion and an outlook in section 5. Hyperboloidal compactification provides a clean solution to the artificial outer boundary problem. It allows us to solve suitable hyperbolic equations accurately on unbounded domains. The underlying idea originates in studies of asymptotic structure of spacetimes in general relativity [5, 7] . It is remarkable that such a practical method is available at the interface between differential geometry, conformal structure, partial differential equations, and numerical analysis.. With the hyperboloidal layer method, in which compactification is restricted to a layer, we can apply arbitrary coordinates in an interior domain, and we gain direct quantitative access to the asymptotic solution. The numerical boundary of the simulation domain corresponds to infinity, therefore no outer boundary conditions are needed. The hyperboloidal time transformation leads to a non-vanishing coordinate speed for outgoing characteristics up to and including infinity in compactifying coordinates.. Some further advantages of the hyperboloidal method are listed as follows. Efficiency of numerical calculations improves by suitable choices of free parameters that modify the wavelength of outgoing radiation. It is straightforward to apply the method to various covariant hyperbolic equations, such as wave, Maxwell, or Yang-Mills equations. It is not necessary to calculate boundary data that typically depend on the particular equations. Including source terms and nonlinearities does not modify the boundary treatment if certain asymptotic fall-off conditions are satisfied. No separate error controlling at the outer boundary is needed because there are no errors beyond the numerical discretization. There is no overhead in software implementation of boundary routines. The method is largely independent of numerical schemes due to its geometric origin.. An important current limitation of the method is the requirement of a spherical or spheroidal grid near the outer boundary. It is not clear whether hyperboloidal compactification works with global Cartesian or cylindrical grid structures. Another limitation is source terms that lead to a singular compactification (Klein-Gordon equation, low power nonlinearities). The applicability of the method in even spatial dimensions, starting with two dimensional problems, should be studied.. The decision to apply hyperboloidal compactification can be made only after detailed comparative studies which go beyond the scope of this paper. A comparison of different boundary treatments (absorbing boundary conditions, perfectly matched layers, and hyperboloidal layers) within the same numerical setting on a simple example, such as the linear, homogeneous wave equation, would be useful. The method should be applied to Maxwell equations and its efficiency should be studied when scatterers with irregular shapes are present in the computational domain. Generalizations of the method, such as adapting the coordinate representation of the hyperboloidal layer to the shape of scatterers, might lead to very efficient computations.. The relevance of hyperboloidal compactification depends on the problem at hand. The method in its current form does not apply to non-covariant problems, however, it may be extended to deal with heterogeneous media, anisotropic elastic waves, optical waveguides, Euler equations, Navier-Stokes equations, among others.. The idea of boosting the time direction may find applications beyond compactification. It seems that transformations of the characteristic cone for hyperbolic equations are largely unexplored outside numerical relativity. The presented solution of the outer boundary problem may be just one application of such transformations. Others might be useful, for example in numerical studies of high frequency waves. The total energy is radiated to infinity leaving the zero solution behind in accordance with Huygens' principle for Maxwell equations in one dimensional flat spacetime. Therefore, a good measure of the quality of the boundary treatment is the value of the unknown after the initial wave packet leaves the simulation domain. This value is related to the numerical reflection coefficient of the boundary. The analytical reflection coefficients at the interfaces and at the boundaries are zero in the hyperboloidal method. Numerically, however, the interfaces and the finite differencing at the outer boundaries cause reflections. Consider the L 2 norm of the solution as a function of time plotted in figure 11 . The plots depict the field with respect to the hyperboloid foliation (left) and with respect to the layer (right). The errors are larger for the layer. Red, green, and blue curves represent solutions calculated with 4th, 6th, and 8th accurate finite difference operators, respectively. To each order the solution is calculated with and without dissipation. We observe that artificial dissipation strongly reduces numerical errors. Figure 11 shows that the error at late times with 4th order stencils with dissipation has the same order of magnitude as the one with 8th order stencils without dissipation. The main result from figures 10 and 11 is that the numerical error is reduced drastically by higher resolution, higher order discretization, and artificial dissipation-without changing the boundary treatment. The boundary treatment does not introduce errors into the solution that are put in by hand. To emphasize this point, we plot in figure 12 two solutions using the hyperboloid foliation and the hyperboloidal layer. For each method, one solution is obtained with 4th order finite differencing, without dissipation, and 100 grid cells (red curve); the other solution is obtained with 8th order finite differencing, with dissipation, and 200 grid cells (blue curve). The combined effect of these improvements demonstrates that accuracy with hyperboloidal compactification is not restricted by the boundary treatment but by the numerical accuracy in the simulation domain. . Spatial truncation errors converge exponentially in a pseudospectral code. We show such spectral convergence in figure 14 for a solution of the homogeneous linear wave equation with off-centered initial data. We calculate convergence in the auxiliary variable of the first order reduction v i := ∂ i v. Figure 14 shows that the numerical difference between v i and ∂ i v converges in the L 2 norm geometrically with N collocation points in each subdomain direction. The constraint error grows slowly in time but the evolution is stable. Convergence of the constraint error is not disturbed by the presence of the boundary located at ρ = 20. A stringent test for boundary methods is the inclusion of nonlinear terms in the equations. The boundary treatment for semilinear wave equations is a difficult problem, especially in three spatial dimensions [4] . Therefore, we present also results for the wave equation with a cubic source term (with p = 3 in (38)). The backscatter due to the self interaction of the field violates Huygens' principle which makes the solution more interesting. Transparency boundary conditions must not eliminate all reflections from the outer boundary but only spurious ones. This makes the treatment of the artificial boundary difficult.. The backscatter by the nonlinear term leads to a late time polynomial decay of the solution: as t −2 at a finite distance, as t −1 at infinity. It is difficult to obtain the correct decay rates with an artificial outer boundary. The decay rate at infinity is not even accessible with standard methods. Figure 15 shows that the rates are calculated accurately with the hyperboloidal method. The local decay rate is defined as d ln |v(ρ, τ)|/d ln τ. The invariance of the time direction under hyperboloidal transformations implies that the rates calculated with the hyperboloidal method are equivalent to the rates calculated with the untransformed equations. Any numerical method fails after a certain time due to accumulation of numerical errors. In figure 15 the bottom curve that corresponds to the fastest decay bends up after about τ = 500. This behavior is expected, and is due to accumulated numerical errors. We can delay its appearance by increasing the numerical resolution of the simulation. The long time accuracy of the solution is not limited by the boundary treatment, but solely by the accuracy of the interior calculation.

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


Statistical and Information-Theoretic Analysis of Resolution in Imaging Abstract-In this paper, some detection-theoretic, estimation-theoretic, and information-theoretic methods are investigated to analyze the problem of determining resolution limits in imaging systems. The canonical problem of interest is formulated based on a model of the blurred image of two closely spaced point sources of unknown brightness. To quantify a measure of resolution in statistical terms, the following question is addressed: "What is the minimum detectable separation between two point sources at a given signal-to-noise ratio (SNR), and for prespecified probabilities of detection and false alarm ( and )?". Furthermore, asymptotic performance analysis for the estimation of the unknown parameters is carried out using the Cramér-Rao bound. Although similar approaches to this problem (for one-dimensional (1-D) and oversampled signals) have been presented in the past, the analyzes presented in this paper are carried out for the general two-dimensional (2-D) model and general sampling scheme. In particular the case of under-Nyquist (aliased) images is studied. Furthermore, the Kullback-Liebler distance is derived to further confirm the earlier results and to establish a link between the detection-theoretic approach and Fisher information. To study the effects of variation in point spread function (PSF) and model mismatch, a perturbation analysis of the detection problem is presented as well. T HE problem of resolution historically has been of significant interest in different communities in science and engineering, for example in astronomy, optics, different applications in physics, array processing and imaging. In this paper, we focus on the problem of achievable resolution in imaging practice for the following reasons. First, the problem of resolving point sources can be considered as a canonical case study to investigate the performance of imaging systems and image restoration techniques. Second, developing different techniques to resolve point sources is indeed a concern in real-world application, e.g., astronomy. Our approach to this problem in this paper is based on statistical detection and estimation theory, since we assume the measured signal from which the sources are to be resolved, is noise-corrupted.. To begin, let the original image of interest consist of two impulse functions positioned at points and with a small distance of apart ( Fig. 1 ). That is, the signal model is (1) We consider the following two-dimensional (2-D) model for the measured (discrete) signal (2) where is the blurring kernel 1 (representing the overall point spread function (PSF) of the imaging system) and is assumed to be a zero-mean Gaussian white noise 2 process with variance .. According to the classical Rayleigh criterion, two point sources are "barely resolvable" when the central peak of the pattern generated by one point source falls exactly on the first zero of the pattern generated by the second one [6] . See Fig. 2 for a depiction. But in fact under certain conditions related to signal-to-noise ratio (SNR), resolution beyond the Rayleigh limit is indeed possible. This can be called the "super-resolution" limit.. The statistical analysis presented in this paper is formulated based on the ability to distinguish whether the measured image is generated by one point source or two point sources. Specifically, for the proposed model the equivalent question is whether the parameter is equal to zero or not. If , then we only have one peak and if then there are two well resolved peaks according to the Rayleigh criterion. 3 So the problem of interest revolves around values of in the range of . This can be posed as a hypothesis testing problem, i.e., One point source Two point sources (3) or equivalently (see (4) at the bottom of the page). The problem of resolution from the statistical viewpoint has been well studied in the past [25] , [2] , [1] , [7] , [9] . The majority of researchers have used the Cramér-Rao bound to analyze the problem of resolution to study the mean-square error of unbiased estimators for the distance between the sources [8] , [9] , [1] , [21] , [22] , [13] , [16] . In addition, some papers [8] , [2] have considered a hypothesis testing approach to determine the resolvability of sources in some limited cases for one-dimensional signals.. As for novel contributions in this paper, we formulate the problem of interest as a composite hypothesis test and derive an explicit relationship between the minimum detectable separation and the required SNR for any sampling strategy including aliased (sub-Nyquist) images. As a result, a quantitative measure of resolution is obtained which can be used to understand the effect of model parameters (e.g., PSF and sampling parameters). The following question is addressed: what is the minimum separation between two point sources (maximum attainable resolution) that is detectable with high probability at a given SNR. Besides, general analytical results are derived for arbitrary sampling schemes. Our previous work has been focused on the case of one-dimensional (1-D) signals and over-Nyquist sampling [14] , [20] , [19] . In this paper, we present extensions of these results to 2-D and to the under-sampled (aliased) case (for a general PSF).. We put forward a solution to (4) for its most general case where all the parameters involved in the signal model are unknown to the detector. Furthermore, we consider two cases, where the value of noise variance is known and where it is unknown. For both cases we develop corresponding detection strategies and obtain the explicit relationship between SNR and the parameters in the model. Also, to further analyze the problem of resolution. such frameworks will be explained in the following. 3 Throughout this paper, we assume without loss of generality that d = 1 corresponds to the Rayleigh limit.. A significant question which has not been addressed in the past is to analyze the effect of a known or unknown perturbation of the PSF on the detection performance. Variation in PSF can be caused by other blurring elements in the system, for example an out-of-focus lens or atmospheric or underwater turbulence. We first address the problem of finding the change in the required SNR due to a variation of the PSF required for resolvability.. This will help us to analyze sensitivity to model inaccuracies.. In the interest of completeness, we also compute the Fisher Information matrix (and Cramér-Rao (CR) bound) in closed form for two different cases. The Cramér-Rao lower bound formulation is used to study the limits to attainable precision of estimated distances of the two point sources. We carry out this analysis for the case of under-sampled images and for a general PSF.. Another appealing and informative analysis is to compute the symmetric Kullback-Liebler Distance (KLD) or Divergence [12, p. 26] . KLD is a measure of discriminating power between two hypotheses, and is directly related to the performance figure of a related optimal detector. To accurately compute the KLD for the underlying problem, we make some essential extensions to the conventional formula of approximating the KLD. We shall see an interesting and important connection between the KLD analysis and the detection-theoretic analysis.. As for the application of the presented analysis in this paper, we shall see that the framework of addressing the fundamental relationship in resolving point sources (i.e., the tradeoff between SNR and resolvability) will also suggest a corresponding detector which can be well used in practice. In other words, the established performance expression is simply the result of applying the proposed local detector.. The organization of the paper is as follows. In Section II, we will present and formulate our detection-theoretic approach. The asymptotic performance of maximum likelihood estimate of the unknown parameters in terms of the Fisher Information Matrix will be discussed in Section III. In Section IV, we will study an information-theoretic criterion for the problem of interest. We will explain our perturbation and variational analysis for the underlying detection problem and discuss the effect of model mismatches in Section V. We will also discuss some of the practical applications and possible extensions of the proposed framework in Section VI. Finally, some comments and conclusion will be presented in Section VII. In this paper we investigated the problem of resolving two closely spaced point sources (beyond the Rayleigh limit) from noise-corrupted, blurred, and possibly undersampled images. We have studied three different frameworks to derive performance limits on minimum resolvability, namely, a detectiontheoretic approach, an estimation-theoretic analysis, and an information-theoretic approach. We have discussed the case of under-Nyquist sampling in several parts of this paper because of its significance in, for example, image super-resolution reconstruction [3] , [15] , [4] . To our knowledge, this paper presents the first direct analysis of resolution limits in imaging for undersampled images.. For the detection-theoretic approach, we have put forward a hypothesis testing framework and derived the relationship between the required SNR and the detectable distance between point sources. We explicitly found a general, fundamental, and informative relationship to quantify a (statistical) measure of resolution, and also to reveal the effect of point spread function and sampling parameters on resolvability. The established expressions are in general applicable to any point spread function and any arbitrary sampling scheme. The analysis for the undersampled images demonstrates explicitly how the performance is affected by the sampling phases (sampling offset), while in the over-Nyquist case the performance is independent from the sampling offset.. In the process of addressing the fundamental relationship between SNR and resolvability, we also proposed a corresponding detector which can be implemented in practice to estimate the locations and intensities of point sources from a degraded (noisy and blurred) image. As an example of implementing such a technique, one can first apply a running windowed energy detector to determine whether a pattern (generated by a point source) within the window is present or not. Then as a refinement step, the proposed detector will be applied to resolve the overlapping patterns originated by two closely-spaced point sources.. We studied the asymptotic performance of the ML estimate of the unknown parameters, using the Fisher information matrix. Deriving a lower bound on the variance of the parameter , in particular, was rather helpful in confirming the results of detection-theoretic analysis and also justifying the effect of estimation accuracy on the performance of the proposed detector. We also derived the symmetric Kullback-Liebler distance for the underlying problem by extending its standard form to higher order terms. This analysis provides an upper bound on the detection performance we have derived in Section II and also connects the Fisher information matrix with this performance bound.. As a practical matter, we presented a sensitivity/variational analysis to find the effect of known or unknown variation of PSF on the performance of the detector discussed in Section II. The results of this part are useful to compute the effect of a known variation of PSF on performance of an imaging system and to quantify the degradation of such a system due to imperfect knowledge of the model of the imaging system. Furthermore, we have briefly discussed some of the practical uses of the proposed methodology to other fields such as astronomical data analysis tasks and circuit inspection applications.. As for future directions, one open area is to extend this framework to a more complicated scenario where the PSF is partially known or is associated with some stochastic parameters. Another direction is to study the case of unknown motion between frames. Also, as mentioned before, the problem of resolving point sources is a canonical case study in imaging and image processing. We believe more general imaging problems can be treated with the same techniques employed in this paper, with proper extension (and complication) of the imaging models.. Resolution limit in photon-limited imaging systems is another important case study that needs to be investigated. The noise in these systems is no longer additive readout noise and follows the Poisson process. Also, in many cases the underlying image itself may contain different forms of noise and in particular clutter or unwanted interference which are also described by a probabilistic function.. Another challenging direction is to investigate the performance of the imaging systems with time-varying characteristics (for example, time-varying PSF particularly in astronomical applications). Such behavior may be modeled by some probabilistic descriptions as well. The analysis here requires a somewhat more sophisticated machinery.. We have briefly studied the resolution problem for the case where the measured signal is contaminated by Poisson noise (for example in photon-limited imaging) in [17] . This is an important case study in astronomical applications and SEM imaging.

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


Constraint functional logic programming over finite domains In this paper, we present our proposal to Constraint Functional Logic Programming over Finite Domains (CF LP (FD) ) with a lazy functional logic programming language which seamlessly embodies nite domain (FD) constraints. This proposal increases the expressiveness and power of constraint logic programming over nite domains (CLP (FD) ) by combining functional and relational notation, curried expressions, higher-order functions, patterns, partial applications, non-determinism, lazy evaluation, logical variables, types, domain variables, constraint composition, and nite domain constraints.We describe the syntax of the language, its type discipline, and its declarative and operational semantics. We also describe T OY(FD), an implementation for CF LP (FD) , and a comparison of our approach with respect to CLP (FD) from a programming point of view, showing the new features we introduce. And, nally, we show a performance analysis which demonstrates that our implementation is competitive with respect to existing CLP (FD) systems and that clearly outperforms the closer approach to CF LP (FD) . In this paper we have presented CF LP (FD), a functional logic programming approach to FD constraint solving. CF LP (FD) is not only a declarative alternative to CLP (FD) but also extends its capabilities with new characteristics unusual or not existing in CLP (FD) such as functional and curried notation, types, curried and higher-order functions (e.g., higher-order constraints), constraint composition, higher-order patterns, lazy evaluation and polymorphism, among others. As a consequence, CF LP (FD) provides better tools, when compared to CLP (FD), for a productive declarative programming as it implicitly enables more expressivity, due to the combination of functional, relational and curried notation as well as type system. Moreover, lazy evaluation allows the use of structures hard to manage in CLP (FD), such as in nite lists.. A CF LP (FD) language is also presented by describing its syntax, type discipline and both declarative and operational semantics. FD constraints are integrated as functions to make them rst-class citizens and allow their use in any place where a data can (e.g., as arguments of functions). This provides a powerful mechanism to de ne higher-order constraints.. We have also reported an implementation of the CF LP (FD) proposal which connects a F LP language to a FD constraint solver, that provides both lazy computation and FD constraint solving. The FD solver is required to hold termination, soundness and completeness properties. T OY(FD) is our implementation of the CF LP (FD) language previously described, that connects the functional logic language T OY to the e cient FD constraint solver of SICStus Prolog. The result is that T OY(FD) is a lazy functional logic system with support for FD constraint solving.. We have also explained the most important contributions by showing the extra capabilities of CF LP (FD) with respect to CLP (FD) . This comparison points out the main bene ts of integrating F LP and FD in a declarative language.. Moreover, we have also shown that constraint solving in T OY(FD) is fairly e cient as, in general, behaves closely to SICStus Prolog, which means that the wrapping of SICStus Prolog by T OY does not increase signi cantly the computation time. In addition, T OY(FD) clearly outperforms existing CLP (FD) systems such as SWI Prolog and Ciao Prolog and also is competitive with respect to GNU Prolog, one of the most e cient CLP (FD) systems. Furthermore, T OY(FD) is around one and three times faster than PAKCS, its closer CF LP (FD) implementation. Practical applications of T OY(FD) can be found in (Fern andez et al. 2002; Fern andez et al. 2003) .. Throughout the paper it should have been clear that one inherent advantage of the CF LP (FD) approach is that it enables to solve all the CLP (FD) applications as well as other problems closer to the functional setting.. We claim that the integration of FD constraints into a F LP language receive bene ts from both worlds, i.e., taking functions, higher-order patterns, partial applications, non-determinism, lazy evaluation, logical variables, and types from F LP and domain variables, constraints, and propagators from the FD constraint programming.. In addition, we claim that the idea of interfacing a F LP language and constraint solvers can be extended to other kind of interesting constraint systems, such as non-linear constraints, constraints over sets, or Boolean constraints, to name a few. Observe that T OY(FD) can be thought of as a constraint solving procedure integrated into a sophisticated, state-of-the-art execution mechanism for lazy narrowing. Operationally speaking, T OY(FD) compiles CF LP (FD)-programs into Prolog-programs in a system equipped with a constraint solver. This makes both lazy evaluation and constraint solving be inherent features of the system.. The proof of theorem 1 can be done distinguishing several cases from the declarative semantics of each primitive function symbol given in Table 1 and the requirements of each constraint solver rule or failure rule in Tables 2-5: Table 2 We examine for example the rst rule in Table 2 : seq t s !! R; S ' ' (t == s; S 1 1 ) _ (t n = s; S 2 2 ) with R = 2 , 1 = fR 7 ! trueg and 2 = fR 7 ! f alseg (the rest of rules in Table 2 are analogous). We prove that Sol F D (seq t s !! R; S ) = Sol F D (t == s; S 1 1 ) [ Sol F D (t n = s; S 2 2 ): All the benchmarks were performed on the same Linux machine (under Fedora Core system, 2.69-1667) with an Intel(R) Pentium 4 processor running at 2.40 GHz and with a RAM memory of 512 Mb. For the sake of brevity, we only provide the results for rst solution search. Table 9 shows the results using na ve labeling. The meaning for the columns is as follows. The rst column gives the name of the benchmark used in the comparison, and the next six columns show the running (elapsed) time (measured in milliseconds) to nd the rst answer of the benchmark for each system. Table 10 shows the results shown in Table 9 in terms of the speed-up of T OY(FD) with respect to the rest of the systems (that is, the result of dividing the time of a given system by the time of T OY(FD)). Table 11 shows the results of solving the same benchmarks by using rst-fail labeling. Note that the current versions of SWI Prolog and Ciao Prolog do not provide rst-fail labeling. Also, Table 12 shows the speed-up corresponding to the results in Table 11 and again displays the performance of T OY(FD) with respect to the rest of the systems. The meaning for the columns is as in Table 10 , but a last column is added in order to show the speed-up of T OY(FD) using rst-fail labeling with respect to the same system with na ve labeling.. Tables 13 and 14 display corresponding results for optimization. Particularly, Table 13 shows the (elapsed) time measured in milliseconds to solve the optimization problem considered in the benchmarking process, whereas Table 14 shows the speedup of our system with respect to the rest of the systems.. In these tables, all numbers represent the average of ten runs. The symbol ?? means that we did not receive a solution for the benchmark in a reasonable time and (?) indicates a non-determined value. The symbol N in the PAKCS, SWI Prolog and Ciao Prolog columns mean that we could not formulate the benchmark because of insu cient provision for constraints.. Also the notation OGS in the SWI column indicates that we received an error of Out Of Global Stack and, consequently, no answer was returned. In the GNU Prolog column, the notation (SO) number means that, in the rst execution of the program no answer was calculated because a Stack Over ow error was raised, and that, after increasing signi cantly the corresponding (cstr and trail) environment variables, in further executions we obtained an answer in the (average) time indicated by number.. The notation RE in the SICStus Prolog column indicates that we also did not compute an answer because a Resource Error by Insu cient Memory was returned. The dash (-) in the Ciao Prolog column means that we received an incorrect answer for this benchmark 5 . As already declared, whenever possible we maintained the same formulation for all the benchmarks in each system. However, this was not always possible in the magic series benchmark. In the T OY(FD), PAKCS and SICStus Prolog systems, this problem was coded by using speci c constraints (i.e., count=4, sum=3 and scalar product=4 -see formulation in Example 9). However, the GNU Prolog system lacks these constraints and, therefore, we used a classical formulation that requires to use rei ed constraints (Codognet and Diaz 1996) . This classical formulation is somewhat di erent in T OY(FD) since rei cation applies to Boolean types (whilst in GNU Prolog, as in general in CLP (FD) languages, the Boolean values f alse and true correspond to the numerical values 0 and 1 respectively). On the other hand, it was not possible in PAKCS as rei ed constraints are not available in this system. However, since SICStus Prolog allows rei ed constraints, the two formulations were considered in this system. Then, in the SICStus column and for the magic series benchmark row, we show between parentheses the (elapsed solving) time associated with the rei ed constraints-based formulation followed by the time associated to the alternative formulation based on the use of speci c constraints.. In the speed-up tables, in those cases in which for a particular system either a problem could not be expressed (e.g., for PAKCS, SWI Prolog or Ciao Prolog), or an error was returned avoiding to compute a rst answer, or an incorrect answer was returned, we use the symbol 1 to express that our system clearly outperforms that system since our system provides constraint support to formulate a solution for the benchmark and compute an answer. Also, a result. x.00 indicates that T OY(FD) computed an answer in 0.0 milliseconds and thus no speed-up can be calculated; in these cases, x.00 indicates that T OY(FD) is, at least, x times faster than the compared system. In addition to already cited related work, in this section we discuss some more related work. As already said, most of the work to integrate constraints in the declarative programming paradigm has been developed on LP (Codognet and Diaz 1996; Carlsson et al. 1997) . However, there exist some attempts to integrate constraints in the functional logic framework. For instance, (Arenas et al. 1996; L opez-Fraguas and S anchez-Hern andez 1999) show how to integrate both linear constraints over real numbers and disequality constraints in the F LP language T OY. Also, (Lux 2001 ) describes the addition of linear constraints over real numbers to the F LP language Curry (Hanus 1999) . Our work is guided to the FD constraint, instead of real constraints (although they are preserved), which allows to use non-linear constraints and adapts better to a range of FD applications.. With respect to FD, the closer proposal to ours is that described in that indicated how the integration of FD constraints in F LP could be carried out. As already indicated, PAKCS is an implementation that follows these indications.. T OY(FD) may also be considered from a multiparadigmatic view that means to combine constraint programming with another paradigms in one setting. In this context, there are some similarities with the language Oz (Van Roy et al. 2003; Van Roy and Haridi 2004) as this provides salient features of F P such as compositional syntax and rst-class functions, and features of LP and constraint programming including logic variables, constraints, and programmable search mechanisms. However, Oz is quite di erent to T OY(FD) because of a number or reasons: (1) Oz does not provide main features of classical functional languages such as explicit types or curried notation; (2) functional notation is provided in Oz as a syntactic convenience; (3) The Oz computation mechanism is not based on rewriting logic as that of T OY(FD) ; (4) Oz supports a class of lazy functions based on a demand-driven computation but this is not an inherent feature of the language (as in T OY(FD) ) and functions have to be made lazy explicitly (e.g., via the concept of futures); (5) functions and constraints are not really integrated, that is to say, they do not have the same category as in T OY(FD) (i.e., constraints are functions) and both coexist in a concurrent setting, and (6) Oz programs follow a far less concise program syntax than T OY(FD). In fact, Oz generalizes the CLP and concurrent constraint programming paradigms to provide a very exible approach to constraint programming very di erent to our proposal for CF LP (FD).. Also, LIFE (A it-kaci and Podelski 1993) is an experimental language aiming to integrate logic programming and functional programming but, as Oz, the proposal is quite di erent to T OY(FD) as rstly, it is considered in the framework of object-oriented programming, and, secondly, LIFE enables the computation over an order-sorted domain of feature trees by allowing the equality (i.e., uni cation) and entailment (i.e., matching) constraints over order-sorted feature terms.. There exist other constraint systems that share some aspects with T OY(FD) although they are very di erent. One of those systems is FaCiLe (Barnier and Brisset 2001) , a constraint programming library that provides constraint solving over integer nite domains, HO functions, type inference, strong typing, and user-de ned constraints. However, despite these similarities, FaCiLe is very di erent to T OY(FD) as it is built on top of the functional language OCaml that provides full imperative capabilities and does not have a logical component; also OCaml is a strict language, as opposed to lazy ones. In fact, as Oz , it allows the manipulation of potentially in nite data structures by explicit delayed expressions, but laziness is not an inherent characteristic of the resolution mechanism as in T OY(FD). Moreover, FaCiLe is a library and thus it lacks programming language features.. Other interesting systems are OPL (Van Hentenryck 1999) and AMPL (Fourer et al. 1993 ) that cannot be compared to our work because they are algebraic languages which therefore are not general programming languages. Moreover, these languages do not bene t neither from complex terms and patterns nor from non-determinism as T OY(FD) does.. Finally, we mention here another CF LP scheme proposed in the Phd Thesis of M. Marin (Marin 2000) . This approach introduces CF LP (D; S; L), a family of languages parameterized by a constraint domain D, a strategy S which de nes the cooperation of several constraint solvers over D, and a constraint lazy narrowing calculus L for solving constraints involving functions de ned by user given constrained rewriting rules. This approach relies on solid work on higher-order lazy narrowing calculi and has been implemented on top of Mathematica (Marin et al. 1999; Marin et al. 2000) . Its main limitation from our viewpoint is the lack of declarative semantics.. Generally speaking, T OY(FD) is, from its nature, di erent to all the constraints systems discussed above since T OY(FD) is a pure F LP language that combines characteristics of pure LP and pure F P paradigms, and its operational mechanism is the result of combining the operational methods of logic languages (i.e., uni cation and resolution) and functional languages (i.e., rewriting).

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A

New

Lower

Bound

on

the

Maximum

Number

of

Satisfied

Clauses

in

Max-SAT

and

Its

Algorithmic

Applications

A

pair

of

unit

clauses

is

called

conflicting

if

it

is

of

the

form

(x),

(x).

A

CNF

formula

is

unit-conflict

free

(UCF)

if

it

contains

no

pair

of

conflicting

unit

clauses.

Lieberherr

and

Specker

(J.

ACM

showed

that

for

each

UCF

CNF

formula

with

m

clauses

we

can

simultaneously

satisfy

at

leastφm

clauses,

whereφ

We

improve

the

Lieberherr-Specker

bound

by

showing

that

for

each

UCF

CNF

formula

F

with

m

clauses

we

can

find,

in

polynomial

time,

a

subformula

F

with

m

clauses

such

that

we

can

simultaneously

satisfy

at

least

ϕmWe

consider

two

parameterized

versions

of

MAX-SAT,

where

the

parameter

is

the

number

of

satisfied

clauses

above

the

bounds

and

m(

The

former

bound

is

tight

for

general

formulas,

and

the

later

is

tight

for

UCF

formulas.

Mahajan

and

Raman

(J.

Algorithms

[335][336][337][338][339][340][341][342][343][344][345][346][347][348][349][350][351][352][353][354]

showed

that

every

instance

of

the

first

parameterized

problem

can

be

transformed,

in

polynomial

time,

into

an

equivalent

one

with

at

most

variables

and

clauses.

We

improve

this

to

variables

and

clauses.

Mahajan

and

Raman

conjectured

that

the

second

parameterized

Algorithmica

problem

is

fixed-parameter

tractable

(FPT).

We

show

that

the

problem

is

indeed

FPT

by

describing

a

polynomial-time

algorithm

that

transforms

any

problem

instance

into

an

equivalent

one

with

at

most

variables.

Our

results

are

obtained

using

our

improvement

of

the

Lieberherr-Specker

bound

above.

A

CNF

formula

I

is

t-satisfiable

if

any

subset

of

t

clauses

of

I

can

be

satisfied

simultaneously.

In

particular,

a

CNF

formula

is

unit-conflict

free

if

and

only

if

it

is

Let

r

t

be

the

largest

real

such

that

in

any

t-satisfiable

CNF

formula

at

least

r

t

-th

fraction

of

its

clauses

can

be

satisfied

simultaneously.

Note

that

r

and

r

Lieberherr

and

Specker

and,

later,

Yannakakis

proved

that

r

Käppeli

and

Scheder

proved

that

r

and,

thus,

r

Král

For

general

t,

Huang

and

Lieberherr

showed

that

lim

t→∞

r

t

and

Trevisan

proved

that

lim

t→∞

r

t

(a

different

proof

of

this

result

is

later

given

by

Král

In

the

preliminary

version

of

this

paper

published

in

the

proceedings

of

IPEC

we

asked

to

establish

parameterized

complexity

of

the

following

parameterized

problem:

given

a

CNF

formula

F

(V

C),

decide

whether

sat(F

k,

where

k

is

the

parameter.

This

question

was

recently

solved

in

by

showing

that

the

problem

has

a

kernel

with

a

linear

number

of

variables.

Unlike

this

paper,

uses

the

Probabilistic

Method.

Similar

question

for

any

fixed

t

remains

open.

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 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 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


Social and ethical checkpoints for bottom-up synthetic biology, or protocells Abstract An alternative to creating novel organisms through the traditional ''top-down'' approach to synthetic biology involves creating them from the ''bottom up'' by assembling them from non-living components; the products of this approach are called ''protocells.'' In this paper we describe how bottom-up and top-down synthetic biology differ, review the current state of protocell research and development, and examine the unique ethical, social, and regulatory issues raised by bottom-up synthetic biology. Protocells have not yet been developed, but many expect this to happen within the next five to ten years. Accordingly, we identify six key checkpoints in protocell development at which particular attention should be given to specific ethical, social and regulatory issues concerning bottom-up synthetic biology, and make ten recommendations for responsible protocell science that are tied to the achievement of these checkpoints. Protocell science and technology ) is a bottom-up form of synthetic biology that is synthesizing novel kinds of living systems from materials that have never been alive. So, part of society's evaluation of synthetic biology in general must focus on the ethical, social, and regulatory issues that specifically involve protocell science and technology, or bottom-up synthetic biology. Many of the broader issues involving protocells have parallels in top-down synthetic biology; for example, the trajectory of development in both areas will be increasingly driven by potential commercial applications. But some broader issues are unique to protocells.. We have identified six checkpoints in protocell science and technology that call for significant ethical, social, and regulatory attention. These checkpoints are specifically relevant only for bottom-up synthetic biology. We have also proposed ten specific recommendations about the ethical, social, and regulatory aspects of protocells. The way in which specific checkpoints trigger specific recommendations is summarized in Table 1 .. One of the lessons of this paper is that the details of the social and ethical implications of protocells often largely hinge on specific information that will not be available until science and technology have progressed further. So, the details of our analysis should be treated as a work in progress and continually revisited as we learn more about protocells. Nevertheless, some broad lessons can be gleaned from the overall structure of Table 1 . Table 1 vividly shows that a number of the social and ethical implications of protocells require attention right now. Although fully autonomous protocells are not yet technically feasible, the time is already ripe to communicate the state of protocell science to the general public (Recommendation 1), to address cultural and religious concerns about assembling new forms of life in the laboratory (Recommendation 2), to fund further investigation of the ethical, legal, and social implications of protocells (Recommendation 3), and to start creating a curriculum covering these issues (Recommendation 4). In addition, we should also start laying the foundation for subsequently evaluating IP regulations, potential misuse, safety mechanisms, and regulatory scope. We should stress that these recommendations are what we believe should happen, but in general they are not yet happening. So, we are sounding a call to action. Table 1 also vividly conveys how the social and ethical implications of bottom-up and top-down synthetic biology differ. The six checkpoints for protocell research and development have no particular significance in top-down synthetic biology. The social and ethical implications of bottom-up synthetic biology unfold over time in connection with specific scientific and technical achievements in a distinctive way, summarized in Table 1 . This directly results from the fact that protocells are produced in a bottom-up fashion, rather than starting with existing forms of life and then modifying them.. What animates most of the social and ethical issues concerning protocells is that protocells are for all intents and purposes autonomous forms of life. This is what raises many of the social and religious concerns; this is what makes safety mechanisms so vital; this is what drives the need for thinking through their environmental consequences; and this is what makes the ultimate impact of protocells so hard to predict.

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


Fast self-healing gradients We present CRF-Gradient, a self-healing gradient algorithm that provably reconfigures in O(diameter) time. Selfhealing gradients are a frequently used building block for distributed self-healing systems, but previous algorithms either have a healing rate limited by the shortest link in the network or must rebuild invalid regions from scratch. We have verified CRF-Gradient in simulation and on a network of Mica2 motes. Our approach can also be generalized and applied to create other self-healing calculations, such as cumulative probability fields. • Execution happens in partially synchronous rounds, once every ∆t seconds; each device has a clock that ticks regularly, but frequency may vary slightly and clocks have an arbitrary initial time and phase.. • Devices communicate via unreliable broadcasts to their neighbors (all other devices within r meters distance).. Broadcasts are sent at most once per round.. • Devices are provided with estimates of the distance to their neighbors, but naming, routing, and global coordinate services are not provided. 1 Excessive expenditure of either is still bad, and memory is an important constraint for the Mica2 implementation.. • Devices may fail, leave, or join the network at any time, which may change the connectedness of the network.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O