The Nonparanormal: Semiparametric Estimation of High Dimensional Undirected Graphs Recent methods for estimating sparse undirected graphs for real-valued data in high dimensional problems rely heavily on the assumption of normality. We show how to use a semiparametric Gaussian copula-or "nonparanormal"-for high dimensional inference. Just as additive models extend linear models by replacing linear functions with a set of one-dimensional smooth functions, the nonparanormal extends the normal by transforming the variables by smooth functions. We derive a method for estimating the nonparanormal, study the method's theoretical properties, and show that it works well in many examples. The linear model is a mainstay of statistical inference that has been extended in several important ways. An extension to high dimensions was achieved by adding a sparsity constraint, leading to the lasso (Tibshirani, 1996) . An extension to nonparametric models was achieved by replacing linear functions with smooth functions, leading to additive models (Hastie and Tibshirani, 1999) . These two ideas were recently combined, leading to an extension called sparse additive models (SpAM) (Ravikumar et al., 2008b,a) . In this paper we consider a similar nonparametric extension of undirected graphical models based on multivariate Gaussian distributions in the high dimensional setting. Specifically, we use a high dimensional Gaussian copula with nonparametric marginals, which we refer to as a nonparanormal distribution.. If X is a p-dimensional random vector distributed according to a multivariate Gaussian distribution with covariance matrix Σ, the conditional independence relations between the random variables X 1 , X 2 , . . . , X p are encoded in a graph formed from the precision matrix Ω = Σ −1 . Specifically, missing edges in the graph correspond to zeroes of Ω. To estimate the graph from a sample of size n, it is only necessary to estimate Σ, which is easy if n is much larger than p. However, when p is larger than n, the problem is more challenging. Recent work has focused on the problem of estimating the graph in this high dimensional setting, which becomes feasible if G is sparse. Yuan and Lin (2007) and Banerjee et al. (2008) propose an estimator based on regularized maximum likelihood using an ℓ 1 constraint on the entries of Ω, and Friedman et al. (2007) develop an efficient algorithm for computing the estimator using a graphical version of the lasso. The resulting estimation procedure has excellent theoretical properties, as shown recently by Rothman et al. (2008) and Ravikumar et al. (2009) . While Gaussian graphical models can be useful, a reliance on exact normality is limiting. Our goal in this paper is to weaken this assumption. Our approach parallels the ideas behind sparse additive models for regression (Ravikumar et al., 2008b,a) . Specifically, we replace the Gaussian with a semiparametric Gaussian copula. This means that we replace the random variable X = (X 1 , . . . , X p ) by the transformed random variable f (X) = (f 1 (X 1 ), . . . , f p (X p )), and assume that f (X) is multivariate Gaussian. This semiparametric copula results in a nonparametric extension of the normal that we call the nonparanormal distribution. The nonparanormal depends on the functions {f j }, and a mean µ and covariance matrix Σ, all of which are to be estimated from data. While the resulting family of distributions is much richer than the standard parametric normal (the paranormal), the independence relations among the variables are still encoded in the precision matrix Ω = Σ −1 . We propose a nonparametric estimator for the functions {f j }, and show how the graphical lasso can be used to estimate the graph in the high dimensional setting. The relationship between linear Comparison of regression and graphical models. The nonparanormal extends additive models to the graphical model setting. Regularizing the inverse covariance leads to an extension to high dimensions, which parallels sparse additive models for regression.. regression models, Gaussian graphical models, and their extensions to nonparametric and high dimensional models is summarized in Figure 1 .. Most theoretical results on semiparametric copulas focus on low or at least finite dimensional models (Tsukahara, 2005) . Models with increasing dimension require a more delicate analysis; in particular, simply plugging in the usual empirical distribution of the marginals does not lead to accurate inference. Instead we use a truncated empirical distribution. We give a theoretical analysis of this estimator, proving consistency results with respect to risk, model selection, and estimation of Ω in the Frobenius norm.. In the following section we review the basic notion of the graph corresponding to a multivariate Gaussian, and formulate different criteria for evaluating estimators of the covariance or inverse covariance. In Section 3 we present the nonparanormal, and in Section 4 we discuss estimation of the model. We present a theoretical analysis of the estimation method in Section 5, with the detailed proofs collected in an appendix. In Section 6 we present experiments with both simulated data and gene microarray data, where the problem is to construct the isoprenoid biosynthetic pathway.

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Convex and Semi-Nonnegative Matrix Factorizations Abstract-We present several new variations on the theme of nonnegative matrix factorization (NMF). Considering factorizations of the form X ¼ F G T , we focus on algorithms in which G is restricted to containing nonnegative entries, but allowing the data matrix X to have mixed signs, thus extending the applicable range of NMF methods. We also consider algorithms in which the basis vectors of F are constrained to be convex combinations of the data points. This is used for a kernel extension of NMF. We provide algorithms for computing these new factorizations and we provide supporting theoretical analysis. We also analyze the relationships between our algorithms and clustering algorithms, and consider the implications for sparseness of solutions. Finally, we present experimental results that explore the properties of these new methods. M ATRIX factorization is a unifying theme in numerical linear algebra. A wide variety of matrix factorization algorithms have been developed over many decades, providing a numerical platform for matrix operations such as solving linear systems, spectral decomposition, and subspace identification. Some of these algorithms have also proven useful in statistical data analysis, most notably the singular value decomposition (SVD), which underlies principal component analysis (PCA).. Recent work in machine learning has focused on matrix factorizations that directly target some of the special features of statistical data analysis. In particular, nonnegative matrix factorization (NMF) [1] , [2] focuses on the analysis of data matrices whose elements are nonnegative, a common occurrence in data sets derived from text and images. Moreover, NMF yields nonnegative factors, which can be advantageous from the point of view of interpretability.. The scope of research on NMF has grown rapidly in recent years. NMF has been shown to be useful in a variety of applied settings, including environmetrics [3] , chemometrics [4] , pattern recognition [5] , multimedia data analysis [6] , text mining [7] , [8] , DNA gene expression analysis [9] , [10] , and protein interaction [11] . Algorithmic extensions of NMF have been developed to accommodate a variety of objective functions [12] , [13] and a variety of data analysis problems, including classification [14] and collaborative filtering [15] . A number of studies have focused on further developing computational methodologies for NMF [16] , [17] , [18] , [19] . Finally, researchers have begun to explore some of the relationships between matrix factorizations and K-means clustering [20] , making use of the least square objectives of NMF; as we emphasize in the current paper, this relationship has implications for the interpretability of matrix factors. NMF with the Kullback-Leibler (KL) divergence objective has been shown [21] , [13] to be equivalent to probabilistic latent semantic analysis [22] , which has been further developed into the fully probabilistic latent Dirichlet allocation model [23] , [24] .. Our goal in this paper is to expand the repertoire of nonnegative matrix factorization. Our focus is on algorithms that constrain the matrix factors; we do not require the data matrix to be similarly constrained. In particular, we develop NMF-like algorithms that yield nonnegative factors but do not require the data matrix to be nonnegative. This extends the range of application of NMF ideas. Moreover, by focusing on constraints on the matrix factors, we are able to strengthen the connections between NMF and K-means clustering. Note in particular that the result of a K-means clustering run can be written as a matrix factorization X ¼ F G T , where X is the data matrix, F contains the cluster centroids, and G contains the cluster membership indicators. Although F typically has entries with both positive and negative signs, G is nonnegative. This motivates us to propose general factorizations in which G is restricted to be nonnegative and F is unconstrained. We also consider algorithms that constrain F ; in particular, restricting the columns of F to be convex combinations of data points in X we obtain a matrix factorization that can be interpreted in terms of weighted cluster centroids.. The paper is organized as follows: In Section 2, we present the new matrix factorizations, and in Section 3, we present algorithms for computing these factorizations. Section 4 provides a theoretical analysis, which provides insights into the sparseness of matrix factors for a convex variant of NMF. In Section 5, we consider extensions of Convex-NMF and the relationships of NMF-like factorizations. In Section 5.1, we show that a convex variant of NMF has the advantage that it is readily kernelized. In Section 6, we present comparative experiments that show that constraining the F factors to be convex combinations of input data enhances their interpretability. We also present experiments that compare the performance of the NMF variants to K-means clustering, where we assess the extent to which the imposition of constraints that aim to enhance interpretability leads to poorer clustering performance. Finally, we present our conclusions in Section 7. We have presented a number of new nonnegative matrix factorizations. We have provided algorithms for these factorizations and theoretical analysis of the convergence of these algorithms. The ability of these algorithms to deal with mixed-sign data makes them useful for many applications, particularly given that covariance matrices are often centered.. Semi-NMF offers a low-dimensional representation of data points which lends itself to a convenient clustering interpretation. Convex-NMF further restricts the basis vectors to be convex combinations of data points, providing a notion of cluster centroids for the basis. We also briefly discussed additional NMF algorithms-Kernel-NMF and Cluster-NMF-that are further specializations of Convex-NMF.. We also showed that the NMF variants can be viewed as relaxations of K-means clustering, thus, providing a closer tie between NMF and clustering than has been present in the literature to date. Moreover, our empirical results showed that the NMF algorithms all outperform K-means clustering on all of the data sets that we investigated in terms of clustering accuracy. We view these results as indicating that the NMF family is worthy of further investigation. We view Semi-NMF and Convex-NMF as particularly worthy of further investigation, given their native capability for handling mixed-sign data and their particularly direct connections to clustering. We first present the results of an experiment on synthetic data, which aims to verify that Convex-NMF can yield factorizations that are close to cluster centroids. We then present experimental results for real data comparing K-means clustering and the various factorizations.

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


Biological Sequence Alignment On The Computational Grid Using The Grads Framework In spite of the existence of several Grid middleware projects, developing and executing programs on the computational Grid remains a user intensive process. The goal of the Grid Application Development Software (GrADS) project is to make the Grid simpler to use despite the dynamically changing status of Grid resources. Protein and genome sequence alignment is a basic operation in bioinformatics, and it requires large datasets and tends to be highly compute intensive. In this paper, we present work done to grid-enable a biological sequence alignment package (FASTA) and to run it under the GrADS framework. We discuss the advantages of using GrADS framework for FASTA. The Grid Application Development Software (GrADS) project [5] has defined a framework to make it simpler to prepare and execute programs on a computational Grid. In order to guide the development of this framework, an implementation known as GrADSoft [8, 9] was developed together with a set of software packages that use the framework. This set of packages includes the numerical linear algebra library ScaLAPACK [25] , the astrophysics problem solving environment Cactus [2, 19] and satisfiability solvers for circuit design [7] . This paper describes the work done to enable a parallel, master-worker implementation of the biological sequence alignment application FASTA [24] to run on the GrADSoft framework. There have been many implementations of grid based sequence alignment applications. Our implementation is designed to demonstrate several things. Firstly, it demonstrates the ease with which the GrADS infrastructure can be used to grid-enable a legacy code. Secondly, it acts be an example of a grid application bound by data locality, where the computation must be scheduled at the site of the data.. Several projects provide generic facilities to run master-worker applications on a grid, such as the Condor MW (Master-Worker) implementation [15] , the AppLeS Master Worker Application Template (AMWAT) [28] and the NetSolve system [1] . The Condor MW and the AMWAT approaches require that the application provide a specific programming interfaces that can be called by a scheduler. The NetSolve interface is simpler, but it is designed only for problems that can be decomposed into a bag-of-tasks that are executed using a simple remote procedure call interface. All of these approaches would require substantial changes to a pre-existing application, and none of them is designed to schedule an application using the data-locality constraints (i.e., distributed, partial data sets) that are addressed in this work.. Spring and Wolski [30] discuss scheduling a Master-worker implementation of FASTA on a metacomputer using application specific performance models. A static schedule based on run time resource information and application specific performance models resulted in a faster execution than simple self-scheduling. But the best execution time was obtained with an adaptive approach that uses resource information to schedule most of the work, and then self-scheduling to complete. However, this work did not consider data-locality constraints when making scheduling decisions.. There are many other projects aimed at enabling Grid resources to be used for bioinformatics, including the Japanese BioGrid [6] , North Carolina BioGrid [22] , MyGrid [21] , and APBioGrid [4] . These projects generally provide a por- tal interface from which a user can run a specific set of programs on Grid resources. In contrast, the GrADS project is designed to enable the end user to run any arbitrary executable on the Grid resources in an transparent and efficient manner. The goal of this work was to demonstrate that the GrADS architecture can be adapted to handle applications with data-locality constraints, and that these applications can be grid-enabled with ease.. GrADSoft greatly reduces the burden on the end user of finding, selecting and using the appropriate grid resources for their application. Porting and running FASTA, an master-worker sequence-matching application, was greatly simplified. A performance model and mapper had to be constructed for the application and minor changes were made to the FASTA code, in order to have the workers use local databases rather than have the master distribute all the data. After that, GrADSoft was able to handle all the details of scheduling, executing and monitoring the application.. The GrADS architecture provides a plausible method for providing large amounts of computing power to applications on demand. In this demonstration, the logistic tasks of scheduling, moving the application to the site of the data and gathering the results are handled by the framework. With the rapid growth of large, distributed data collections in fields such as biology, astronomy, and physics, the techniques described here could find wide applicability.

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Change patterns and change support features - Enhancing flexibility in process-aware information systems Companies increasingly adopt process-aware information systems (PAISs), which offer promising perspectives for more flexible enterprise computing. The emergence of different process support paradigms and the lack of methods for comparing existing approaches enabling PAIS changes have made the selection of adequate process management technology difficult. This paper suggests a set of 18 change patterns and 7 change support features to foster the systematic comparison of existing process management technology in respect to process change support. While the proposed patterns are all based on empirical evidence from several large case studies, the suggested change support features constitute typical functionalities provided by flexible PAISs. Based on the proposed change patterns and features, we provide a detailed analysis and evaluation of selected approaches from both academia and industry. The presented work will not only facilitate the selection of technologies for realizing flexible PAISs, but can also be used as a reference for implementing flexible PAISs. In today's dynamic business world the economic success of an enterprise depends on its ability to react to changes in its environment in a quick and flexible way [1] . Causes for these changes can be manifold and include the introduction of new laws or changes in customers' attitudes. For these reasons companies have recognized business agility as a competitive advantage, which is fundamental for being able to cope with business trends like increasing product and service variability, faster time-to-market, and business-ondemand.. Process-aware information systems (PAISs) offer promising perspectives in this respect, and a growing interest in aligning information systems in a process-oriented way can be observed [2, 3] . In contrast to data-or function-centered information systems, PAISs are characterized by a strict separation of process logic and application code. In particular, most PAISs describe process logic explicitly in terms of a process model providing the schema for process execution. Usually, the core of the process layer is built by a process management system, which provides generic functionality for modeling, executing, and monitoring processes. This allows for a separation of concerns, which is a well established principle in computer science for increasing maintainability and for reducing cost of change [4] . Changes to one layer often can be performed without affecting the other layers. For example, modifying the application service which implements a particular process step (i.e., activity) does usually not imply any change to the process layer as long as interfaces remain stable (i.e., the external observable behavior of the service remains the same). In addition, changing the execution order of activities or adding new activities to the process can, to a large degree, be accomplished without touching any of the application services.. The ability to efficiently deal with process change has been identified as one of the critical success factors for any PAIS [5, 6] . PAISs facilitate changes significantly through the above described separation of concerns. According to a recent study conducted among several Dutch companies, however, enterprises are reluctant to change PAIS implementations once they are running properly [7] . High complexity and high cost of change are mentioned as major reasons for not fully leveraging the potential of PAISs. To overcome this problem flexible PAISs are needed, enabling companies to capture real-world processes adequately without leading to a mismatch between the computerized business processes and those running in reality [8, 5] . In particular, the introduction of a PAIS must not freeze existing business processes [9] . Instead, authorized users must be able to flexibly deviate from the predefined processes as required (e.g., to deal with exceptions) and to evolve PAIS implementations over time (e.g., to continuously adapt the underlying process models to process optimizations). Such changes must be possible at a high level of abstraction without affecting consistency and robustness of the PAIS [10] . In addition, PAISs must support users to deal with uncertainty by deferring decisions to run-time as required. Patterns were first used by Christopher Alexander [67] to describe solutions to recurring problems and best practices in architectural design. Patterns also have a long tradition in computer science. Gamma et al. [50] applied the same concepts to software engineering and described 23 design patterns. In the workflow area, patterns have been introduced for analyzing the expressiveness of process meta models [19, 20, 68] . In this context, control flow patterns describe different constructs to specify activities and their ordering. In addition, workflow data patterns [21] provide ways for modeling the data aspect in PAISs, and workflow resource patterns [22] describe how resources can be represented in workflows. Furthermore, patterns for describing service interactions and process choreographies were introduced [69] .. The introduction of workflow patterns has had significant impact on PAIS design as well as on the evaluation of PAISs and process languages like BPEL [20] , BPMN [70] , EPC [20] , and UML [71] . To evaluate the powerfulness of a PAIS regarding its ability to cope with change, the existing workflow patterns are important, but not sufficient. In addition, a set of patterns addressing the aspect of process change is needed. Although workflow pattern support allows for reducing the need for modifying process instances, these patterns require flexibility to be entirely built into the process model. By contrast, patterns for changes to predefined regions allow deferring decisions from build-to runtime to better cope with uncertainty. In addition, adaptation patterns will allow for structural process modifications to deal with non-anticipated exceptions and evolving needs. Exception handling patterns [24] like Rollback only change the state of a process instance (i.e., its behavior), but not its schema (i.e., its structure). Thus they are well suited for dealing with expected situations. Non-anticipated situations, in turn, might require structural adaptations as well [23] . Change patterns support this by allowing to modifiy the observable behavior of a process instance and its structure. In addition, change patterns can be applied at the process type level decreasing the efforts needed for accomplishing a change. To provide a complete evaluation framework for PAIS flexibility, expected and unexpected exceptions as well as schema evolution must be considered.. Exception handling often requires combined use of several exception handling patterns resulting in rather complex routines. The Exlet approach [47, 48] addresses this problem by allowing for the combination of different exception handling patterns to an exception handling process called Exlet. Generally, Exlets are executed in parallel to the process instance to be modified and can be reused when similar exceptions re-occur. Exlets allow for "simulating" several of the adaptation patterns. However, as Exlets are executed independently of the process instance without structurally modifying it, end users have to suspend the process instance manually if required. In contrast, change patterns hide this complexity from users by providing high-level change operations.. In [72] it has been shown that patterns can additionally be used for facilitating process modeling. It proposes 9 patterns for business functions (e.g., approval, notification) and shows that modeling efforts can be decreased when using this pattern set as building blocks. This approach speeds up process modeling, change patterns likewise allow for reducing the efforts of accomplishing process changes.. In [73] , some of the proposed change patterns are used to implement refactoring techniques, which will be behaviour-preserving if certain pre-and post conditions are met. In particular, refactorings allow PAIS engineers to keep process models maintainable and understandable over time and consequently reduce costs of future process changes.. Most systems considered by our evaluation, model processes in a procedural or imperative way. An exception is provided by Pockets of Flexibility (PoF) [16] , which use a combination of imperative and declarative modeling style. The process itself is modeled imperatively, but the placeholder activities are specified in a declarative way using constraints. Other declarative approaches not considered in our evaluation are MOBILE [74] and DECLARE [15, 75] . Instead of requiring designers to specify how the process shall be executed, they only have to state what shall be done during process execution. With declarative approaches changes become less frequent. However, run-time adaptations still can be an issue (e.g., a constraint might have to be violated for a particular instance due to an unforeseen situation). Further, constraints themselves may evolve over time, which raises the challenge of propagating changes to ongoing instances. A promising approach towards this direction is offered by DECLARE [15] . Another challenging issue is the maintenance and testing of constraint-based process models, particularly in case of large constraint sets.. Different frameworks exist for comparing specific aspects related to process change. In [10] the authors provide one such framework for elaborating strengths and weaknesses of adaptive PAISs along typical dynamic change problems. Main emphasis is on investigating formal properties and correctness criteria in connection with dynamic process changes. In [76] graph-and rule-based languages along dimensions along dimensions like flexibility, adaptability, complexity, and expressiveness.. Several approaches target the automatic handling of process exceptions. Some of these approaches [77] [78] [79] 52] apply structural adaptations to process instances to deal with the exceptions. By using the adaptation patterns offered by existing systems. In ADEPT2, for example, respective patterns are not only accessible via a process editor, but can also be invoked via a powerful application programming interface. Several approaches have utilized ADEPT2's interface to implement agents for automated exception handling [77] [78] [79] 52 ]. This adaptation pattern can be realized by transforming the high level insertion operation into a sequence of low level change primitives (e.g., add node, add edge).. This adaptation pattern can be implemented based on patterns AP1 and AP2 (insert / delete process fragment) or be based on change primitives (e.g., add/remove node, add/remove edge). Adaptation Pattern AP4: Replace Process Fragment. This pattern enables the replacement of a process fragment by another one (cf. Fig. 9 ). Like AP3, pattern AP4 can be implemented based on patterns AP1 and AP2 (Insert/Delete Process Fragment) or be directly based on change primitives.. For implementing pattern AP6 graph aggregation techniques can be used. When considering data aspects as well, variables which constitute input / output for the selected process fragment have to be determined and must be considered as input / output for the created sub process.. The implementation of this adaptation pattern can be based on other adaptation patterns (e.g., AP1). When considering data aspects as well, the data context of the sub process and its current mapping to the parent process have to be transferred to the parent process schema. . This adaptation pattern can be implemented based on Patterns AP1 (insert process fragment), AP11, and AP12 (add / remove control dependency). Alternatively, implementation can be based on change primitives.. This adaptation pattern can be implemented based on Patterns AP3 (move process fragment), AP11 and AP12 (add / remove control dependency) or be based on change primitives.. This adaptation pattern can be implemented based on patterns AP1 (insert process fragment), AP11, and AP12 (add / remove control dependency) and AP13 (update condition) or be based on change primitives. An additional control edge (e.g., for synchronizing the execution order of two parallel activities) is added to process schema S.. This adaptation pattern can be implemented based on Pattern AP1 (insert process fragment) or by using change primitives. . By selecting the respective sub process or activity program, a reference to it is dynamically set and the sub-process or activity program is invoked.

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


A self-adaptable query allocation framework for distributed information systems Abstract In large-scale distributed information systems, where participants are autonomous and have special interests for some queries, query allocation is a challenge. Much work in this context has focused on distributing queries among providers in a way that maximizes overall performance (typically throughput and response time). However, preserving the participants' interests is also important. In this paper, we make the following contributions. First, we provide a model to define the participants' perception of the system regarding their interests and propose measures to evaluate the quality of query allocation methods. Then, we propose a framework for query allocation called Satisfaction-based Query Load Balancing (SQLB, for short), which dynamically trades consumers' interests for providers' interests based on their satisfaction. Finally, we compare SQLB, through experimentation, with two important baseline query allocation methods, namely Capacity based and Mariposa-like. The results demonstrate that SQLB yields high efficiency while satisfying the participants' interests and significantly outperforms the baseline methods. We consider distributed information systems with a mediator that allows consumers to access information providers through queries [19, 32] . Consumers and providers (which we refer to participants) are autonomous in the sense that they are free to leave the mediator at any time and do not depend on anyone to do so. In the context of a single mediator, leaving the mediator is equivalent to depart from the system, but it could be that, in a multi-mediator system, a participant registers to another competing mediator.. Providers can be heterogeneous in terms of capacity and data. Heterogeneous capacity means that some providers are more powerful than others and can treat more queries per time unit. Data heterogeneity means that providers provide different data and thus produce different results for the same query. Providers declare their capabilities for performing queries to the mediator. Then, the main function of the mediator is to allocate each incoming query to the providers that can satisfy it. Much work in this context has focused on distributing the query load among the providers in a way that maximizes overall performance (typically throughput and response time), i.e. query load balancing (qlb) [2, 8, 20, 31, 36] . Nevertheless, participants usually have certain expectations with respect to the mediator, which are not only performance-related (see Example 1) . Such expectations mainly reflect their preferences to allocate and perform queries. Consumers' preferences may represent, e.g. their interests towards providers (based on reputation for example), preferred providers, or quality of service. Providers' preferences may represent, for example, their topics of interests, relationships with other participants, or strategies.. Example 1 Consider a provider that represents a courier company. During the promotion of its new international shipping service, the provider is more interested in treating queries related to international shipments rather than national ones. Once the advertising campaign is over, its preferences may change. Similarly, consumers expect the system to provide them with information that best fits their preferences.. In this context, because of participants' autonomy, dissatisfaction is a problem since it may lead participants to leave the mediator. Thus, it is important to have a query allocation strategy that balances queries such that participants are satisfied. Participant's satisfaction means that the query allocation method meets its expectations. To make this possible, the participants' preferences must be taken into consideration when balancing queries. However, preferences are usually considered as private data by participants (e.g. in an e-commerce scenario, enterprises do not reveal their business strategies). In addition, preferences are static data, i.e. long-run, while the desire of a participant to allocate and perform queries may depend on its context and thus is more dynamic, i.e. short-run. For instance, in Example 1, even if the provider (the courier company) prefers to perform queries related to international shipments during its advertising campaign, it is possible that, at some time, it may not desire to perform such queries because of other local reasons, e.g. by overload. Thus, participants are required to express their desire to allocate and perform queries via their intention, which may stem, e.g. from combining their preferences and other local consideration such as load and reputation (see Fig. 1 ).. In such distributed information systems, query allocation is a challenge for several reasons.. -There is no definition of satisfaction to reflect how well the system meets the participants' expectations in the long-run. Economic approaches consider utility and individual rationality, but utility is not normalized and is commonly related to economic notions (e.g. money), and individual rationality does not capture long-run aspects. -Participants' expectations may be contradictory among them as well as with respect to the system performance. -The query allocation process should be adaptable to applications and self-adaptable to changes in the participants' expectations because such expectations usually change in the course of time. -Unlike several economic models [5, 6, 38] , queries must be always treated whenever possible (if there exits at least one provider to perform it) even if providers do not desire to deal with them. This is because consumers that do not get results may become dissatisfied and thus simply leave the system, which may hurt providers as well. -Participants' departures may have consequences on the functionalities provided by the system. The providers' departure may mean the loss of important system capabilities and the consumers' departure is a loss of queries for providers.. To our knowledge, this problem has not been addressed completely before. Thus, our first objective is to propose a model that provides a satisfaction notion to reflect how well the mediator meets the participants' expectations in the long-run. Then, our second objective is to propose a query allocation framework that considers both satisfaction and intentions of participants. We considered distributed information systems where participants are autonomous to leave the system at will. In this context, it is crucial to consider the participants' intentions to allocate and perform queries so that their expectations, response times, and system capacity are ensured. We presented, in this paper, a general and complete solution to allocate queries among providers by considering the participants' intentions and query load balancing (qlb). Our work carried out fourth main contributions.. First, we characterized the participants' expectations in a new model, which allows to evaluate a system from a satisfaction point of view. The definitions that we proposed are original, considering the long-run notions of adequation and satisfaction. They are independent of the way participants compute their intentions and how the mediator considers them. This model facilitates the design and evaluation of new query allocation methods for these environments. The proposed model is general, and thus, can be used for any distributed systems architecture. Second, we proposed three different measures to evaluate the quality of qlb methods:. -The mean measure reflects the effort that a query allocation method does for equally either maximizing or minimizing a given set of values. -The fairness measure evaluates how fair a query allocation method is. -The balance measure measures the Min-Max values.. We proved that using these proposed measures together, one can predict possible consumers' and providers' departures from the system. Third, we presented the SQLB framework for balancing queries in these environments. The originality of SQLB is to perform all query demand while satisfying participants' expectations. SQLB strongly differs from the related work in several ways: -It allows providers to trade their preferences for their utilization while keeping their strategic information private. -It affords consumers the flexibility to trade their preferences for providers' reputation. -It allows trading consumers' intentions for providers ' intentions. -It strives to balance queries at runtime via the participants' satisfaction, thus reducing starvation. -It affords the mediator to regulate the system with respect to some predefined function and can adapt the query allocation process to the kind of application. -It can ensure good levels of satisfaction as far as the system is adequate to participants and vice versa, which allows participants to reach their expectations in the system whether they correctly work out their intentions.. Fourth, we evaluated and compared SQLB with two baseline query allocation methods (Capacity based and Mariposa-like), in two kinds of environments: captive and autonomous. We showed through experimentation that, by considering together the qlb and satisfaction of participants, SQLB significantly outperforms both baseline methods. We observed that participants are, in general, very satisfied with SQLB and Mariposa-like, which is not the case for Capacity based that suffers from several providers' departures due to dissatisfaction. However, Mariposa-like has serious problems for balancing queries correctly. On the one hand, we showed that, unlike the baseline methods, SQLB maintains the high-interest, high-adaptation, and high-capacity providers in the system. On the other hand, the results show that while baseline methods lose more than 20% of consumers (for all workloads), SQLB has no consumer's departures! We showed the self-adaptability of SQLB to the expectations and satisfaction of participants. We also discussed its adaptability to different kinds of applications. All these results demonstrate that SQLB can scale up with autonomous participants, while Capacity based and Mariposa-like cannot.. As future work, we plan to address fault-tolerant issues so that SQLB ensures good system performance even in the presence of participants' failures. We also plan to study the impact, on performance and participants' satisfaction, of introducing the random providers selection phase (discussed in Sect. 5.3.2). In our experimentations, we also noted that self-organizing phenomenons occur. When the system is composed of several mediators and the participants have divergent interests, the participants that share the same interests gather on the same mediator. We wish to explore this phenomenon that we not observe with the other approaches. Finally, the problem addressed in this paper as that addressed by economic approaches is to regulate a distributed information system while satisfying participants. We then plan to develop an economic version of SQLB, based on [14] , to scale up to several mediators and to analyze in detail the contributions of the use of money for allocating queries. The proposed model can be applied with different purposes. First, to evaluate how well a query allocation method satisfies the participants' expectations. Second, to try to explain the reasons of the participants' departures from the system. For example, to know if they are leaving the system because (i) they are dissatisfied with the queries they perform, (ii) they are dissatisfied with the mediator's job, or (iii) the system is inadequate to them. To do so, one has to apply measures, which reflect a global behavior, over all concepts of the model: adequation, satisfaction, and allocation efficiency (see Sect. 4). Third, to design new self-adaptable query allocation methods that meet the participants' expectations in the long-run (see Sect. 5).. As noted earlier, even if the model can be applied to the preferences and intentions of participants, the interpretation of results is not the same. Thus, two different levels of satisfaction exist: at the preferences' and intentions' level. On the one hand, the satisfaction at the preferences' level reflects the happiness of a participant with what it is doing in the system. On the other hand, it is with the satisfaction at the intentions' level that a participant evaluates if the mediator generally gives to it the queries it asks for. Thus, a participant can know if it is properly computing its intentions by evaluating both satisfactions. For instance, a participant can observe that its expressed intentions do not allow it to be satisfied at its preferences' level even if the mediator does a good job for it and then it is satisfied at its intentions' level.. Moreover, notice that several possibilities to compute participants' satisfaction may exist. For example, participants' satisfaction may decrease with the time or consider the number of received queries. However, to explore and explain all the possibilities to compute participants' satisfaction is well beyond the scope of this paper. In fact, this could be the subject of a full paper. We thus report this to future work.. As final remark, reputation does not directly appear, but it is clear that it has a major role to play in the manner that participants work out their intentions. Thus, it is taken into account as much as participants consider it important.. We pointed out in Sects. 5.1 and 5.2 that there exist several ways a participant can compute its preferences. To the best of our knowledge, there is no work that proposes a comparison study of these different preference functions and hence it is still an open problem. We believe that such a study may be quite interesting to allow a participant knowing which strategy it can adopt to compute its preferences. Similarly, several manners to compute the consumers' and providers' intentions exist. This is also an open problem that should be explored so as to identify the best ways for a participant to adapt their intentions to their context and application. Improving on these functions is not the focus of our work. Instead, our framework is designed so it can leverage any existing preference and intention function.. Moreover, the score function of a query allocation method is usually based on specific demands, which are given by the application challenges that one wants to solve. Thus, a large number of specific query allocation methods with different behaviors may exist. For example, the score function of a qlb method is designed for those applications whose goal is to ensure good system performance. However, when the behavior of a query allocation method is specific to an application, it cannot be applied elsewhere, and worse, it cannot perform in environments where participants change their interests on the fly. To experimentally explore some of these implications is out of the scope of this paper.. Therefore, we proposed a score function that makes no assumption about either the kind of application nor the way in which a participant obtains its preferences. It just allocates queries based on the participants' intentions. But, we are aware that sometimes a mediator, or even the system administrator, is required to satisfy some constraint, e.g. to ensure a specific Quality of Service, no matter what the participants prefer. This is why we also proposed a strategy that allows the query allocation method to regulate the system with regards to a given function. As a result, conversely to specific query allocation methods, SQLB is quite general, self-adaptable to the interests of participants, and adaptable to the application. This allows SQLB to perform in many kinds of environments and to perform as well as any specific query allocation method by tuning its parameters or if participants desire so.. We assumed a mono-mediator system, i.e. a system that contains only one mediator to allocate queries. Clearly, a mediator may become a single point of failure and a performance bottleneck and thus one may desire to have more than one mediator in the system to allocate queries. In this case, SQLB does not scale well because it considers current participants' satisfaction, which a mediator can no longer compute itself as it also depends on the query allocations made by other mediators. Hence, when allocating a query, a mediator should keep informed all other mediators of the mediation result to update participants' satisfaction. This tends to significantly increase the network traffic. A way to avoid such a traffic overhead between mediators is that providers express their interest for queries through "monetary" bids so that mediators no longer consider the providers' satisfaction but only their bids. This requires introducing some "virtual" money to be used by providers and mediators. In this context, we are currently doing some work to show how to adapt SQLB to use virtual money and demonstrate that such an economic version can easily scale up to several mediators. However, a further discussion on this subject is not the focus of this paper (see [29, 30] ).. Finally, in large-scale distributed information systems participants may fail, usually because of network failures. Nevertheless, we do not deal with fault-tolerance issues in this paper since the main focus of this work is to evaluate SQLB from a satisfaction point of view. To make SQLB faulttolerant is the focus of one of our forthcoming work. Generally speaking, the idea is to enable a mediator to set, in a predictive way, the number of replicas that should be created for an incoming query based on participants' satisfaction. We start, in Sect. 6.3.1, by evaluating the quality of the three query allocation methods with regards to satisfaction and qlb. In Sect. 6.3.2, we evaluate how well these methods deal with the possible participants' departure by dissatisfaction, starvation, or overutilization. Then, in Sect. 6.3.3, we show the self-adaptability of SQLB to participants' expectations. In these three first sections, we assume that k n = k , i.e. set K n denotes set P q considering that k = N . Finally, in Sect. 6.3.4, we study the adaptability of SQLB to the kind of application by varying parameter k n . The query allocation problem, which appears as a subproblem of query processing [12] , is very general and is addressed in many domains such as distributed databases, networking systems, grid systems, and multi-agent systems. The assumptions and techniques to allocate queries often differ depending on the context and the system goals. To the best of our knowledge, the problem of allocating queries by considering qlb and the participants' intentions has not received much attention and is still an open field. In the remainder of this section, we discuss five main approaches related to our query allocation framework: economics, data mediators, multi-agent systems, load balancing approaches, and web services. Notice that the scope of this paper goes well beyond related work by characterizing the participants' expectations in the long-run, proposing measures to analyze them and new algorithms to exploit them.

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


Network Capacity Region and Minimum Energy Function for a Delay-Tolerant Mobile Ad Hoc Network Abstract-We investigate two quantities of interest in a delaytolerant mobile ad hoc network: the network capacity region and the minimum energy function. The network capacity region is defined as the set of all input rates that the network can stably support considering all possible scheduling and routing algorithms. Given any input rate vector in this region, the minimum energy function establishes the minimum time average power required to support it. In this work, we consider a cellpartitioned model of a delay-tolerant mobile ad hoc network with general Markovian mobility. This simple model incorporates the essential features of locality of wireless transmissions as well as node mobility and enables us to exactly compute the corresponding network capacity and minimum energy function. Further, we propose simple schemes that offer performance guarantees that are arbitrarily close to these bounds at the cost of an increased delay. Two quantities that characterize the performance limits of a mobile ad hoc network are the network capacity region and the minimum energy function. The network capacity region is defined as the set of all input rates that the network can stably support considering all possible scheduling and routing algorithms that conform to the given network structure. The minimum energy function is defined as the minimum time average power (summed over all users) required to stably support a given input rate vector in this region. Here, by stability we mean that the input rates are such that for all users, the queues do not grow to infinity and average delays are bounded. In this paper, we exactly compute these quantities for a specific model of a delay-tolerant mobile ad hoc network.. Asymptotic bounds on the capacity of static wireless networks and mobile networks are developed by [2] , [3] . The work in [3] shows that for networks with full uniform mobility, if delay constraints are relaxed, a simple 2-hop relay algorithm can support throughput that does not vanish as the number of network nodes N grows large. Recent work in [4] generalizes this model and investigates capacity scaling with non-uniform This work was supported in part by the DARPA IT-MANET Program under Grant W911NF-07-0028 and the National Science Foundation (NSF) under Grant OCE 0520324 and Career Grant CCF-0747525. This work was presented in part at the 4th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt), Boston, MA, April [3] [4] [5] [6] [7] 2006 .. The authors are with the Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 USA (e-mail: urgaonka@usc.edu; mjneely@usc.edu).. node mobility and heterogeneous nodes. Capacity-delay tradeoffs in mobile ad hoc networks are considered in [8] - [12] . Flow-based characterization of the network capacity region is presented in several works (e.g., [7] , [13] , [14] ).. However, little work has been done in computing the exact capacity and energy expressions for these networks. Exceptions include a closed form expression for the capacity of a fixed grid network in [5] , an expression for the exact information theoretic capacity for a single source multicast setting in a wireless erasure network [6] , and an expression for the capacity of a mobile ad hoc network in [8] that uses a cellpartitioned structure. The work in [8] quantizes the network geography into a finite number of cells over which users move, and assumes that a single packet can be transmitted between users who are currently in the same cell, while no transmission is possible between users currently in different cells. 1 In this work, we extend this model to more general scenarios allowing adjacent cell communication and different ratepower combinations. Specifically, we extend the simplified cell-partitioned model of [8] (which only allows same cell communication and considers i.i.d. mobility) to treat adjacent cell communication. We establish exact capacity expressions for general Markovian user mobility processes (possibly nonuniform), assuming only a well-defined steady-state location distribution for the users. Our analysis shows that, similar to [8] , the capacity is only a function of the steady-state location distribution of the nodes and a 2-hop relay algorithm is throughput optimal for this extended model as well. Further, our analysis illuminates the optimal decision strategies and precisely defines the throughput optimal control law for choosing between same cell and adjacent cell communication. We then use this insight to design a simple 2-hop relay algorithm that can stabilize the network for all input rates within the network capacity region. We also compute an upper bound on the average delay under this algorithm. (Sec. III). We next compute the exact expression for the minimum energy required to stabilize this network, for all input rates within capacity. Our result demonstrates a piecewise linear structure for the minimum energy function that corresponds to opportunistically using up successive transmission modes. Then we present a greedy algorithm whose average energy can be pushed arbitrarily close to the minimum energy at the cost of an increased delay. (Sec. IV) Before proceeding further, we emphasize that the network capacity and minimum energy function derived in this paper are subject to the scheduling and routing constraints of our model as described in the next section. Specifically, in this work, we do not consider techniques that "mix" packets, such as network coding or cooperative communication, which can increase the network capacity and reduce energy costs. In fact, in Sec. V, we present an example scenario that shows how network coding in conjunction with the wireless broadcast advantage can increase the capacity for this model. Calculating the network capacity region and the minimum energy function when these strategies are allowed is an open problem in general in network information theory and is beyond the scope of this paper. In this work, we investigated two quantities of fundamental interest in a delay-tolerant mobile ad hoc network: the network capacity and the minimum energy function. Using a cell-partitioned model of the network, we obtained exact expressions for both these quantities in terms of the network parameters (number of nodes N and number of cells C) and the steady-state location distribution of the mobility process. Our results hold for general mobility processes (possibly nonuniform and non-i.i.d.) and our analytical technique can be extended to other models with additional scheduling constraints.. We also proposed two simple scheduling strategies that can achieve these bounds arbitrarily closely at the cost of an increased delay. Both these schemes restrict packets to at most 2 hops and make scheduling decisions purely based on the current user locations and independent of the actual queue backlogs. For both schemes, we computed bounds on the average packet delay using a Lyapunov drift technique.. In this paper, we have focused on network control algorithms that operate according to the network structure as presented in Sec. II. We assumed that the packets themselves are kept intact and are not combined or network coded. As shown in the example in Sec. V, it is possible to increase the network capacity by making use of network coding and the wireless broadcast feature. An interesting future direction of this research is to determine the exact capacity region with such enhanced control options.

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


Pseudorandomness and Average-Case Complexity Via Uniform Reductions Abstract. Impagliazzo and Wigderson (1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worstcase hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper:• We obtain an optimal worst-case to average-case connection for EXP: if EXP ⊆ BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction 1/2 + 1/t (n) of the inputs by BPTIME(t (n)) algorithms, for t = t Ω(1) .• We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #P-complete problem with these properties.• We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions. Over the past two decades, a rich body of work has investigated the relationship between three basic questions in complexity theory:. 1. The existence of problems of high worst-case complexity in classes such as EXP, One approach to removing nonuniformity in derandomization results is to show that uniform lower bounds imply nonuniform lower bounds, in the spirit of Karp and Lipton [37] . In fact, one of the theorems in [37] , attributed to Albert Meyer, applies to EXP and shows that EXP = Σ 2 implies EXP ⊆ P/poly; the hypothesis was subsequently improved to EXP = MA in [5] . Thus, since we know how to get nontrivial derandomizations of BPP under the assumption EXP ⊆ P/poly, we also get the same derandomizations assuming EXP = MA [6] . However, this does not fully achieve the goal of removing nonuniformity, because the MA is not a uniform analogue of P/poly; the two classes seem incomparable. Indeed, Impagliazzo and Wigderson [35] take a much more indirect approach to get a derandomization of BPP under only the assumption EXP = BPP.. Still, this leads one to wonder whether one can prove a "super Karp-Lipton" theorem for EXP, showing that EXP = BPP if and only if EXP ⊆ P/poly, or more generally, that if EXP ⊆ BPTIME(t(n)) then EXP is not contained in circuit size t(n) Ω(1) . Such a result would be extremely important beyond the application to derandomization, and it requires a non-relativizing proof [20] . Interestingly, it is plausible that research in derandomization could lead to such a result. Indeed, Impagliazzo, Kabanets, and Wigderson [32] use derandomization techniques to prove that NEXP = MA if and only if NEXP ⊆ P/poly. Already, this is major progress towards a "super Karp-Lipton" theorem, because it establishes an equivalence between a uniform lower bound and a nonuniform lower bound. However, it still considers a uniform class (MA) that seems incomparable to the nonuniform class (P/poly).. Another interpretation of the Impagliazzo-Kabanets-Wigderson result is that any nontrivial derandomization of MA implies circuit lower bounds for NEXP, because a derandomization of MA (such as MA ⊆ NSUBEXP) would presumably separate MA from NEXP by the Hierarchy Theorem for Nondeterministic Time (or a variant). It also implies circuit lower bounds from derandomizations of promise-BPP, because these imply derandomizations of MA via the observation that MA = NP promise-BPP (cf. [26] ). Considering that Impagliazzo and Wigderson [35] prove that EXP = BPP implies a sub-exponential time derandomization of promise-BPP (for most inputs, on infinitely many input lengths), the two results together come close to proving that if EXP = BPP then NEXP ⊆ P/poly. Unfortunately, the result of [35] only implies a derandomization that works on most inputs of length n, for infinitely many n, which is not enough to derive a derandomization of MA = NP promise-BPP and apply [32] . Still, the combination of the results of [35, 32] come tantalizingly close to a "super Karp-Lipton" result, and provide motivation to further investigate uniform versus non-uniform results in derandomization and related topics.. cc 16 (2007) In this paper, our goal is to provide uniform versions of the known nonuniform trade-offs between worst-case complexity of problems in EXP, averagecase complexity of problems in EXP, and derandomization.. The uniform result by Impagliazzo and Wigderson [35] uses many previous theorems in complexity theory, some of which do not appear related to derandomization. In addition, unlike previous (and subsequent) derandomization results in the nonuniform setting, it was not stated as giving a continuous tradeoff between hardness and randomness. It also was not proved by (explicitly) presenting a uniform reduction from worst-case to average-case hardness, which is typically the first step in previous derandomization results. Thus, their work leaves several intriguing open questions:. • What is the best tradeoff between hardness and derandomization in the uniform setting? In particular, can a sufficiently strong uniform lower bound on E yield a polynomial-time deterministic simulation of BPP? By analogy with the nonuniform setting, we might hope to prove that if E ⊂ BPTIME(t(n)), then there is a pseudorandom generator that mapping ≈ n bits into roughly ≈ t(n) bits fooling uniform distinguishers running in time ≈ t(n) (which implies a time 2 O(t −1 (n)) simulation of BPP). Below, we refer to this as the ideal tradeoff.. • Of the several previous results that are being used, how many are really necessary, and what are the properties of EXP that are essential to the proof?. • Is it possible to prove, along similar lines, that if EXP = BPP then EXP contains problems that are hard on average? What is the best tradeoff for this worst-case vs. avg-case problem in the uniform setting?. 1.2. Some remarks on the Impagliazzo-Wigderson proof. We first revisit, in Section 3, the arguments of Impagliazzo and Wigderson [35] , and make the following observations.. 1. Analyzing [35] for general time bounds, we can obtain a pseudorandom generator that stretches ≈ n bits into ≈ t(n) bits that fools distinguishers running in time ≈ t(n), under the assumption that EXP ⊆ BPTIME(t(t(n))).. Recall that the ideal tradeoff, matching what is known in the nonuniform setting, has t(n) instead of t(t(n)).. Uniform reductions 335 2. As observed by Cai, Nerukar, and Sivakumar [15] , the ideal tradeoff for derandomization can be obtained from uniform lower bounds on #P (instead of EXP). That is, the same generator as above can be obtained under the hypothesis that #P ⊂ BPTIME(t(n)). The key property of #P that is used in [35, 15] is that it has a complete problem that is both "random self-reducible" and "downward self-reducible", namely the Permanent [60] .. 3. Result 1 is obtained by constructing two pseudorandom generators, one from an EXP-complete problem and one from the Permanent. If there is a time t(n) distinguisher for both generators, then they can be combined in a sophisticated way (not just by a uniform "reduction") to obtain a time t(t(n)) algorithm for EXP. This step makes crucial use of Toda's Theorem that Σ 2 ⊆ P #P [55] .. 4. Using the techniques of [35] , we can also obtain a uniform worst-case to average-case connection for EXP: if every problem in EXP admits a probabilistic algorithm that runs in t(n) time and solves the problem in a 1/2 + 1/t(n) fraction of inputs of length n, then (roughly) EXP ⊆ BPTIME(t(t(n)). This is another result that cannot be proved via black-box reduction. (Previously, a uniform worst-case/average-case connection was also known for #P, again using special properties of the Permanent [24, 16] .). The observations above set the stage for our work. One main goal is to remove the t(t(n)) lower bounds that are needed above, for they are far from matching the ideal tradeoff, which incurs only a polynomial loss in t rather than a composition. They give nothing, for example, with t(n) = 2 n , and more generally limit t to being at most half-exponential [44] . In terms of derandomization, this means that we cannot get anything near a polynomial-time deterministic simulation of BPP from such results.

B 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 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 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 B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 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 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 B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O


Maximal Profit Dimensioning And Tariffing Of Loss Networks With Cross-Connects In this paper will we present an approach to optimal dimensioning and tari ng of communication networks. We will choose link capacities, tari s and the routing strategy in order to maximise the pro t for the company operating the network. The tari s and grade of service are subject to regulatory constraints. It is assumed that we have an existing network structure consisting of a set of nodes and physical links. By cross-connecting tra c through nodes at a high bandwidth rather than multiplexing and demultiplexing it, a logical link (consisting of capacity on several physical links) is created. However, it may be better for an OD pair to take advantage of existing physical links rather than to initiate its own logical link. Several results will be presented. These include a simple formula for the optimal tari and a conjecture that only one of the possible routes for each OD pair will be used in the optimal solution. A numerical investigation will also be discussed. Loss networks can be used to model circuit switched telephone networks, along with many other practical networks. Loss networks consist of sets of resources that users of di erent types try to access. If the required resources are not available, one or more alternative sets may be tried, but ultimately a user whose request cannot be satis ed is lost from the system. Many references to research on the dimensioning of loss networks are given in Bean and Taylor (1995) (see for example Girard (1990) and Kelly (1991) ). This paper is an extension of the work presented there.. As explained in Bean and Taylor (1995) , the usual method used in loss network dimensioning is that of minimising network cost subject to grade of service constraints. However, the approach used there, and also here, takes into account the fact that telephone companies are now operating in a private enterprise environment, by maximising network pro t. That is, the di erence between the revenue generated by network users and the cost of providing the network is maximised.. Although this approach has been discussed previously (see Kelly (1988) and Girard (1990) ), Bean and Taylor (1995) incorporate the concept of a tra c elasticity function. This function acknowledges the fact that the tra c o ered to the network is a decreasing function of the tari charged to users. Such a tra c elasticity function could re ect factors such as a competitor's tari structure or the grade of service o ered by the network.. In this paper we discuss the introduction of logical links. A logical link consists of reserved capacity on a set of two or more physical links. Tra c using the logical link is cross-connected through intermediate nodes rather than being multiplexed and de-multiplexed at these nodes. For instance, in the Australian network (see Figure 1 ), tra c travelling from Perth to Melbourne on the logical link would use reserved capacity on the Perth to Adelaide and Adelaide to Melbourne links and be cross-connected through Adelaide. Although this may be an advantage in some cases, it also introduces ine ciencies as the reserved capacity is allocated when the network is con gured rather than being used on demand.. In the next section of this paper we de ne our model and discuss methods of network analysis. In Section 3 we explain the network costing, improvements in our model and present our formulation. Section 4 contains a numerical example, Section 5 contains theoretical results concerned with optimal tari ng and route choice. Our conclusions are presented in Section 6. In this paper we have presented a formulation for the determination of the optimal dimensions, tari s and splitting probabilities for a loss network in order to maximise the operating company's pro t. We allow the tra c arrival rate on a stream to depend on the tari charged for that stream. Our formulation uses the Erlang xed point approximation and uses link blocking probabilities rather than link capacities as variables, which provides large computational savings. As a result we are able to solve problems of a realistic size.. By using this formulation we proved a result that explains the optimal tari as the cost to the network of carrying the call plus a term depending only on the elasticity function. Finally, we also conjectured that for any given stream either all calls will use the physical path or all calls will use the logical path. Using these results we hope to be able to develop an algorithm to simplify the optimisation procedure further. This work is currently being extended to allow calls to use paths which consist of a combination of logical and physical links. Similar results are anticipated.

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


Star-Free Languages are Church-Rosser Congruential Abstract. The class of Church-Rosser congruential languages has been introduced by McNaughton, Narendran, and Otto in 1988 . A language L is Church-Rosser congruential (belongs to CRCL), if there is a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S. To date, it is still open whether every regular language is in CRCL. In this paper, we show that every star-free language is in CRCL. In fact, we prove a stronger statement: For every star-free language L there exists a finite, confluent, and subword-reducing semi-Thue system S such that the total number of congruence classes modulo S is finite and such that L is a union of congruence classes modulo S. The construction turns out to be effective. Church-Rosser congruential languages (CRCL) are a nonterminal-free form of Church-Rosser languages (CRL). Both classes have been defined in [9] , and it was shown there that CRCL forms a proper subclass in CRL. Languages in CRL enjoy various nice properties. For example their word problem is decidable in linear time. A detailed discussion with links to further references can be found in the PhD-thesis of Niemann [12] , see also [13] . We content ourselves to define CRCL: A language L ∈ A * is called a Church-Rosser congruential language, if there is a finite, length-reducing, and confluent semi-Thue system S ⊆ A * × A * such that L is a finite union of congruence classes modulo S. This means that L contains a finite set F of shortest words such that we have w ∈ L if and only if every rewriting procedure starting on w and using S terminates in one of the finitely many words in F .. It was also shown in [9] that all deterministic context-free languages are ChurchRosser. However, surprisingly it is not known whether all regular languages are CRCL. The general conjecture is "yes", but so far only partial results have been established as in [14] . The most advanced result has been announced by Reinhardt and Thérien [15] : According to their manuscript, if a regular language has a group as its syntactic monoid, then this language is in CRCL.. In this note we consider the complementary class of group-free regular languages; and we show that they belong to CRCL. A regular language is group-free if its syntactic monoid is group-free. This means it is aperiodic. There are many other characterizations for this class. A fundamental result of Schützenberger says that the class of aperiodic language AP(A) is exactly the same as the class of star-free languages SF(A) [17] . It is the class where the Krohn-Rhodes decomposition leads to a wreath product of the three-element commutative idempotent reset-monoid U 2 [8] . It is also the class FO(A, <) of languages definable in first-order logic [10] ; and this is the same as the class LTL(A) of languages definable in the linear temporal logic [7] .. A proof that FO(A, <) = SF(A) = AP(A) = LTL(A) can be conveniently arranged in a cycle. The inclusion FO(A, <) ⊆ SF(A) can be explained very nicely with Ehrenfeucht-Fraïssé-games [5] . The inclusion SF(A) ⊆ AP(A) follows Schützenberger's original idea. The inclusion AP(A) ⊆ LTL(A) is done in the survey [3] with the concept of local divisors which play a prominent role here, too. The final inclusion LTL(A) ⊆ FO(A, <) is trivial.. Coming back to the class of Church-Rosser congruential languages, our main result shows SF(A) ⊆ CRCL. Actually, we prove a much stronger result. First we define subword-reducing semi-Thue systems which are a proper subclass of finite length-reducing semi-Thue systems. For every language L ∈ AP(A) we effectively construct a finite subword-reducing confluent semi-Thue system S ⊆ A * × A * such that the total number of congruence classes modulo S is finite and L is a union of such classes, see Theorem 6. A main tool in our proof is the notion of local divisor, see Section 3 for a definition.. In the final section of this paper, Section 5, we explain our constructions in a rigorously algebraic framework. This part is mainly intended for possible future work.. In order to give a complete positive solution to the conjecture that all regular languages are CRCL, it remains to combine our approach with the one in [15] . There are however quite a number of obstacles for a fruitful combination. So, we leave the general conjecture as a challenging research problem.

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