What Information Can or Cannot Be Exchanged? What information can or cannot be exchanged between different systems? Since the dawn of the computer-aided design and engineering era, interoperability has been an issue. The exchangeable set of information between different systems has been loosely defined as the intersection of information. Yet, as information flow has directionality, a general definition of "intersection" is inadequate to define an exchangeable set of information. This paper proposes a new directive set operation, semantic intersection, and discusses which information can or cannot be exchanged between different systems using the concept of semantic intersection. This opens up the possibility of predetermining or calculating the exchangeable set of information between two or more systems or the possibility of automated generation of a standard product model between different systems. This paper focuses on data exchange between systems, but the proposed theory is also applicable to data exchange between human beings. The flow of information follows a natural direction just as heat flows from hotter objects to colder ones and water flows down from a height. However, unlike the flow of heat or water, the flow of information is not always unidirectional. Information generally flows from the more informed to the less informed; yet, depending on the contents of the information, it can sometimes flow in the opposite direction. In some cases, a set of information can be exchanged freely, with little constraint, between two agents. Conversely, in other cases, the exchange of information in one direction is practically impossible unless forced by making artificial assumptions. For example, although it is possible to derive twodimensional representations from a three-dimensional representation, it is generally very difficult to reverse the process without making wild guesses to fill the missing information gaps.. Since the dawn of the computer-aided design ͑CAD͒ and engineering era, interoperability ͑data exchange capability͒ has been an issue ͑Goldstein et al. 1998͒. An early solution for interoperability issues was the single standard product model, which resulted in ISO 10303 STEP ͑ISO TC 184/SC 4 1994͒, CIS/2 ͑Crowley 2003͒, and industry foundation class ͑IFC͒ ͑buildingS-MART 2009a͒. Many successful case studies were conducted on the deployment of such standard data models ͑AISC 2009; Smith 2002͒. However, the same number of interoperability problems has since been reported ͑Amor 2006; Gielingh 2008; Jeong et al. 2009; Kiviniemi et al. 2008; Lim et al. 2008; Pazlar and Turk 2008͒ . Even though some of these problems can be attributed to the standard data model issues and system bugs, some information is not meant to be exchanged between two different systems.. The sources of exchange problems can be classified into the following four groups. ͑1͒ Incomplete coverage of a data model (or an exchange format): A data model ͑or an exchange format͒ such as ISO STEP and IFC does not fully support the intended data exchange scenarios. For example, if a data model does not include a definition of a reinforcing bar, reinforcing bar information cannot be exchanged using the data model. Gielingh ͑Gielingh 2008͒ identified this as one of the major causes of interoperability problems. ͑2͒ Translator problems: This is also a common cause of the interoperability issues today ͑Pazlar and Turk 2008͒. Translators are the modules of a system that import nonnative file formats or export the native file format to other formats. Translator problems occur because each translator developer develops translators without specific development guidelines. This type of problem can be relieved by the use of specific translator development guidelines; one such effort is the information delivery manual ͑buildingSMART 2009b͒. ͑3͒ System bugs: Data exchange problems can occur simply due to system bugs. For example, a data set that is read into a system correctly can be displayed different from the original data set simply because of a bug in the visualization module. ͑4͒ Software domain problems: A software application does not support the domain or the object. For example, a design software application usually does not support structural loads. In such cases, even if structural load information is read into a design system, the information cannot be interpreted and will be abandoned.. Given that some types of information are meant not to be exchanged between two systems that are different by nature, what information can or cannot be exchanged between different systems? Intersection has been used to loosely define an exchangeable set of information ͑Gielingh 2008͒. Obviously, the intersection of two information sets is exchangeable between the systems. For example, if one system includes "project code" information and the other system includes the same information, the two systems can exchange the information project code. In reality, each system defines a data field in its own way. For example, project code may mean the "identifier number of a project" in one system, but "short project name" in the other. Furthermore, if System A stores the "start date" and "end date" of a project and System B stores only the "project duration," System A can derive the project duration by subtracting end date from start date, which can be passed to System B, but not vice versa. Here, we see that the traditional set-theoretic notion of intersection is not adequate to define the exchangeable set of information between different systems. Hence, we propose a generalized and formal definition of such exchangeable and nonexchangeable sets of information between two different systems by introducing the concept of "semantic intersection." We argue that three types of information sets can be exchanged between two different systems: ͑1͒ the intersection of two sets; ͑2͒ a set of information in the synonym relation; and ͑3͒ a set of information that is functionally dependent on another set. However, note that the semantic intersection does not guarantee the validity of exchanged information. The issue of a valid set has been discussed in length in another paper ͑Lee 2009b͒ focusing on information sets specified in EXPRESS language. In order to define an exchangeable set of information between different systems that satisfy all these criteria, we proposed a formal definition of an exchangeable set of information and its characteristics using a new set-theoretic set operation semantic intersection based on the theory of FD. We also defined the semantic subset and the "scope of a standard product model" based on the definition of the semantic intersection. The specified definitions imply several apparent yet important facts about data exchange between two systems. First, there is always a set of information that cannot be exchanged between two systems regardless of the size of the coverage of an exchange data format or how well translators are developed. The information that is not the semantic intersection of two application data models cannot be exchanged. Therefore, there cannot be complete lossless data exchange between two different types of applications. Furthermore, a standard product model also cannot guarantee lossless data exchange. Second, we can also conclude that the more similar two application types are, the more information they can exchange. Third, the set of information that can be exchanged between two systems differs depending on the direction of data exchange. Fourth, with all these limitations a standard product model is still useful in supporting data exchange between various systems because data exchange in practice is not the data exchange of even the entire semantic intersection of two information sets, but only a part of it, i.e., what is required to perform targeted work.. The specified definitions and theorems provide an explicit definition of an exchangeable set of data between different applications. This opens up the possibility of predetermining or calculating the exchangeable set of information between two or more systems and also the possibility of automating the generation of a standard product model between different systems based on Definition 5: the scope of a standard product model. As an application example, this paper demonstrated the development process of a product model that could support the exchange of beam information between four different systems.. There remain, however, many more things to be done. Data dictionaries and tools that can support mapping between homonyms, synonyms, and derived items are required to accelerate the semantic mapping process. Once such tools and dictionaries are developed, the theorems can be used to validate the mapping relations. Although the mapping process may still be tedious, once the mapping is completed, millions of data instances can be exchanged automatically. One effort to develop a data dictionary for the AEC industry is the international framework for dictionaries ͑IAI 2007͒. In the long term, we hope that the proposed definitions and theorems can contribute to the understanding of data exchange between both systems and human beings.

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


Elastic DVS Management in Processors With Discrete Voltage/Frequency Modes techniques to real-time systems running on processors with discrete voltage/frequency modes causes a waste of computational resources. In fact, whenever the ideal speed level computed by the DVS algorithm is not available in the system, to guarantee the feasibility of the task set, the processor speed must be set to the nearest level greater than the optimal one, thus underutilizing the system. Whenever the task set allows a certain degree of flexibility in specifying timing constraints, rate adaptation techniques can be adopted to balance performance (which is a function of task rates) versus energy consumption (which is a function of the processor speed).In this paper, we propose a new method that combines discrete DVS management with elastic scheduling to fully exploit the available computational resources. Depending on the application requirements, the algorithm can be set to improve performance or reduce energy consumption, so enhancing the flexibility of the system. A reclaiming mechanism is also used to take advantage of early completions. To make the proposed approach usable in real-world applications, the task model is enhanced to consider some of the real CPU characteristics, such as discrete voltage/frequency levels, switching overhead, task execution times nonlinear with the frequency, and tasks with different power consumption. Implementation issues and experimental results for the proposed algorithm are also discussed. The problem of minimizing energy consumption while guaranteeing real-time constraints has been widely addressed in the real-time literature. However, most of the achieved results were derived under simplified system models, where the processor, for example, can change its voltage and frequency within a continuous range. Unfortunately, most of the current commercial processors [2] , [11] only provide a limited number of operating modes, each characterized by a given voltage, clock frequency, and power consumption. Adapting a continuous model to a discrete DVS system clearly causes a waste of computational resources, because, in order to guarantee the feasibility of the task set with a single speed level, the processor speed must be set to the nearest level greater than the optimal one [21] .. Other authors focused on energy-aware scheduling for specific task models. The most investigated task model is the periodic one [3] , [4] , [13] , [28] , but energy-aware algorithms have also been proposed and analyzed for aperiodic tasks [25] , sporadic tasks [22] , and mixed task sets [26] . To deal with processors having discrete speed levels, some authors proposed to split tasks into parts and run each part at a different frequency [12] , [15] , [28] .. Mejia-Alvarez et al. [19] proposed an approach where each task is assigned a different frequency; however, the processor model is simpler than the one used in this paper, and the frequency assignment problem is NP-hard; thus, it can be treated online only by a heuristic algorithm.. More recently, Bini et al. [6] presented a method for approximating any speed level with two given discrete values, which are properly switched as a pulse width modulation signal to obtain its average value. Schedulability analysis to guarantee the feasibility of real-time task sets was also presented.. In order to guarantee task timing constraints, most of the algorithms for hard real-time systems perform the analysis assuming that each task executes for its worst-case execution times (WCETs). This is usually a strong conservative hypothesis that may cause a waste of computational resources. To exploit the additional slack coming from early completions, some authors [5] , [13] proposed to mix an offline approach based on WCETs with an online method, which is in charge of reclaiming the unused computation time for a further reduction of the CPU frequency.. Reclaiming algorithms can be distinguished in two categories, known as inter-task and intra-task methods. Inter-task algorithms decide the working frequency on a task-by-task basis, while intra-task algorithms may adjust the frequency, even within a single task [20] . Some algorithms mixing intra-task and inter-task approaches have also been proposed [13] .. The use of elastic scheduling has already been proposed for improving DVS management [16] , but the method is based on WCET estimations and is only applicable offline; thus, all the unused computation time leads to a waste of energy. Moreover, the algorithm is presented without a set of tests to experimentally validate the approach.. In this paper, an online reclamation algorithm works in combination with an offline method to achieve a further reduction of the processor speed whenever a job completes earlier than expected. A set of experimental results are presented to show various aspects of the algorithm and describe the effects of some user-defined parameters.

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


A priori experimental design for inverse identification of magnetic material properties of an electromagnetic device using uncertainty analysis Purpose – The purpose of this paper is to determine a priori the optimal needle placement so to achieve an as accurate as possible magnetic property identification of an electromagnetic device. Moreover, the effect of the uncertainties in the geometrical parameter values onto the optimal sensor position is studied.Design/methodology/approach – The optimal needle placement is determined using the stochastic Cramer‐Rao lower bound method. The results obtained using the stochastic method are compared with a first order sensitivity analysis. The inverse problem is solved starting from real local magnetic induction measurements coupled with a 3D finite element model of an electromagnetic device (EI core inductor).Findings – The optimal experimental design for the identification of the magnetic properties of an electromagnetic device is achieved. The uncertainties in the geometrical model parameters have a high effect on the inverse problem recovered solution.Originality/value – The solution of the inverse prob... Electromagnetic devices (EMDs) are widely used in industry applications. The accurate knowledge of the magnetic material characteristic of the EMD is an essential aspect for the precise design and analysis of such applications. The magnetic material characteristic of an EMD, before fabricating it, is classically determined by means of Epstein frame or single sheet tester on a separate sheet of the same material as the EMD ( Jiles, 1991) . However, the magnetic material properties are changed dramatically after manufacturing process (Belahcen and Arkkio, 2008) . So, performing the traditional measurements techniques on a separate sheet of the same material as the EMD, which are mostly not available, are not useful. Therefore, it is convenient to characterize the magnetic properties on the specific geometry of the EMD itself. This can be done using simultaneous local magnetic field-induction measurements, at a same position. However, the non-uniform electromagnetic field patterns make this task much more difficult . So that an inverse approach needs to be solved where the measured signals are interpreted through the use of a numerical model (Ioan and Rebican, 2002) . It has been proposed in such inverse approaches that the material properties can be reconstructed starting from local magnetic measurements in conjunction with a numerical model of the EMD, by iteratively minimizing the error between the measured and simulated quantities .. However, a key question that arises in solving such inverse problems is what the optimal experimental design is (where to place the sensors, i.e. needles), so that the errors in the recovered magnetic properties are limited as much as possible since the measurements contain noise. In addition, the numerical model contains geometrical parameter values uncertainties, which may affect the accuracy of the inverse problem. The above question can be answered by solving the inverse problem several times for different needles positions; however, this is time consuming. So, determining a priori the optimal needles position is an important preliminary task in order to enhance the accuracy and to reduce the computational time of the inverse problem.. The research presented in this paper aims at finding a priori the optimal needles placement for the most accurate identification of the magnetic properties of an EMD. Moreover, the effect of the uncertainties in the geometrical parameter values of the EMD on the optimal sensor position is introduced. Finally, the presented results are compared with a first-order sensitivity analysis (FOSA). The optimal experimental design (optimal needles position) for identifying the magnetic material properties of an EMD, by solving an inverse problem starting from the local induction measurements, is presented. The presented methodology uses the sCRB method as an optimal criterion. Moreover, the effect of the measurement noise and uncertainties for two geometrical model parameters on the accuracy of the recovered material parameters is introduced. Furthermore, the proposed stochastic methodology is validated and compared numerically using a deterministic technique based on a FOSA. From the results presented in this study, it is obvious that the better prediction of the material characteristics is obtained when the measurements are carried out at positions with less stray fields. The behavior of a magnetic system can be represented by a mathematical model with a set of partial differential equations. This model is parameterized by the following model parameters: the unknown parameters u [ R p£1 and the known parameters b [ R q£1 . The values of the parameters b are assumed uncertain. In order to estimate the unknown parameters u, an inverse problem has to be solved by iteratively minimizing the sum of the quadratic residuals between the experimental observations of the magnetic system W [ R K£1 and the modeled ones F [ R K£1 , with K being the total number of discrete experimental observations. In other words, the functional G(u) needs to be minimized with G given as:. The resolution of the inverse procedure highly depends on both measurements and modeling accuracy. G can depend on the definition of the inverse problem, i.e. objective function to be minimized. This is due to measurement noise and available uncertainties in the forward model parameter values. Below, two uncertainty analysis techniques are introduced; stochastic, which is based on a Cramér-Rao bound (CRB) method (Sections 2.1 and 2.2), and a deterministic technique, which is based on the FOSA (Section 2.3).

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


A spatial temporal decision framework for adaptation to sea level rise There is a strong link between decision making and environmental stresses. Two dilemmas confront decision makers: how and when to adapt to sea level rise, due to complexities of environmental systems and the changing nature of the decision making process. This process is inherently complex and often involves many stakeholders with conflicting views. Considering the complexity and dynamic nature of coastal systems, this paper introduces a Spatial Temporal Decision framework to assess coastal vulnerability, and the adaptation alternatives to SLR. The STD is based upon a combination of: System Dynamics modelling; Geographical Information Systems modelling; and multicriteria analyses of stakeholders' views using the Analytical Hierarchy Process. For case study analyses, the City of the Gold Coast located in Southeast Queensland, Australia has been selected. The results of the vulnerability assessment indicate that, at the end of a 100 year simulation period, approximately 6% of the landscape in the study area will be gradually inundated over time, with 0.5?cm rise per year. However, the percentage of the vulnerable area leapt to about 34% for Scenario 2, and 56% for Scenario 3, which represent 1?cm and 1.5?cm rise per year. Using the information obtained from vulnerability assessments, three stakeholder groups (Politicians, Experts and Residents) were consulted to determine the goal, criteria and adaptation alternatives for the multicriteria analyses. Analyses of survey data reveal that across the three stakeholder groups, Effectiveness and Sustainability are the criteria of highest priority. An innovative characteristic of the STD approach is its ability to evaluate the decision choices prior to their implementation. This is achieved by incorporating the DSM simulation results into the decision making process and, then, retesting the information, obtained from this process, using the DSM. The model's ability to pre-evaluate decision choices is an important feature; its legacy is that communities can avoid or minimise their decision error, and increase their chance of obtaining better decisions.. The STD approach, in summary, provides a critical tool for obtaining quantitative information for managing and making choices with the aim of effective decisions. This There is overwhelming scientific consensus over the causes and impacts of climate change (IPCC, 2007) . Sea level rise (SLR) is one of the most recognized possible impacts of changing climate. Coastal areas are economically productive and three times more densely populated than the global average (Small and Nicholls, 2003) . Clearly, while communities have benefited from the many advantages of living in these areas, inevitably they also face the threat of natural disasters and specifically from SLR via permanent inundation of low-lying regions, inland extension of episodic flooding, increased beach erosion and saline intrusion of aquifers (McLean et al., 2001 ).. Coastal communities have been adapting to changing conditions throughout history.. However, faced with increased threats due to SLR, coastal communities must act faster to develop more effective management policies. Moreover, the impacts of SLR are not expected to be spatially uniform across the world (Nicholls et al., 2007) . It is, therefore, essential for decision makers (DM) to consider the dynamic and spatial characteristics of these changes in assessing the impacts of SLR when making decisions about the future.. There is a range of analytical tools available to improve decision makers' (DM) ability to understand and evaluate environmental management problems such as simulation models, GIS, and experts systems. However, although these tools provide invaluable information for decision making, each tool addresses only one aspect of a management problem. Therefore, effective decision making, in a dynamic complex environment, requires the expansion of the mental modelling boundaries and the development of additional tools to help DMs better understand how complex systems behave. Thus, DMs need to integrate each tools' analytical results into a rational choice about what to do, where to do it, and when to do it (Schmoldt, 2001 ).

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


On the complexity of quantified linear systems In this paper, we explore the computational complexity of the conjunctive fragment of the firstorder theory of linear arithmetic. Quantified propositional formulas of linear inequalities with (k − 1) quantifier alternations are log-space complete in Σ P k or Π P k depending on the initial quantifier. We show that when we restrict ourselves to quantified conjunctions of linear inequalities, i.e., quantified linear systems, the complexity classes collapse to polynomial time. In other words, the presence of universal quantifiers does not alter the complexity of the linear programming problem, which is known to be in P. Our result reinforces the importance of sentence formats from the perspective of computational complexity. In the first-order theory of linear arithmetic, the sentences are arbitrary propositional combinations of linear inequalities. The theory of linear arithmetic is itself a sub-theory of the theory of real closed fields. Both these theories have been widely studied in the literature [1, 2, 3] , since they find applications in a number of domains including but not limited to constraint databases [4] and model checking [5] . This paper examines the computational complexity of the conjunctive fragment of the first-order theory of linear arithmetic. Consider the sub-class of formulas in the theory of reals with addition and order starting with an existential quantifier and with (k − 1) quantifier alternations:. where Q is ∃ for k odd and ∀ for k even, and F(x 1 , . . . , x k ) is a propositional combination of linear inequalities over variable sets x 1 , . . . , x k . This sub-class has been shown to be log-space complete in Σ P k [6] . It is legitimate to ask ourselves whether this complexity result is attributable to the fact that F is a propositional combination or, rather, to an intrinsic complexity of quantification over linear inequalities. We answer this very question in this paper. We show that when F is a conjunction of linear inequalities, i.e., a linear system, then the complexity of deciding whether the formula above holds is in P. It is well-known that linear programming is decidable in P [7, 8, 9] . Our results in this paper, extend the work in [7] to the case of quantified linear systems. All the proofs provided in this paper are constructive, with algorithms based on solving a number of linear programming problems.. The rest of the paper is organized as follows. We first recall background notation in Section 1.1, and related work from the literature in Section 1.2. Section 2 provides a formal description of the problem under consideration. Section 3 discusses the principal complexity result in this paper. We conclude in Section 4, by summarizing our contribution and describing avenues for future work. We have investigated the computational complexity of the conjunctive fragment of the theory of linear arithmetic, called quantified linear systems. This class extends the satisfiability problem of linear systems, which is known to be in P, and is included in the theory of real numbers with addition and order, which is PSPACE-hard. We have shown that the decision problem for QLSes is in P. Our proofs are constructive, in that we Turing-reduce the QLS decision problem to linear programming. This result demonstrates that the complexity of the theory of real numbers with addition and order is attributable to the presence of propositional connectives and of quantifiers, rather than only to quantifiers. Hence, it reinforces the importance of sentence formats from the perspective of computational complexity.. In quantified linear systems, variables are unbounded on both sides, i.e., there is no upper or lower bound imposed on their values. A question that arises naturally is whether the presented complexity results change, when bounds are placed on the variables. The case where each universally quantified variable has an upper and a lower bound has been studied in [28] . The class of ∀∃ formulas for this problem is coNP-complete and the problem itself is in PSPACE; establishing its hardness however remains open. We are particularly interested in the computational complexity of Quantified Linear Systems, in which each universally quantified variable has an upper bound or a lower bound (but not both). Since Tarski [1] established a decision procedure for the full elementary theory of real closed fields with addition (+), multiplication (·) and order (<, =), a number of quantifier elimination methods have been proposed [11, 12] together with approaches efficient-in-practice, including partial cylindrical algebraic decomposition [2] provided in the QEPCAD/QEPCAD-B systems [13] , virtual substitution of test terms [14] provided in the REDLOG system [15] , and the RSolver algorithm [16] , whose implementation is publicly available [17] . Although it has been improved several times [3, 18] , the complexity of quantifier elimination is, in the worst case, doubly exponential in the number of quantifier alternations and exponential in the number of variables [12, 19] .. It is important to note that the real numbers themselves cannot be fully axiomatized by a first-order theory. This can be seen in Tarski's axiomatization of the reals, which requires a nonfirst-order axiom to express the Dedekind completeness of the real numbers. Briefly, Dedekind completeness is the property that all bounded subsets of real numbers must have a real least upper bound and real greatest lower bound. The axiom in question involves universal quantification over subsets of the real numbers, which cannot be done with first-order logic.. Any field which satisfies the same first-order properties as the real numbers is called a real closed field. One example of a real closed field is the set of real algebraic numbers, i.e., the set of all real numbers that are the roots of non-zero polynomials with rational coefficients. While the real algebraic numbers are a real closed field, they are not Dedekind complete, since it is possible to construct a set of rational numbers, which are all algebraic, that have a supremum of π which is not. It is also easy to see that the rationals do not form a real closed field; for instance, the solution of the algebraic equation x 2 − 2 = 0 is not a rational number. The complexity of sub-classes of the full elementary theory of the reals has been considered. The existential theory of the reals is obtained by restricting to existentially quantified formula ∃x F(x) where F(x) is a quantifier-free formula. The computational complexity of the decision problem is singly exponential in the number of quantified variables x [20] .. The theory of reals with addition and order is obtained by restricting to the function symbol + and predicate symbols < and =. In [21] , a quantifier elimination procedure is provided for sentences in this theory that is singly exponential in space and doubly exponential in time. The procedure replaces unbounded quantified variables with variables that are quantified over a finite set of rational numbers. An exponential time lower bound is shown in [22] , where Berman studies Pressburger arithmetic (which is the first-order theory of natural numbers with addition) and determines the time and space complexity of both theories at various levels of quantifier alternation. Consider a formula in the theory of reals with addition and order in prenex normal form with (k − 1) quantifier alternations. where Q is ∃ for k odd and ∀ for k even, while F(x 1 , . . . , x k ) is a quantifier-free formula. Sontag [6] shows that this class of formulas is log-space complete in Σ P k . He does so by converting any such formula into a disjunction of conjunctions of equalities and strict inequalities, and using the fact that the extreme points of polyhedra are polynomial in the size of the input.. In this paper, we deal with a sub-theory of the reals with addition and order, where the F(x 1 , . . . , x k ) above consists of conjunctions of linear inequalities, namely a linear system. For k = 1, the decision problem for such a formula boils down to to satisfiability of linear systems, which is known to be in P. We adhere to standard notation of linear algebra [10] . R is the set of real numbers. Small bold letters (a, b, . . . ) denote column vectors, while capital bold letters (A, B, . . . ) denote matrices. 0 is a column vector with all elements equal to 0. The transposed vector of a is denoted by a T . The inner product is denoted by a T · b. We write A · x ≤ b for a system of linear inequalities over the variables in x, also called a linear system. We assume that the dimensions of vectors and matrices in inner products and linear systems are of the appropriate size. A polyhedron is the set of solution points of a linear system:. is the problem of finding max {c. is empty, the problem is said to be infeasible. If the linear function c T · x has no upper bound, the problem is said to be unbounded.

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 B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 B I I 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 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 B I I 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 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 B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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 review of diffusion tensor magnetic resonance imaging computational methods and software tools In this work we provide an up-to-date short review of computational magnetic resonance imaging (MRI) and software tools that are widely used to process and analyze diffusion-weighted MRI data. A review of different methods used to acquire, model and analyze diffusion-weighted imaging data (DWI) is first provided with focus on diffusion tensor imaging (DTI). The major preprocessing, processing and post-processing procedures applied to DTI data are discussed. A list of freely available software packages to analyze diffusion MRI data is also provided. Whereas conventional magnetic resonance imaging (cMRI) provides methods to map the anatomy or tissue volume, diffusion-weighted imaging (DWI) of random translational water molecules offers quantitative anisotropy and orientation information that has been utilized to map the integrity or architecture of the soft tissue in the central nervous system [1] [2] [3] [4] [5] [6] . Contributors to diffusion tensor anisotropy include cellular membranes, axons, myelin sheaths, and other factors [7] . Water molecular diffusion in cerebral white matter is less restricted along the axon than perpendicular to the compact bundles and hence it is termed anisotropic (see Figure 1 ). Gray matter is less anisotropic, while diffusion in barrier free tissue (e.g. edema, cerebrospinal fluid) is isotropic [8] [9] [10] . In this work, we have provided an up-to-date review of computational methods applied to diffusion MRI data with focus on the single diffusion tensor model. We have also listed a host of software packages that have been reported in the diffusion MRI literature. To-date there has been no general consensus on the optimal diffusion tensor acquisition protocol (e.g. ∼ 7 minute single tensor vs. ∼ 30 minute DSI). Attention must be paid to data acquisition details (e.g. SNR and spatial resolution) while minimizing image distortion and motion artifacts and keeping the scan time tolerable for imaging children and patients. Quality assurance measures to ensure consistency and accuracy of the DTI data need to be implemented before comparing healthy control and patient data collected for cross-sectional and serial analyses. There are trade-offs, strengths and weaknesses associated with the choice of diffusion data analysis strategy (e.g. deterministic vs. probabilistic tracking, ROI, atlas-based, VBA vs. TBSS). Illustration of diffusion anisotropy using the ellipsoid representation of the single tensor model.

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


Maximum Renamable Horn Sub-CNFs The NP-hard problem of finding the largest renamable Horn sub-CNF of a given CNF is considered, and a polynomial time approximation algorithm is presented for this problem. It is shown that for cubic CNFs this algorithm has a guaranteed performance ratio of 40 67 . Let us consider n propositional variables V = {x 1 , ..., x n }, and let L = {x 1 , x 1 , ..., x n , x n } denote the set of corresponding literals, where x i = 1 − x i denotes the complement of x i . Let B = {0, 1}, and let us denote by x = (x 1 , ..., x n ) ∈ B n the vectors of binary assignments to the propositional variables. A clause is the disjunction of a subset of the literals, and a conjunctive normal form (or CNF in short) is a Boolean formula of the form. where C ⊆ 2 L is a family of clauses. To simplify notation we shall identify clauses with the corresponding subsets of literals, as well as CNFs with the corresponding families of subsets of literals, i.e. we shall also refer to the CNF (1) simply as a family C, and to its clauses as subsets C ∈ C of literals.. The satisfiability problem (SAT) consists of finding a binary assignment x * ∈ B n satisfying all the clauses of a given CNF C, i.e. for which Φ C (x * ) = 1.. The satisfiability problem is a frequently studied important problem, which is well-known to be computationally difficult (see e.g. [13, 20] ). Computationally efficient algorithms are only available for some very special classes of Boolean formulae.. Several algorithms for SAT exploits the special structures for which there are polynomial time solutions available, either by recognizing that the initial CNF simplifies to one of these special type after some partial assignment (i.e. after fixing some of the variables at some binary values), or by discovering that a subset of the clauses form a CNF belonging to one of these special classes. For a brief overview of these type of results and techniques we refer the reader to [14] .. In this paper we consider the second type of use of the special classes. The fact that a subfamily C ⊆ C belongs to a class for which SAT is "easy" can be used in several different ways. If it happens that C \ C = ∅, then the SAT problem for C happens to be easy to solve. Even if C = C, the inconsistency of C (i.e. if Φ C ≡ 0), can always be used to prove that C is also inconsistent. Finally, the equality Φ C (x) = 1 may imply that some of the variables can only have one possible value, which then can reduce the size of the original SAT problem.. The trivial class of linear CNFs (i.e. when |C| = 1 for all C ∈ C) is in fact used in most SAT algorithms in the above sense, under the name of unit literal rule. The class of quadratic CNFs (i.e. when |C| ≤ 2 for all C ∈ C) is also know to be solvable efficiently (see e.g. [4, 13, 16] ), and is used successfully in many of the fastest exact SAT solvers (see e.g. [9] ). The satisfiability problem for Horn CNFs (i.e. for which |C ∩ V| ≤ 1 for all C ∈ C) is well-known to be solvable in linear time (see e.g. [15, 23, 24] ). Horn subformulae were used systematically in a SAT solver by [19] .. There are a few other extensions of the above special classes, for which SAT is known to be solvable in polynomial time, see e.g. [5, 6, 11, 18, 25, 27] . None of these classes however, as far as the author knows, were put in a systematic use in the above sense, except some negative results appearing in [14] .. In this paper we concentrate on the class of renamable Horn formulae (for definitions see next section), and study the problem of finding the largest sub-CNF of a given CNF which is renamable Horn. Since this problem turns out to be an NP-hard optimization problem, we present a polynomial time approximation algorithm, and show that for cubic CNFs this algorithm always finds a renamable Horn sub-CNF, the size of which is not smaller than 59.7% of the size of a largest renamable Horn sub-CNF. We have shown in the previous sections that for the NP-hard problem of finding a maximum renamable Horn sub-CNF of a given CNF polynomial time approximations with guaranteed performance ratio are possible. It seems a challenging task to further improve this ratio, even for cubic inputs. Let us remark that for higher degree CNFs the guaranteed performance ratio is getting worse, and it perhaps requires different methods to find more reasonable bounds, in general. From the limited computational experience with problem MRH we feel that much better bounds could perhaps be guaranteed. We have generated random cubic CNFs, in which clauses were drawn at a uniform distribution, without repetitions allowed. Table 1 summarizes the computational results. Each row represents the average of 10 instances for each of the sizes of n variables and m = cn clauses with c = 2, .., 6 and with n = 50, 100, 150, 200, 250, 300. Columns 3 and 4 displays the range and the average proportion of Horn clauses in the generated formulae. This is indeed around 50%, as expected. Columns 7 and 8 displays the same for the value of the linear programming relaxation, λ(C), and as we can see this value is as high as 80-85% of the number of clauses. In columns 5 and 6 we can see the number of Horn clauses and their average proportion compared to the number of clauses after the switch obtained by the proposed approximation algorithm. Intuitively, this proportion should be close to 100% for CNFs with only a few clauses, since in this case almost all of them could be turned into Horn, while for CNFs containing almost all possible cubic clauses this proportion must be about 50%. It is interesting to see that for CNFs in the range 2n ≤ m ≤ 6n, which contains the statistically most difficult cubic satisfiability problems, this proportion is still as high as about 70%, on average, with a small variation.. In the last column a crude lower bound on the achieved performance rate is displayed. As it can be seen this is well above 80%, much higher than the 59% guaranteed by the theory. This indicates that, perhaps with a different technique, one may improve on the guaranteed performance rate.. We also plan to do a more thorough computational study with this method applied as preprocessing within a SAT solver.. Let us remark finally that there are several other classes of Boolean formulae for which efficient SAT algorithms are known, and for which therefore an analogous problem could be considered. Perhaps the most natural would be to consider q-Horn formulae, introduced in [5] . A CNF C is called q-Horn, if there is a subset of the literals S ⊆ L, S ∩ S = ∅ such that. for all C ∈ C. It is easy to see that both quadratic and renamable Horn formulae are included in this class, and SAT is known to be linearly solvable for q-Horn formulae, too (see [7] ). The problem of finding a maximum q-Horn sub-CNF of a given CNF was shown to be an NP-hard optimization problem in [14] , and therefore it would be interesting to find a good approximation algorithm with some non-trivial guaranteed performance ratio. Table 1 Computational results with randomly generated cubic CNFs. Each entry is the average of 10 instances. The last column shows a lower bound for the achieved performance rate.

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


Distinguishing the rates of gene activation from phenotypic variations Background: Stochastic genetic switching driven by intrinsic noise is an important process in gene expression. When the rates of gene activation/inactivation are relatively slow, fast, or medium compared with the synthesis/degradation rates of mRNAs and proteins, the variability of protein and mRNA levels may exhibit very different dynamical patterns. It is desirable to provide a systematic approach to identify their key dynamical features in different regimes, aiming at distinguishing which regime a considered gene regulatory network is in from their phenotypic variations. Results: We studied a gene expression model with positive feedbacks when genetic switching rates vary over a wide range. With the goal of providing a method to distinguish the regime of the switching rates, we first focus on understanding the essential dynamics of gene expression system in different cases. In the regime of slow switching rates, we found that the effective dynamics can be reduced to independent evolutions on two separate layers corresponding to gene activation and inactivation states, and the transitions between two layers are rare events, after which the system goes mainly along deterministic ODE trajectories on a particular layer to reach new steady states. The energy landscape in this regime can be well approximated by using Gaussian mixture model. In the regime of intermediate switching rates, we analyzed the mean switching time to investigate the stability of the system in different parameter ranges. We also discussed the case of fast switching rates from the viewpoint of transition state theory. Based on the obtained results, we made a proposal to distinguish these three regimes in a simulation experiment. We identified the intermediate regime from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations.We proposed a simulation experiment to distinguish the slow, intermediate and fast regimes, which is the main point of our paper. In order to achieve this goal, we systematically studied the essential dynamics of gene expression system when the switching rates are in different regimes. Our theoretical understanding provides new insights on the gene expression experiments. In this paper, we proposed a methodology to distinguish the regime of genetic switching rates from phenotypic variations in a simulation experiment. To accomplish this task, we put forward different reduction and investigation methods to reveal the essential dynamics of the genetic switching system when the switching rates are relatively slow, fast, or medium compared with the degradation rates of proteins. Although in the fast and slow regimes, the observed steady state distributions of the mRNAs and proteins are similar, we showed that the underlying mechanisms are qualitatively different. Based on our results, we provided a simulation experiment to distinguish these three regimes. The intermediate regime can be identified from the fact that the strength of cellular memory is lower than the other two cases, and the fast and slow regimes can be distinguished by their different perturbation-response behavior with respect to the switching rates perturbations. We also discussed the robustness of our results with respect to different choices of parameters and feedback function forms. This study provides insights to the experimental analysis of the gene expression system. It will be interesting to investigate similar problems in different experimental setups in the future. Understanding the dynamical features and the energy landscapes of the gene regulatory network is an interesting and challenging topic in quantitative biology. Some approaches focused on the regime of fast switching rates have been developed in recent years, whereas the slow regime which many eukaryotic cells correspond to attracts less attention. In this paper, besides the theoretical analysis of the essential dynamics in the slow, intermediate and fast regimes, we provide a method to distinguish these three regimes. The result is meaningful and enlightening for the design of corresponding biological experiments.. With regard to a previous work about stochastic dynamics of single regulating genes [19] , we provide a systematic study to understand the genetic switching dynamics when the switching rates are in different regimes. Although the global quasi-potential energy landscape and other properties of the system can be obtained from brutal force through numerical simulations, the theoretical reductions can reveal the intrinsic mechanisms. We analyze the model theoretically in the slow and fast regimes and successfully reconstruct the global energy landscape which reflects the inherent pattern of system behavior. But we obtain little theoretical results in the intermediate regime, since the nonexistence of time-scale separation makes this problem more challenging. Recent new ideas proposed in this field may be helpful in the study of this regime [36, 37] .. A related but different concept with multi-layer landscapes for the DNA fast switching case was mentioned in a recent literature [36] . However, we are mainly concerned with the two-layer case when the system is in slow regime. Besides understanding the system as decoupled layers, we further study the properties about transition paths and average switching time, and we construct the global potential energy landscape through simple Gaussian mixture and modified Gaussian mixture approximations. These pictures are quite different from the case of fast regime.. The MST is a key characterization of gene regulatory networks and has been studied analytically with respect to different genetic switching rates in [38] . Compared with the detailed theoretical analysis for a typical onedimensional self-activating switch model, we can not give fully theoretical results in the fast regime, i.e. the strictly adiabatic regimes considered in [38] , since our model involves both mRNA and protein explicitly. However, we present efficient numerical methods to make the quantitative estimates. The results for the extremely slow regime are also consistent with those obtained for non-adiabatic regime (i.e. Eq. 7). We also suggested the landscape as an intuitive tool to understand the essential dynamics of our model. Overall, the shape of the MST curve we obtained as a function of κ is qualitatively similar as in [38] as well.. The proposed quasi-potential energy function S(x) in the fast regime has close connection with the potential function considered in previous studies. In Wang et al's approach [14, 39] , they proposed to define the energy landscape as U(x) = −ln P ss (x), where P ss (x) is the steady state distribution. We have the connection that S(x) = lim V →∞ V −1 U(x) [15, 40] since U(x) itself depends on the system size V. We also note that the proposed quasi-potential S(x) satisfies the same Hamilton-Jacobi equation as the potential defined in [41, 42] in the diffusive approximations. But in addition, we provided a variational minimization approach to explicitly construct the S(x). Nonetheless, this construction is only valid for the fast regime and it is quite different from the picture we showed in the slow regime.. Although our four-state genetic switching model is based on chromatin remodeling, this is no longer expressed in the final simplified two-state model. In fact, some prokaryotic cells also have similar gene regulatory networks like Fig. 1b [17] . Our study mainly focuses on the two-state model, which can be suitable for a wide variety of cellular systems rather than the motivating chromatin remodeling process. Our simulation experiment to distinguish the slow, intermediate and fast regimes is an important point of the paper. So far, we are not able to perform real experiments to make the verification. The real experimental systems may be also more complicated than the model considered here. But our simulation experiment is logically true and self-consistent, and there have been related experiments performing well in Th2 cells [16] . Finding suitable system and validating our proposal will be an interesting task in the future.. Our study has potential application in single-cell experiments [43] [44] [45] . In the single-cell gene expression analysis, the protein level of each cell at different time points can be obtained through tracking and recording the behavior of each cell by fluorescence microscopy. With these data, one can obtain the switching time as the time difference between two consecutive states of a cell. Thus the MST can be roughly estimated. Similar as the procedure in flow cytometry experiments, the information about MST can be utilized to identify the intermediate regime, and the fast and slow regimes can be distinguished by perturbing κ. However, these points remain to be investigated in detail in the follow-up researches. Stochasticity is an inherent property of living cells. Stochastic perturbation comes not only from the noisy environment that cells live in, but also from all processes going on inside cells [1] . Especially when the copy number of molecules like mRNA or protein gets smaller, stochastic fluctuations can have significant effect on the *Correspondence: tieli@pku.edu.cn 1 LMAM and School of Mathematical Sciences, Peking University, No. 5. Yiheyuan Road, 100871 Beijing, China Full list of author information is available at the end of the article behavior of a cell. There have been many studies focused on discovering the effect of noise in living systems [2, 3] , and some of the results show that cells do not only evolve to resist noise perturbation better, but also form massive mechanisms which are the benefits of stochasticity [4, 5] . Genetic switching is a typical example of noisedriven cellular behavior which can be performed spontaneously under particular conditions in living cells, such as the switch of lac operon in Escherichia coli [6] , the formation of biofilm [7] , and the entrance into competence state [8] in Bacillus subtilis, etc.. Several recent studies discovered strong connections between noise and gene regulation mechanisms through measuring variability in protein and mRNA levels [9, 10] . Both experimental and theoretical results suggested that the switching rates of gene between its active and inactive states have significant effects on the noise of gene expression [11, 12] and are highly responsible to different gene expression dynamics. Previous theoretical studies mainly focused on the regime with fast switching rates [13] [14] [15] , however, recently some unique phenomena are observed especially when switching rates are slow, either due to the chromatin remodeling precess [16] or the transportation of transcription factor into nucleus [17] . Thus, it is important and desirable to provide an approach which can effectively identify the key dynamical features of gene expression system when switching rates are slow, and to distinguish the regime of gene regulatory networks from different phenotypic variations.. In this paper, we aim at providing a method to distinguish the rates of genetic switches from phenotypic variations. First, we systematically investigated the dynamical and stochastic properties of different genetic regulatory models in a comparative way when biochemical rates for gene activation/inactivation, transcription, translation, and mRNA/protein degradation vary over a wide range. We focused on a two-state gene-expression model with positive feedback loop, which is simplified from a four-state gene-expression network taking account of chromatin opening/closing step and regulatory protein binding/unbinding step [16, 18] . In the regime when the rates of gene activation/inactivation are relatively small compared with the synthesis/degradation rates of mRNAs and proteins, we found that the effective dynamics can be reduced to independent evolutions on two separate layers. We reconstructed the energy landscape of the system through Gaussian mixture approximations, and it is qualitatively different from the previously known picture in [15, 19] . In the regime with relatively fast switching rates, we briefly discussed about the system based on our previous study with large deviation theory [15] . In the regime with intermediate switching rates, we utilized the mean switching time (MST) to investigate the stability of the system in different parameter ranges. In the following paragraphs, we will call these three regimes as slow, fast, and intermediate regimes for convenience. Finally, based on our theoretical understanding of the essential dynamics of the slow, intermediate and fast regimes, we designed a simulation experiment to distinguish the three regimes. We identified the intermediate regime from the fact that cellular memory is much weaker in this regime than the other two regimes. And we distinguished the slow and fast regimes through their different response behavior with respect to switching rates perturbations. Our study gives a global characterization of the genetic switching dynamics in various cells and has potential guidance in real experiments. We consider a four-state gene-expression model with positive feedback loop ( Fig. 1a ). Proteins are self-activators in this system, since they can bind to DNA and activate the transcription step. Chromatin opening becomes relatively easy when DNA is bounded by a monomer regulatory protein. Open chromatin structure is permissible for transcription whereas closed chromatin is not. The rate of mRNA production reaches its maximum on the bounded and open state of DNA. Additionally, proteins and mRNAs are subject to degradation, which is not shown in Fig. 1a . Assuming that binding and release of regulatory proteins are in fast equilibrium [20, 21] , this four-state model can be simplified into a two-state model ( Fig. 1b ) through the quasi-steady-state approximation (QSSA) (see Additional file 1 for derivation details) [22, 23] . We will say that the gene is open (closed) when the chromatin is open (closed), and the system switches between open and closed gene states through gene activation/inactivation steps.. In the two-state model, there are six reactions occurring among four chemical species. All of the reactions involved in this system are summarized in Table 1 . Here m and n are the numbers of protein molecules and mRNA molecules, respectively. The jump rate into the open state and the transcription rate are controlled by the relative occupancy of protein at DNA binding site (i.e. O P ), whereas the jump rate into the closed state is not affected by the number of proteins. O P has the form O P = n/(n + K), where K is a constant by reduction and represents the strength of positive feedback. It is actually the conditional probability of state (iv) in Fig. 1a given that the chromatin is open. The rates k G O P + k G0 and k R O P + k R0 characterize the gene activation and mRNA synthesis, which are dependent of the number of proteins. The constants d G , k P , d R and d P correspond to the rates of gene inactivation, protein synthesis, mRNA and protein degradation. In fact, if we assume that dimer proteins (or other multimer-type of proteins) regulate the gene instead of the monomer protein, O P will have the form O P = n k /(n k + K k ). The value of k will not influence the essential dynamics of the gene expression model, on condition that we choose proper range of parameters with respect to different k. Under some reasonable assumptions, most of the derivations and results in the next section are still correct (see Additional file 1 for details).. Due to the intrinsic noise, the dynamics of the system can be described by the chemical master equation (CME) [1, [24] [25] [26] [27] as . where P α (m, n) stands for the probability that there are m mRNA molecules and n protein molecules in the system when the gene is open (α = 1) or closed (α = 0). One possibility to study the behavior of the system is to numerically solve Eqs. 1-(2) by truncating the domain and setting the boundary conditions P α (m, n) = 0 when m M, n N (M, N is large enough, α = 0, 1). But simply obtaining this solution does not mean we understand the essential mechanism of the dynamics, and the computational effort may be huge when the mean number of molecules is quite large in this system. Thus, we aim to find other ways to learn about the dynamics.. We define a key parameter κ to compare two characteristic timescales of the problem: the timescale d −1 P on which the gene state changes and the timescale d −1 G on which the number of protein and mRNA molecules changes, where d −1 P is the lifetime of a single protein and d −1 G is the lifetime of the open gene state. The ratio κ = d G /d P can take any positive value, which will lead to notable differences in the dynamics. Small ratios (κ 0.01) describe long timescales of the change in gene states, compared with the timescale on which mRNA and protein numbers change. In this regime, there is enough time for the system to reach a steady state when gene is either open or closed. Transition from one steady state to another always occurs immediately after the genetic switches. Large values of the ratio (κ 1) indicate that the gene states change very rapidly, whereas the synthesis and degradation of protein and mRNA molecules are relatively slow. As κ goes to infinity, the system exhibits the deterministic features [13, 15] . When κ is neither large nor small enough, genetic switches, transcription and translation will take place on the same timescale. The dynamics is complicate in this regime, and analytical results are hard to reach. So we resort to numerical simulations and extract useful information. In all of the three regimes, metastable states may occur with proper choice of parameters. Indeed, we will utilize a set of biologically relevant parameters in later computations (see Additional file 1 for details). For convenience, we will call the steady state with relatively large number of proteins as on state, whereas the steady state with relatively small number of proteins is off state.

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


Estimating Mutation Distances from Unaligned Genomes Alignment-free distance measures are generally less accurate but more efficient than traditional alignment-based metrics. In the context of genome sequence analysis, the efficiency gain is often so substantial that it outweights the loss in accuracy. However, a further disadvantage of alignment-free distances is that their relationship to evolutionary events such as substitutions is generally unknown. We have therefore derived an estimator of the number of substitutions per site between two unaligned DNA sequences, K r . Simulations show that this estimator works well with ''ideal'' data. We compare K r to two alternative alignment-free distances: a k-tuple distance and a measure of relative entropy based on average common substring length. All three measures are applied to 27 primate mitochondrial genomes, eight whole genomes of Streptococcus agalactiae strains, and 12 whole genomes of Drosophila species. In each case, the cluster diagrams based on K r are equivalent to or significantly better than those based on the two alternative measures. This is due to the fact that in contrast to the alternative measures K r is derived from an explicit model of evolution. The computation of K r is efficiently implemented in the program kr, which can be downloaded freely from the internet. Phylogenies are the central metaphor of evolutionary biology (Archibald, 2008) . Any particular phylogeny implies distances between the taxa investigated. Since the advent of molecular biology, taxa are increasingly represented by a sample of their genome sequence, and the computation of distances between molecular sequences has attracted a correspondingly large amount of research interest over the years (Yang, 2006) . Alignment-free distances have for some time now complemented the more widely used alignment-based measures, but it is the current availability of genome-scale sequence data that has sparked renewed interest in alignment-free distances (Vinga and Almeida, 2003) . Any such measure needs to fulfil two necessary conditions for usefulness: biological relevance and efficient implementation.. Biological relevance is ensured by explicitly stating the connection between the metric and some known measure of evolutionary distance. We constructed K r as an estimator of the expected number of substitutions per site, K, by deriving the probability density function of shustring lengths as a function of the number of mismatches per site, d. The resulting equation (7) can be used to compute the expected average shustring length. By equating an observed average shustring length with its expectation, we can solve numerically for d. The relationship between d and the number of substitutions, finally, is given by the oldest distance measure for DNA sequence data, the Jukes-Cantor equation.. The Jukes-Cantor equation has been refined in various ways since its inception, particularly by taking into account nucleotide-specific mutation rates and rate variation between sites (Yang, 2006) . In principle, it might be possible to re-derive equation (7) for more complex mutation models. However, Figure 2 shows that at least for the simplest of evolutionary models K r approaches the number of substitutions per site much better than the other two measures investigated.. The distance measures we compared K r to are representative of two great classes of alignment-free distance measures: those based on word frequencies (d ktup ) and those based on the information-theoretical concept of relative entropy (d acs ). k-Tuple distances are easy to compute and are used in various contexts including phylogeny reconstruction in bacteria (Pride et al., 2003) , guide tree reconstruction for multiple sequence alignment (Larkin et al., 2007) , and phylogeny reconstruction for highly divergent sequences (Yang and Zhang, 2008) . However, for the small to moderate evolutionary distances considered in this study, they turned out to be less useful (Figs. 4A and 6B) .. The average common substring length is very similar to the average shustring length on which K r is based. Both measures are therefore restricted to closely related DNA sequences as they saturate with evolutionary distance (Fig. 2B) . The comparison between K r and d acs demonstrated the importance of expressing a quantity such as the average common substring length as a function of specific evolutionary events: for primate mitochondrial DNA and for Drosophila genomes K r outperformed d acs (Figs. 4 and 6) . However, when applied to the genomes of S. agalactiae, both methods obtained the same topology (Fig. 5B,C) . This is an instructive case, as the S. agalactiae strains were the most closely related of the taxa analyzed. In fact, they were so closely related that d ktup failed to resolve any phylogeny at all. In contrast, both d acs and K r returned distances whose topology agreed better with the multilocus sequence data than a published analysis based on gene presence=absence (Tettelin et al., 2005) .. Alignment-free distance measures generally trade speed for precision. This is the reason why, for instance, a k-tuple distance is used in clustalw for fast guide tree construction (Larkin et al., 2007) . Guide tree construction in ''quicktree'' mode of clustalw is fast indeed, taking only 0.5 s on a 3-GHz Intel Xeon system to compute a tree of the 27 primate mitochondrial genomes. This is orders of magnitudes faster than the default guide tree construction mode based on all pairwise alignments. However, the quality of the resulting tree is approximately as low as that of our d ktup tree shown in Figure 4A . Given that the quality of the K r tree (Fig. 4C) is much better than that, K r might be useful as a fast guide tree construction method. This brings us to the question, how fast is K r compared to other distance measures?. Word-frequency measures have very fast run times that are linear in the combined lengths of the input sequences, O(l). In addition, they only need to store the word frequency table, so their memory requirement is O(4 k ). In contrast, the run times of both d acs and K r are dominated by suffix array construction. For the algorithm underlying the suffix array implementation used by us (Manzini and Ferragina, 2002) , this would be O(cÂl log l), where c depends on the time it takes to compare two suffixes. When sorting identical suffixes, this time can become large so that the worst case run time is O(l 2 log l), but in most applications it is much closer to O(l log l). In addition to computing a suffix array, K r also requires the estimation of d from equation (7), which makes it somewhat slower than d acs . Here are a few example run times: On a 3 GHz Intel Xeon system it took 0.15 s to compute d ktup for the 27 primate mitochondrial genomes, 8.99 s to compute d acs , and 10.83 s to compute K r . In other words, our implementation of d ktup is over 70 times faster than our implementation of K r . While this may make our distance measure seem slow, it is still fast compared to the computation of the corresponding multiple sequence alignment which takes 8977 s using muscle (Edgar, 2004) and 8657 s using clustalw (Larkin et al., 2007) . These programs are designed for protein sequence alignment. In contrast, MAVID is a program designed for aligning multiple long sequences (Bray and Pachter, 2004) and it analyzes the primate data in 16.94 s, which is close to the run time of our more approximate K r measure.. As expected from the complexity formulas, the run time difference increased between d ktup on the one hand and d acs =K r on the other when applied to the Drosophila data. On our test system d ktup was computed in 6 min and 43 s, while the computation of d acs took roughly 403 times as long and the computation of K r 481 times as long amounting to 2 d, 5 h, 45 min, and 46 s. We tried to align the Drosophila data using MAVID, but ran out of memory on a 64 GB computer. Moreover, given that the Drosophila data consists of unordered contigs, it is not clear how a global multiple sequence alignment as computed by MAVID might even sensibly be defined.. Notice that the difference in run time between d acs and K r is appreciable. This would make it desirable to approximate the sum in equation (6) by a probability density function. Still, it is remarkable that 2 days suffice to analyze the full genomes of the Drosophila dozen and return with the accepted branching order. In their current form the run times of the measures investigated can be summarized as. It is important to realize that the run time of K r depends on details of how shustrings are looked up and how equation (7) is solved. These might be revised in the future, particularly since efficient algorithms for the construction of suffix arrays are a lively topic of current research (Puglisi et al., 2007) . However, this does not materially affect the central contribution of our study, which is to construct an alignment-free estimator of the number of pairwise substitutions. This estimator is not only applicable to unaligned genomes but also to unordered contigs, as was the case with the Drosophila dozen.. The work presented here can be extended in at least two ways: by revising the calculation of K r , and by estimating other statistics from the average shustring length. Figure 3 shows that the moment-based estimation of K developed in this paper fails for K > 0.55 because then the average shustring length is essentially constant. A maximum likelihood estimator might be more powerful and allow distance estimation for more divergent sequences. This would be important if K r were to be applied in guide tree reconstruction.. As to estimating other statistics, population parameters such as the scaled neutral mutation rate (Hartl and Clark, 1997, p. 319) , y, are a primary target, since genetic diversity is so low that average shustring length would be a very sensitive indicator. We expect that there is going to be heightened interest in alignment-free estimation of such population genetic parameters in the wake of ongoing comparative sequencing projects such as the 1000 genomes projects for Arabidopsis, Drosophila, and human. At the heart of the computation of our distance measure lies the calculation of the average shustring length. This is done by indexing query and subject in a generalized suffix tree to look up individual shustring lengths. We used a suffix array (Manzini and Ferragina, 2002) to simulate suffix tree traversal (Abouelhoda et al., 2002) . Details of the corresponding algorithm are given in Haubold et al. (2008) . The resulting program, kr, can be accessed via a web interface at http:==guanine.evolbio.mpg.de=kr=. The site allows submission of a file containing the sequences to be compared in any format and returns the K r -based phylogeny. In addition, the C source code of kr is freely available from this web site under the GNU General Public License.

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