Lookahead Actions in Dispatching to Parallel Queues Applying the first policy iteration (FPI) to any static dispatching (task assignment) policy yields a new improved dynamic policy that takes into account the particular cost structure and the expected future arrivals. However, it is generally hard to go beyond that due to the complex state space and the resulting difficulty in computing the value function for a dynamic policy. For example, applying FPI to identical FCFS servers with Bernoulli split gives the Least-Work-Left (LWL) policy, for which no closed-form value function is known. In fact, LWL with identical servers is equivalent to an M/G/k queue, the performance measures of which have remained as open problems. The situation gets even more complicated with heterogeneous servers. In this paper, we take an intermediate approach and consider lookahead actions that concern not only the current job but also the job arriving next, after which a basic (static) policy is assumed to take over. This is important as the benefits from some decisions can only be reaped with appropriate subsequent actions. The lookahead enables sound estimates also for marginal admission costs, e.g., with respect to LWL. The superior performance of the new near-optimal dispatching policies is demonstrated numerically. For a system of parallel servers, one fundamental resource allocation question is the so-called dispatching problem (i.e., the task assignment or routing problem), where the task is to choose a server for each arriving job so as to minimize the mean waiting time, energy consumption, or some other performance metric of interest. This general problem has many applications within ICT such as routing in data networks (where different links are seen as servers) and task assignment in web server farms, distributed parallel computing, (mobile) cloud computing and data centers. The dispatching problem arises also in many other contexts such as car toll stations in a highway, cashiers in a supermarket, manufacturing systems, etc.. In the basic setting, all tasks (i.e., jobs or customers) are subject to assignment, i.e., all tasks are flexible [1] . In general, servers could also receive dedicated inflexible tasks that cannot be served elsewhere due to technical or policy reasons. The so-called cycle stealing is a related concept where only the tasks of the beneficiary system can be re-routed to a donor system given that the latter is idle [2, 3] . We assume flexible customers, but note that inflexible customers can be introduced without any technical difficulty. The optimal dispatching policy depends on the objective (cost function), and on the available information about the jobs and the state of the system (service times, number of jobs in each queue, etc.). Jobs arriving in future are typically assumed to constitute a Poisson process, which is also our assumption. Moreover, we assume a size-aware setting where the service time of a job becomes known upon arrival. For a more detailed overview of task assignment problems and policies, we refer to the recent book [4] .. Three approaches to solve a dispatching problem can be identified. First, one can pick a promising dispatching policy and just show its optimality. For example, Winston [5] has shown the optimality of the join-the-shortest-queue (JSQ) for identical first-come-first-served (FCFS) servers when the number in queue is available and exponentially distributed jobs arrive according to a Poisson process. Similarly, the round-robin (RR) has been shown to be the optimal policy for identical FCFS servers, when the queues were initially in the same state [6] [7] [8] .. Second, one can choose a parameterized dispatching policy α(h), derive the mean cost rate for it, and choose the optimal h. For example, the Size-Interval-Task-Assignment (SITA) bases the dispatching decision solely on the size of the job [9, 10] : the range of job sizes is divided to non-overlapping intervals [ξ i−1 , ξ i ) by the cutoff thresholds ξ i and a job with size x ∈ [ξ i−1 , ξ i ) is routed to Server i. The cutoff thresholds ξ i are the parameters of SITA. As the dispatching decision does not depend on the state of the queues, SITA is a static (state-independent) policy and, assuming FCFS and Poisson arrivals, an expression for the mean waiting time follows from the Pollaczek-Khinchine mean value formula. SITA has been shown to be the optimal static size-aware policy for Poisson arrivals with respect to the mean waiting and sojourn time [11] .. The third approach is based on the policy improvement of the Markov decision processes (MDP) [12] [13] [14] . Given the so-called value function, one carries out the first policy iteration (FPI) step, which typically yields the greatest improvement towards the optimal policy. In the context of dispatching problems, this approach has been utilized to minimize the blocking probability, see Krishnan [15, 16] and Leeuwaarden et al. [17] , and the sojourn time (i.e., delay or latency) or its generalization by arbitrary holding costs, see, e.g., Krishnan [18] , Sassen et al. [19] , Bhulai et al. [20] and Hyytiä et al. [21] [22] [23] . Most dispatching systems considered have a rather complex state space (e.g., infinite number of waiting places, a continuous range of remaining service time, etc.) and therefore the application of the policy improvement has been limited to static policies, in case of which, the system decomposes and it is sufficient to analyze isolated single server queues. In this paper, we dig a bit deeper and take an intermediate approach, where also the next decision(s) can be dynamic, after which a static basic policy takes over. The decision for the next job is only tentative (i.e., a plan), and re-evaluated upon an actual arrival. There is a strong analogy to a computer program that builds a decision tree when playing a game, with the distinction that we have a continuous time dimension and a non-discrete state space. Therefore, the decision tree in our case is only two levels deep (lookahead of two), the leaves of which are evaluated assuming a static basic policy. Note that almost all commonly used policies are index-policies, where an index is computed independently for each queue and the queue with the smallest index is chosen. This includes the Least-Work-Left (LWL), JSQ, Bernoulli split, SITA and all FPI policies based on static basic policies. However, as we consider not only one but two (or more) subsequent decisions, the corresponding policy, referred to as the Deep (FPI) policy, is not an index-policy, but belongs to a more general family of dispatching policies.. The rest of the paper is organized as follows. Section 2 describes the system model, notation and earlier results for the basic FPI. In Section 3, we derive the marginal cost for the static, and in Section 4 for the dynamic second actions. Section 5 illustrates how these lookahead actions improve the performance due to the more "accurate" estimates for marginal costs, and Section 6 contains the conclusions. The proposed lookahead approach enables a finer control over the actions in the near future, and (partially) overcomes the obstacles that arise when analyzing dynamic dispatching policies in the MDP framework. In particular, the lookahead policy improvement approach (Deep in short) makes it possible to mimic dynamic dispatching policies such as LWL and Myopic, yielding sound estimates for the corresponding marginal costs. Conceptually, we seek lower costs by considering also the subsequent decisions that apply tentatively to the later arriving jobs. That is, we consider decisions such as choosing Queue i for the current job and Queue j for the next, where the specifics of the latter job are currently unknown. The lookahead step is important as the benefits from some actions can only be reaped with appropriate subsequent actions. In the numerical experiments, we observed significant improvements in performance over the existing dispatching policies in many cases. When improvements were smaller, the reference policies are assumed to be already close to optimal (e.g., LWL is known to be optimal with identical servers and fixed job sizes). Furthermore, we only experimented with the mean delay, while the gain from FPI and Deep in particular can be expected to be higher with more diverse cost structures (by non-constant holding costs). With static decisions, it seems to be straightforward to go beyond the two actions, e.g., by expanding the lookahead to three steps. For example, a decision for the third job according to LWL can be easily deduced by conditioning on the size of the second job. Alternatively, one can evaluate all triplets (i 0 , i 1 , i 2 ), or apply some pruning if the number of queues is too large. Unfortunately, with dynamic actions the combinatorial complexity appears to become a formidable challenge immediately and more advanced techniques and/or alternative approaches should be considered.

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


Mechanised wire-wise verification of Handel-C synthesis The compilation of Handel-C programs into net-list descriptions of hardware components has been extensively used in commercial tools but never formally verified. In this paper, we first introduce an extension of the compilation schema that allows the synthesis of the prioritised choice construct. Then we present a variation of the existing semantic model for Handel-C compilation that is amenable to mechanical proof and detailed enough for analysing properties of the hardware generated. We use this model to prove the correctness of the wiring schema used to interconnect the components at the hardware level and propagate control signals among them. Finally, we present the most interesting aspects of the mechanisation of the model and the correctness proofs in the HOL theorem prover. Handel-C [14] is a Hardware Description Language (HDL) based on the syntax of the C language extended with constructs for dealing with CSP-based [10] parallel behaviour and process communications.. Handel-C's compilation into hardware components (synthesis) was initially formulated to be used in a hardware/software co-design project [20] and later adopted in commercial tools [15] . The synthesis process is described graphically and it covers most of the basic constructs excluding the prioritised choice operator. The approach is based on the fact that each synthesised circuit is controlled by a pair of signals (start and finish). Compositionality, even though informally addressed, is achieved by means of associating a circuit with each construct and then considering it as a black box driven only by means of the above described simple controlling interface. The correctness of the compilation approach is based on the assumption that the start signal will be given to a circuit only if all the previous circuits have already finished. The translation mechanism is designed in a way that, in principle, ensures the validity of this assumption, leading to the satisfaction of a system-level assumption: ''the environment will start the hardware program running just once and will not attempt to start it again before the program has completed'' [20] . There is, however, no formal evidence of any of these claims as no semantics or formal proofs are provided.. A formal model of the compilation and denotational semantics for the hardware generated has been proposed as a first step towards verifying the compilation scheme [21, 24] . The semantics relies on the concept of state-transformers and uses branching sequences of atomic hardware actions as its semantic domain. However, when trying to prove a correctness result from the semantics, we observed that the semantics fails to capture the hardware parallelism at the branching sequence level, making any correctness proof very difficult.. One of the objectives of this work is then to provide a semantics that is capable of capturing the parallelism the hardware is capable of and that is suitable for mechanically verifying the assumptions the synthesis process is based on. As we are interested in a bigger subset of Handel-C than the one addressed in [20] , we also need to define a compilation approach for the priAlt construct (prioritised choice) as well as providing the semantics corresponding to it. Finally, given the fact that we focus on the correctness of the synthesis process, we exclude any efficiency/optimisation analysis from this work.. The work presented in this paper is a first-order logic theory that describes the syntax (by means of a deep embedding of the constructs as datatypes) and semantics for our subset of Handel-C. The syntax and semantics are then embedded in higher order logics to allow the verification of correctness properties associated with the assumptions the synthesis process is based on. The theory and its verification have been mechanised in the HOL theorem prover [9] and further details about its definitions and theorems can be found in [22] . 1 In this paper we intend to give a self-contained and accessible account of that work. In doing so, we deliberately omit all proofs and consequently many technical details in favour of a clearer presentation of the work as a whole.. The rest of the paper is organised as follows. We start by outlining the compilation approach presented in [20] together with our approach for compiling the priAlt construct. We then propose a reformulation of the semantic domain in [21] in terms of a slightly higher level semantic domain that allows us to capture parallelism at the sequence level. We then redefine the semantics in terms of the new domain and prove the existence of fix-point solutions for the recursive equations. Finally, we define the concept of wire-satisfiability in our model and prove the correctness of the wiring schema used in the compilation. The main contributions of this work are a compilation schema for the priAlt construct, an improved semantic model for the hardware components synthesised from Handel-C programs and the mechanical verification of the wiring schema used to handle the control flow among those components.. This work presents a more abstract semantic domain than the one used in previous works and allows a better description of the parallelism exhibited by the synthesised hardware. In particular, we have defined our semantic domain in terms of a deep embedding of sequences of state-transformers in higher order logic. We have also established a partial order relationship over the domain and proved the existence of fix-point solutions to our inductive approximations for recursive constructs.. The synthesis process we are formalising is based on the assumption that no hardware component will be activated unless a precise signal has been given to it through its interface. We have captured this notion by embedding Handel-C's syntactic constructs in HOL and providing a semantic function that maps each construct in the language to its representation in our semantic model. Moreover, the way in which Handel-C's synchronous nature is encoded in the model introduces explicit information about the value held by the wires used to link different components. We have taken advantage of this feature to formally verify the correctness of the wiring schema used in the compilation.. Even though we have proved that each of the hardware components propagates the control token in the right way, we still need to prove that the hardware generated by the compilation rules is correct (i.e., semantically equivalent to its original Handel-C code). This correctness proof will also allow us to discharge the only assumption of this work: that the timing model of the generated hardware is consistent with the one for Handel-C. Towards this end, our next step is to prove the existence of an equivalence relationship using the semantic models for Handel-C [5, 23] and the semantics for the hardware generated that is presented in this paper.

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


Filling holes in meshes using a mechanical model to simulate the curvature variation minimization is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. The presence of holes in a triangle mesh is classically ascribed to the deficiencies of the point cloud acquired from a physical object to be reverse engineered. This lack of information results from both the scanning process and the object complexity. The consequences are simply not acceptable in many application domains (e.g. visualisation, finite element analysis or STL prototyping). This paper addresses the way these holes can be filled in while minimizing the curvature variation between the surrounding and inserted meshes. The curvature variation is simulated by the variation between external forces applied to the nodes of a linear mechanical model coupled to the meshes. The functional to be minimized is quadratic and a set of geometric constraints can be added to further shape the inserted mesh. In addition, a complete cleaning toolbox is proposed to remove degenerated and badly oriented triangles resulting from the scanning process. Reverse Engineering is a powerful technique used to create a digital representation of an existing physical object. The reconstruction process starts with the acquisition of a point cloud from the outer surface of the physical object. A triangle mesh can be created from these dense unorganized data points [14, 12, 2, 4] . BSpline/NURBS surfaces may also be created either directly from the point cloud or from its triangulation [20, 13, 11] . More recently, the use of subdivision surfaces for surface fitting and surface reconstruction has also been explored [21, 17, 30] . Here again, most of the proposed approaches begin with a triangulation of the point cloud. As a consequence, it is crucial to find a triangle mesh that best fits the object outer surface. Depending on both the complexity of the object to be reverse engineered and the adopted data acquisition system technology (e.g. Coordinate Measuring Machines or laser scanning), some areas of the object outer surface may never be accessible. Figure 1 shows the result of an acquisition with a laser from a single point of view. Using the point of view (a), the scanner is unable to reach some portions of the object visible from another point of view (b). This induces some deficiencies in the point cloud and a set of holes in the triangle mesh. This is not acceptable. The presence of undesired holes may induce unexpected results when doing rapid prototyping or finite element analysis for example.. In this paper, we propose a set of models, methods and tools to fill in undesired holes in meshes. The filling process acts in several steps illustrated on the basic example of figure 2. First, the hole contour is identified (a) and cleaned (b) to remove badly oriented and degenerated triangles due to the scanner noise (section 4). A topological grid is then inserted (c) to fill in the hole (section 5). Finally, the inserted mesh is deformed (d) to satisfy various blending criteria with the initial mesh (section 6). We are notably able to minimize the curvature variation across the hole contour while solving a linear equations system. The curvature variation is simulated by the variation between external forces applied to the nodes of a linear mechanical model coupled to the meshes. The functional to be minimized is quadratic and Overall filling process on a simple example: after the identification and cleaning of the hole contour, the missing area is filled in with a topological grid whose final shape results from a deformation process.. a set of geometric constraints can be added to further shape the inserted mesh. These are strong improvements with respect to the existing techniques found in the literature (section 2). The filling process is illustrated with results produced by our prototype software (section 7). The limits and possible upgrades are finally discussed in section 8. In this paper, we propose a complete process to fill in holes in meshes. It goes from the detection and cleaning of the hole contour to the deformation of a topological grid built from the boundary vertices. New methods and tools have been imagined at each of these steps. The deformation itself results from the resolution of an optimization problem where the constraints are optionally specified to shape the interior of the filled area. A new type of objective function is notably proposed and enables the minimization of an approximation of the variation of the curvature between the inner and surrounding meshes. More precisely, the curvature variation is simulated by the variation between external forces applied to the nodes of a linear mechanical model coupled to the meshes. Thus, the equations system to be solved is linear. The whole process has been validated on several examples coming from our prototype software. It gives very good results in configurations where the holes look like a disk. A comparison with the second-ordeer umbrella operator of Liepa is proposed and clearly demonstrates that the proposed minimization enables a smoother connection with the surrounding mesh. Anyhow, additional works should be performed to be able to adapt both the topology (disk, ring and so on) and the shape (circle, rectangle and so on) of the grid to the shape of the hole. At the end, these improvements of the topological filling step should enable the filling of any holes that can be found during the reverse engineering of products. Various techniques have been proposed to fill in undesired holes in meshes. Two main categories can be distinguished: the geometric and non-geometric approaches.. Among the non-geometric approaches, [8] use a volumetric representation to detect the mesh areas that have to be filled in. [9] apply a volumetric diffusion process to extend a signed distance function through this volumetric representation until its zero set bridges whatever holes may be present. This iterative approach is particularly well adapted for complex geometrical and topological holes. Unfortunately, it does not ensure that the inserted mesh smoothly vanish on the surrounding mesh. A similar approach has been developed by [22] for the simplification and the repairing of polygonal meshes. [32] also represent the surface of interest in implicit form, and fill in the holes with a system of geometric partial differential equations derived from image inpainting algorithms. These equations are based on the geometric characteristics of the known mesh (e.g. the curvatures) and are applied only at the holes and a neighbourhood of them. Being these equations anisotropic and geometry based, they lead to a slightly slower algorithm than the one of [9] . [7] use a finite element method to minimize the integral of the squared mean curvature (the so-called Willmore energy) of the filled hole. Their process is iterative and can only ensure a tangency continuity with the surrounding mesh.. Considering the geometric approaches, [19] proposes a filling process quite similar to our ( fig. 2 ). The hole is first detected and filled in with a minimum area triangulation of its 3D contour [3] . The triangulation is refined so that the triangle density agrees with the density of the surrounding mesh triangles [26] . The filled hole is finally smooth with a fairing technique based on an umbrella operator [16] . The faired mesh is obtained while solving a system of linear equations. It solely satisfies tangency blending conditions with the initial mesh. [28] propose a fairing technique based on solving a non-linear fourth order partial differential equation. G 1 boundary conditions are satisfied but the resolution is iterative. Also the implicit fairing of [10] requires an iterative resolution process. Other approaches are based on the Moving Least Squares projection which induces a non-linear minimization process [34, 31] .. To conclude, the main drawbacks of most of these approaches concern the use of iterative resolution processes to find the shape of the inner mesh and/or the non satisfaction of curvature blending conditions between the inner and surrounding meshes. In fact, most of these approaches do not use enough geometric information available on the surrounding mesh. This is not true for the context-based surface completion algorithm of [29] where the inner mesh is defined according to shapes that can be very far from the hole. But their process may lead to very unexpected results. Finally, most of the processes do not enable the prescription of additional constraints inside the inner mesh. These are the limits we try to overcome. We will discuss how successful we are in section 8.

O B I I 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 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 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 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 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 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 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 B I 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


Learning DFA from Simple Examples Abstract. Efficient learning of DFA is a challenging research problem in grammatical inference. It is known that both exact and approximate (in the PAC sense) identifiability of DFA is hard. Science, pp. 231-242, 1996) wherein positive and negative examples are sampled according to the universal distribution conditional on a description of the target concept. Further, we show that any concept that is learnable under Gold's model of learning from characteristic samples, Goldman and Mathias' polynomial teachability model, and the model of learning from example based queries is also learnable under the PACS model. The problem of learning a minimum state DFA that is consistent with a given sample has been actively studied for over two decades. DFAs are recognizers for regular languages which constitute the simplest class in the Chomsky hierarchy of formal languages (Chomsky, 1956; Hopcroft & Ullman, 1979 ). An understanding of the issues and problems encountered in learning regular languages (or equivalently, identification of the corresponding DFA) are therefore likely to provide insights into the problem of learning more general classes of languages.. Exact learning of the target DFA from an arbitrary presentation of labeled examples is a hard problem (Gold, 1978) . Gold showed that the problem of identifying the minimum state DFA consistent with a presentation S comprising of a finite non-empty set of positive examples S + and possibly a finite non-empty set of negative examples S − is N P-hard. Under the standard complexity theoretic assumption P = N P, Pitt and Warmuth (1989) showed that no polynomial time algorithm can be guaranteed to produce a DFA with at most N (1− ) log log(N ) states from a set of labeled examples corresponding to a DFA with N states.. Efficient algorithms for identification of DFAs assume that additional information is provided to the learner. Trakhtenbrot and Barzdin (1973) described a polynomial time algorithm for constructing the smallest DFA consistent with a complete labeled sample i.e., a sample that includes all strings up to a particular length and the corresponding label that states whether the string is accepted by the target DFA or not. Angluin (1981) showed that given a live-complete set of examples (that contains a representative string for each live state of the target DFA) and a knowledgeable teacher to answer membership queries (queries of the form "Does the string y belong to the language of the target DFA?") it is possible to exactly learn the target DFA. In a later paper, Angluin (1987) relaxed the requirement of a live-complete set and has described a polynomial time algorithm (L * ) for learning the target DFA using both membership and equivalence queries (queries of the form "Is the current hypothesis equivalent to the target DFA?"). The L * algorithm tacitly assumes that the learner has the capacity to reset the DFA to the start state before posing each membership query. This assumption might not be realistic in situations wherein it is not feasible to remember the start state or the path taken from the start state to reach the current state while evaluating a membership query. To overcome this limitation of the unknown start state, Rivest and Schapire (1993) have proposed a learning method based on homing sequences that runs N copies of L * in parallel, one for each of the N states of the target DFA. The regular positive and negative inference (RPNI) algorithm is a framework for identifying in polynomial time, a DFA consistent with a given sample S (Oncina & García, 1992) . Further, if S is a superset of a characteristic set (see Section 2.1) for the target DFA then the DFA output by the RPNI algorithm is guaranteed to be equivalent to the target (Oncina & García, 1992; .. Pitt surveyed several approaches for approximate identification of DFA (Pitt, 1989 ). Valiant's distribution-independent model of learning, also called the probably approximately correct (PAC) learning model (Valiant, 1984) , is a widely used framework for approximate learning of concept classes. When adapted to the problem of learning DFA, the goal of a PAC learning algorithm is to obtain in polynomial time, with high probability, a DFA that is a good approximation of the target DFA. We define PAC learning of DFA more formally in Section 2. Angluin's L * algorithm (Angluin, 1987) that learns DFA in polynomial time using membership and equivalence queries can be recast under the PAC framework to learn by posing membership queries alone. Pitt and Warmuth (1988) showed that the problem of polynomially approximate predictability of the class of DFA is hard. They used prediction preserving reductions to show that if DFAs are polynomially approximately predictable then so are other known hard to learn concept classes such as boolean formulas. Further, Kearns and Valiant (1989) showed that an efficient algorithm for learning DFA would entail efficient algorithms for solving problems such as breaking the RSA cryptosystem, factoring Blum integers, and detecting quadratic residues. Under the standard cryptographic assumptions these problems are known to be hard to solve. Thus, they argued that learning DFA from any randomly drawn set of examples is a hard problem.. The PAC model's requirement of learnability under all conceivable distributions is often considered too stringent for practical learning scenarios. Pitt's paper (1989) identified the following open research problem: "Are DFA's PAC-identifiable if examples are drawn from the uniform distribution, or some other known simple distribution?". Using a variant of Trakhtenbrot and Barzdin's algorithm, Lang (1992) empirically demonstrated that random DFAs are approximately learnable from a sparse uniform sample. However, exact identification of the target DFA was not possible even in the average case with a randomly drawn training sample. Several efforts have been made to study the learnability of concept classes under restricted classes of distributions. Li and Vitányi (1991) proposed a model for PAC learning with simple examples called the simple PAC model wherein the class of distributions is restricted to simple distributions (see Section 4). Denis et al. (1996) proposed a model of learning where examples are drawn at random according to the universal distribution conditional on the knowledge of the target concept. This model is known as the PACS learning model. In this paper, we present a method for efficient PAC learning of DFA from simple examples. We will prove that the class of logarithmic Kolmogorov complexity DFA (see Section 4) is learnable under the simple PAC model and the entire class of DFA is learnable under the PACS model. Further, we demonstrate how the model of learning from simple examples naturally extends the model of learning concepts from representative examples (Gold, 1978) , the polynomial teachability model (Goldman & Mathias, 1993) , and the model of learning from example based queries (Angluin, 1988) to a probabilistic framework.. This paper is organized as follows: Section 2 briefly introduces some concepts used in the results described in this paper. This includes a discussion of the PAC learning model, Kolmogorov complexity, and the universal distribution. Section 3 reviews the RPNI algorithm for learning DFA. Section 4 discusses the PAC learnability of the class of logarithmic Kolmogorov complexity DFA under the simple PAC learning model. Section 5 demonstrates the PAC learnability of the entire class of DFA under the PACS learning model. Section 6 analyzes the PACS model in relation with other models for concept learning. Section 7 addresses the issue of collusion that arises because a helpful teacher can potentially encode the target DFA as a labeled training example. Section 8 concludes with a summary of the main contributions of this paper and some directions for future research. The problem of exact learning of the target DFA from an arbitrary set of labeled examples and the problem of approximating the target DFA from labeled examples under Valiant's PAC learning framework are both known to be hard problems. Thus, the question as to whether DFA are efficiently learnable under some restricted yet fairly general and practically useful classes of distributions is clearly of interest. In this paper, we have provided a framework for efficient PAC learning of DFA from simple examples.. We have demonstrated that the class of logarithmic Kolmogorov complexity DFA is polynomially learnable under the universal distribution m (the simple PAC learning model) and the entire class of DFA is shown to be learnable under the universal distribution m r (the PACS learning model). When an upper bound on the number of states of the target DFA is unknown, the algorithm for learning DFA under m r can be used iteratively to efficiently PAC learn the concept class of DFAs for any desired error and confidence parameters 6 . These results have an interesting implication on the framework for incremental learning of the target DFA. In the RPNI2 incremental algorithm for learning DFA, the learner maintains a hypothesis that is consistent with all labeled examples seen thus far and modifies it whenever a new inconsistent example is observed . The convergence of this algorithm relies on the fact that sooner or later, the set of labeled examples seen by the learner will include a characteristic set. If in fact the stream of examples provided to the learner is drawn according to a simple distribution, our results show that in an incremental setting the characteristic set would be made available relatively early (during learning) with a sufficiently high probability and hence the algorithm will converge quickly to the desired target. Finally, we have shown the applicability of the PACS learning model in a more general setting by proving that all concept classes that are polynomially identifiable from characteristic samples according to Gold's model, semi-polynomially T/L teachable according to Goldman and Mathias' model, and learnable in deterministic polynomial time from example based queries are also probably exactly learnable under the PACS model. Li and Vitányi (1991) have shown that the class of simple k-reversible DFA is learnable under the simple PAC model. A k-reversible DFA is simple if each state of the DFA lies on a path (from the initial state to an accepting state) of Kolmogorov complexity O(lg N ) . They have shown that the class of simple k-reversible DFA includes k-reversible DFA whose canonical representations have Kolmogorov complexity O(lg N ) and also some k-reversible DFA whose canonical representations have a higher Kolmogorov complexity, for example, O(lg 2 N ). We have shown that the class of logarithmic Kolmogorov complexity DFA is learnable under the simple PAC model. It is of interest to explore whether classes of DFA of higher Kolmogorov complexity (such as O(lg k N ) where k > 1) are efficiently learnable under the simple PAC model.. The class of simple distributions includes a large variety of probability distributions (including all computable distributions). It has been shown that a concept class is efficiently learnable under the universal distribution if and only if it is efficiently learnable under each simple distribution provided that sampling is done according to the universal distribution (Li & Vitányi, 1991) . This raises the possibility of using sampling under the universal distribution to learn under all computable distributions. However, the universal distribution is not computable. Whether one can instead get by with a polynomially computable approximation of the universal distribution remains an open question. It is known that the universal distribution for the class of polynomially-time bounded simple distributions is computable in exponential time (Li & Vitányi, 1991) . This opens up a number of interesting possibilities for learning under simple distributions. Denis and Gilleron (1997) have proposed a model of learning under helpful distributions. A helpful distribution is one in which examples belonging to the characteristic set for the concept (if there exists one) are assigned non-zero probability. A systematic characterization of the class of helpful distributions would perhaps give us a more practical framework for learning from simple examples.. A related question of interest has to do with the nature of environments that can be modeled by simple distributions. In particular, if Kolmogorov complexity is an appropriate measure of the intrinsic complexity of objects in nature and if nature (or the teacher) has a propensity for simplicity, then it stands to reason that the examples presented to the learner by the environment are likely to be generated by a simple distribution. Against this background, empirical evaluation of the performance of the proposed algorithms using examples that come from natural domains is clearly of interest.

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


Denials leak information: Simulatable auditing Imagine a data set consisting of private information about individuals. The online query auditing problem is: given a sequence of queries that have already been posed about the data, their corresponding answers and given a new query, deny the answer if privacy can be breached or give the true answer otherwise. We investigate the fundamental problem that query denials leak information. This problem was largely overlooked in previous work on auditing. Because of this oversight, some of the previously suggested auditors can be used by an attacker to compromise the privacy of a large fraction of the individuals in the data. To overcome this problem, we introduce a new model called simulatable auditing where query denials provably do not leak information. We present a simulatable auditing algorithm for max queries under the classical definition of privacy where a breach occurs if a sensitive value is fully compromised. Because of the known limitations of the classical definition of compromise, we describe a probabilistic notion of (partial) compromise, closely related to the notion of semantic security. We demonstrate that sum queries can be audited in a simulatable fashion under probabilistic compromise, making some distributional assumptions. Let X = {x 1 , . . . , x n } be a set of n private values from n individuals, where each x i is some real value. We consider the online query auditing problem: Suppose that the queries q 1 , . . . , q t−1 have already been posed about X and the answers a 1 , . . . , a t−1 have already been given, where each a j is either the true answer to the query or "denied". Given a new query q t , deny the answer if privacy may be breached, 2 and provide the true answer otherwise. The classical definition of breach is that there exists an index i such that x i is uniquely determined. In other words, in all data sets consistent with the queries and answers, there is only one possible value of x i . The kinds of queries considered in this paper are sum and max queries. Given a collection of indices S ⊆ [n], sum(S) = i∈S x i and max(S) = max i∈S x i .. One of the first auditing results dates back almost 30 years ago to the work of Dobkin, Jones, and Lipton [9] . That work restricts the class of acceptable queries by their size and overlap, and then demonstrates that a data set cannot be compromised provided that the number of queries is appropriately upper bounded. Following that work, many others considered the auditing problem including [25, 5, 6, 4, 18] (see also [1] for a survey).. In the work of Chin [4] , the online max auditing problem is considered. Given an online sequence of max queries, that paper gives a method of determining whether a value has been uniquely determined using only the queries that were exactly answered. Another example of an online auditing algorithm is due to Kleinberg, Papadimitriou, and Raghavan [18] . In the case that the underlying data set is Boolean valued, a method is given for determining whether a value has been uniquely determined -again using only the queries that were exactly answered.. These online auditing algorithms ignore the queries that were denied and this turns out to be quite problematic since denials can leak information. A simple example illustrates the phenomena. Suppose that the underlying data set is realvalued and that a query is denied only if some value is fully compromised. Suppose that the attacker poses the first query sum(x 1 , x 2 , x 3 ) and the auditor answers 15. Suppose also that the attacker then poses the second query max(x 1 , x 2 , x 3 ) and the auditor denies the answer. The denial tells the attacker that if the true answer to the second query were given then some value could be uniquely determined. Note that max(x 1 , x 2 , x 3 ) ≮ 5 since then the sum could not be 15. Further, if max(x 1 , x 2 , x 3 ) > 5 then the query would not have been denied since no value could be uniquely determined. Consequently, max(x 1 , x 2 , x 3 ) = 5 and the attacker learns that x 1 = x 2 = x 3 = 5 -a privacy breach of all three entries. The issue here is that query denials reduce the space of possible consistent solutions, and this reduction is not explicitly accounted for in existing online auditing algorithms.. This oversight of previous work on auditing [4, 18] is critical: privacy-preserving auditing algorithms that ignore denials can be used to launch massive privacy attacks. We partition related work into online and offline auditing. In the offline auditing problem, one is given a sequence of queries and exact answers and the goal is to determine if a privacy breach has occurred ex post facto. As the initial motivation for work on auditing involves the online auditing problem, we begin with known online auditing results.

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


Improved constructions of mixed state quantum automata Quantum finite automata with mixed states are proved to be super-exponentially more concise rather than quantum finite automata with pure states. It was proved earlier by A. Ambainis and R. Freivalds that quantum finite automata with pure states can have an exponentially smaller number of states than deterministic finite automata recognizing the same language. There was an unpublished ''folk theorem'' proving that quantum finite automata with mixed states are no more super-exponentially more concise than deterministic finite automata. It was not known whether the super-exponential advantage of quantum automata is really achievable.We prove that there is an infinite sequence of distinct integers n, languages L n , and quantum finite automata with mixed states with 5n states recognizing language L n with probability 3 4 , while any deterministic finite automaton recognizing L n needs at least e O(n ln n) states. Unfortunately, the alphabet for these languages grows with n. In order to prove a similar result for languages in a fixed alphabet we consider a counterpart of Hamming codes for permutations of finite sets, i.e. sets of permutations such that any two distinct permutations in the set have Hamming distance at least d. The difficulty arises from the fact that in the traditional Hamming codes for binary strings, positions in the string are independent while positions in a permutation are not independent. For instance, any two permutations of the same set either coincide or their Hamming distance is at least 2. The main combinatorial problem still remains open. A. Ambainis and R. Freivalds proved in [4] that for the recognition of some languages the quantum finite automata can have a smaller number of states than deterministic ones, and this difference can even be exponential. The proof contained a slight non-constructiveness, and the exponent was not shown explicitly. For probabilistic finite automata exponentiality of such a distinction was not yet proved. The best (smaller) gap was proved by Ambainis [2] . The languages recognized by automata in [4] were presented explicitly but the exponent was not. In a very recent paper by Freivalds [8] the non-constructiveness is modified, and an explicit (and seemingly much better) exponent is obtained at the expense of having only a non-constructive description of the languages used. Moreover, the best estimate proved in this paper is proved under assumption of the well-known Artin's Conjecture (1927) in Number Theory [5] . [8] contains also a theorem that does not depend on any open conjectures but the estimate is worse, and the description of the languages used is even less constructive. This seems to be the first result in finite automata depending on open conjectures in Number Theory.. The following two theorems are proved in [8] The two theorems above are formulated in [8] as assertions about reversible probabilistic automata. For probabilistic automata (reversible or not) it was unknown before the paper [8] whether the gap between the size of probabilistic and deterministic automata can be exponential. It is easy to rewrite the proofs in order to prove counterparts of Theorems 1 and 2 for quantum finite automata with pure states. The aim of this paper is to prove a counterpart of these theorems for quantum finite automata with mixed states.. Quantum algorithms with mixed states were first considered by Aharonov, Kitaev, and Nisan [1] . More detailed descriptions of quantum finite automata with mixed states can be found in Ambainis, Beaudry, Golovkins,Ķikusts, Mercer, and Thérien [3] .. The automaton is defined by the initial density matrix ρ 0 . Every symbol a i in the input alphabet is associated with a unitary matrix A i . When the automaton reads the symbol a i the current density matrix ρ is transformed into A * i ρA i . When the reading of the input word is finished and the end-marker $ is read, the current density matrix ρ is transformed into A * end ρA end and separate measurements of all states are performed. After that the probabilities of all the accepting states are totalled, and the probabilities of all the rejecting states are totalled.. Like quantum finite automata with pure states described by Kondacs and Watrous [10] we allow measurement of the accepting states and rejecting states after every step of the computation.. The main result in our paper is:. There is an infinite sequence of distinct integers n such that there are languages L n such that there are quantum finite automata with mixed states with 5n states recognizing the language L n with probability 3 4 while any deterministic finite automaton recognizing L n needs to have at least e O(n ln n) states.. Proof. The proof is delayed till Section 5.. Since the number of the states for deterministic automata and quantum automata with pure states differ no more than exponentially, we have Theorem 4. There is an infinite sequence of distinct integers n such that there are languages L n in a 2-letter alphabet such that there are quantum finite automata with mixed states with 2n states recognizing the language L n with probability 3 4 while any quantum finite automaton with pure states recognizing L n with bounded error needs to have at least e O(n ln n) states.. Unfortunately, the alphabet of the languages of Theorem 3 grows unbounded with n. It is only natural to try to prove a counterpart of Theorem 3 in 2-or 3-letter alphabet. We have developed a methodology for such a proof based on combining ideas of Theorem 3 and the results of [8] . However we need a notion similar to Hamming codes for permutations. Since the Hamming distance between permutations is already considered in several well-known textbooks (e.g. [6] ) it seemed natural that the corresponding theory might be already published. Very far from truth!

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


The impact of the service discipline on delay asymptotics This paper surveys the M/G/1 queue with regularly varying service requirement distribution. It studies the effect of the service discipline on the tail behavior of the waiting-time and/or sojourn-time distribution, demonstrating that different disciplines lead to quite different tail behavior. The orientation of the paper is methodological: We outline four different methods for determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS. Measurements indicate that traffic in high-speed networks exhibits burstiness on a wide range of time scales, manifesting itself in long-range dependence and self-similarity, see for instance [42, 51] . The occurrence of these phenomena is commonly attributed to extreme variability and heavy-tailed characteristics in the underlying activity patterns (connection times, file sizes, scene lengths), see for instance [10, 30, 56] . This has triggered a lively interest in queueing models with heavy-tailed traffic characteristics.. Although the presence of heavy-tailed traffic characteristics is widely acknowledged, the practical implications for network performance and traffic engineering remain to be fully resolved.. While several studies indicate that small buffer sizes, high levels of aggregation, and flow control algorithms limit the impact on packet-scale buffer dynamics (see, e.g. [5, 24] ), heavy-tailed traffic characteristics do dramatically affect flow-level delays experienced by users. A particularly interesting aspect is the role of scheduling and priority mechanisms in controlling the latter negative effect.. In a fundamental paper, Anantharam [4] considered a single-server queue fed by a Poisson arrival process of sessions, whose generic length is distributed as some integer random variable T with B 1 , B 2 , . . . are independent and identically distributed, with distribution B(·) with mean β and LaplaceStieltjes transform (LST) β{·}. A generic service requirement is denoted by B. There is no restriction on the number of customers in the system. We assume that the offered traffic load ρ := λβ < 1, so that the system reaches steady-state. We study the steady-state sojourn-time S of a customer, and in some cases also the steady-state waiting-time W until service begins.. Before surveying the tail asymptotics of the waiting-time and/or sojourn-time distributions for various service disciplines, we first introduce some useful notation and terminology. For any two real functions g(·) and h(·), we use the notational convention g(x) ∼ h(x) to denote lim x→∞ g(x)/ h(x) = 1, or equivalently, g(x) = h(x)(1 + o(1)) as x → ∞. For any stochastic variable X with distribution function F(·), with E{X} < ∞, denote by F r (·) the distribution function of the residual lifetime of X, i.e., F r (x) = 1/E{X} x 0 (1 − F(y)) dy, and by X r a stochastic variable with distribution F r (·). We focus on the class R of regularly varying distributions (which contains the Pareto distribution). This class is a subset of the class of subexponential distributions [39] as treated in Section 6, which includes for example the lognormal and Weibull distributions as well. In this paper, we have surveyed the tail behavior of the waiting-time and/or sojourn-time distributions for several service disciplines. It turns out that, if the service time distribution is regularly varying with index −ν, the waiting-time distribution in FCFS and LCFS-NP is heavier, viz. regularly varying with index 1 − ν. This is in contrast with service disciplines like PS, FBPS, SRPTF, and LCFS-PR: these disciplines all yield a sojourn-time tail of index −ν. These results are reviewed in Section 2, and proved in several different ways in Sections 3-6, where also multiclass disciplines are treated.. The results in this paper raise several questions. First of all, one wonders whether (given a service time distribution that is regularly varying of index −ν) the only possible indices of the sojourn-time distribution in a work-conserving single-server queue are −ν and 1 − ν. We believe that this is not the case; this is a topic for future research.. Another important question is the validity of the asymptotics for moderate values. Results in [1] for the FCFS queue show that the asymptotics may behave poorly as approximation. Moreover, asymptotic estimates tend to underestimate the true value of the exceedance probability. An explanation of this phenomenon is hidden in Section 5 of the present paper: the asymptotics are fully driven by the most likely scenario. However, other scenarios may be relevant as well for moderate values of x. To speed up the convergence of the asymptotic approximations to the true value of the sojourn-time tail probability, one could try to obtain more terms in the expansion. This is a problem which is largely open (an exception is FCFS, see, e.g. [1] ). Another alternative to get sojourn-time tail probabilities is numerical transform inversion: transforms of the distributions are available for many service disciplines, see Section 3 of the present paper.

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


Huge progeny production during the transient of a quasi-species model of viral infection, reproduction and mutation Eigen's quasi-species model describes viruses as ensembles of different mutants of a high fitness "master" genotype. Mutants are assumed to have lower fitness than the master type, yet they coexist with it forming the quasi-species. When the mutation rate is sufficiently high, the master type no longer survives and gets replaced by a wide range of mutant types, thus destroying the quasi-species. It is the so-called "error catastrophe". But natural selection acts on phenotypes, not genotypes, and huge amounts of genotypes yield the same phenotype. An important consequence of this is the appearance of beneficial mutations which increase the fitness of mutants. A model has been recently proposed to describe quasi-species in the presence of beneficial mutations. This model lacks the error catastrophe of Eigen's model and predicts a steady state in which the viral population grows exponentially. Extinction can only occur if the infectivity of the quasi-species is so low that this exponential is negative. In this work I investigate the transient of this model when infection is started from a small amount of low fitness virions. I prove that, beyond an initial regime where viral population decreases (and can go extinct), the growth of the population is super-exponential. Hence this population quickly becomes so huge that selection due to lack of host cells to be infected begins to act before the steady state is reached. This result suggests that viral infection may widespread before the virus has developed its optimal form. It seems that an unavoidable consequence of the increase in complexity of a system is the appearance of parasites. These are entities able to exploit backdoors, bypasses, holes. . . of the system for their own benefit, sometimes even at a cost for the system. We see a huge variety of these parasites in biology, ranging from viruses to humans. Society, in fact, is one of those complex systems amenable to exploitation by free-riders (the paradigm of the Public Goods game [1] is but one prominent acknowledgment of the existence of this social parasitism). More recently, the widespread use of computers and the arrival of Internet has made us witness the emergence and proliferation of computer viruses, trojans, worms, spam, phising, and all kinds of forms of parasitism, which flood the web using the same mechanisms aimed at allowing the transmission of information. Apparently, whenever a complex mechanism emerges, it is soon invaded by its specific parasites.. Parasites need not be complex: on the contrary, by being very specific to a particular mechanism, they are able to do their job with very simple mechanisms. Paradigmatic among parasites for their extreme simplicity are viruses. Their success is such that they are the most abundant life forms on Earth [2] . Their existence is an unavoidable outcome of the very evolutionary process. In fact, the most common strategy of RNA viruses is to have a very high reproductive rate which yields a wide variety of mutants [3] . This ensures their fast adaptation to almost any change.. One of the most important challenges in current medical research is how to fight viruses, and one of the most studied strategies is the design of therapies able to induce viral extinction. Increasing the mutation rate has been successful, at least in experiments in vitro, but there is no consensus as to why the virus loses infectivity at high mutation rates [4, 5, 6, 7] . The pioneering work of Eigen [8] explains viral extinction through a mechanism known as error threshold. According to it, the progeny loses its identity if the mutation rate grows above a given value, which is inversely proportional to the length of the replicating molecule -hence putting an upper bound to the complexity of viruses. This classical theory is currently questioned. The current state of the art of the evolutionary paradigm contradicts some of the basic assumptions of Eigen's theory, crucial for the existence of the error threshold. Alternative mechanisms may lead to viral extinction for reasons other than this hypothetical error threshold (like the presence of defective forms of the virus [9, 10] , the competition induced by geometrical constraints [11, 12, 13, 14, 15] , etc.).. Models of viral evolution need to make simplifying assumptions, and real virus behavior often deviates substantially from their predictions [16] . Current quasi-species models assume high mutation rates that give rise to heterogeneous populations. This is consistent with experimental observations. However, one common approximation is to consider that all new mutations have a deleterious effect on fitness, thus neglecting beneficial and neutral mutations. This is true if, as the theory assumes, there is a unique master sequence of high fitness. But we now know that the genotype-phenotype map is extremely redundant, and that a huge amount of different sequences -forming so-called neutral networks [17] -yield phenotypes that perform equally well. The increase in the rate of beneficial and neutral mutations that this effect brings about invalidates the classical theory of the error threshold [7] and calls for alternative models of viral evolution and extinction.. The aim of this paper is to explore one such model, introduced by Manrubia et al. [18] , with special focus on its transient behavior. We have analyzed the transient behavior of Manrubia et al.'s model (5) for a very large number of classes (R ≫ 1), by transforming it into an infinitely many class model. Although an explicit solution cannot be found, I have obtained the generating function associated to the vector of class populations. The singularities of this function provide the time asymptotic behavior of the total population of the virus, valid as long as the number of generations is smaller than R. Surprisingly we find that viral population grows super-exponentially, unlike in the steady state.. Analyzing eq. (25) more closely, we notice that s 1 can have very large values and thus induce an initial decay of the population. However, this decay gets dominated by the factorial (t + r − 1)! as soon as t > t d ≡ es 1 − r + 1. During this decay time t d (which is shorter the larger r) fluctuations of the branching process can lead the virus to extinction. Beyond that interval the virus population starts to recover and grows at a faster than exponential rhythm.. A standard assumption in studies of viral quasi-species evolution is that their population is in the exponential asymptotic state. But if R ≫ 1 the time to reach this state can be very long (in fact, it requires at least R − r generations to reach the optimal class, let alone to attain a stationary distribution among classes). Before that we have the virus population growing faster than exponential and it is plausible that resources get exhausted during this transient period. This means that selection starts playing a role when the steady distribution is not yet established, leading to a behavior different from what is to be expected in the asymptotic regime. The effects of this phenomenon are as yet unexplored.

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


KISS: Stochastic Packet Inspection Classifier for UDP Traffic Abstract-This paper proposes KISS, a novel Internet classification engine. Motivated by the expected raise of UDP traffic, which stems from the momentum of Peer-to-Peer (P2P) streaming applications, we propose a novel classification framework that leverages on statistical characterization of payload. Statistical signatures are derived by the means of a Chi-Square ( 2 )-like test, which extracts the protocol "format," but ignores the protocol "semantic" and "synchronization" rules. The signatures feed a decision process based either on the geometric distance among samples, or on Support Vector Machines. KISS is very accurate, and its signatures are intrinsically robust to packet sampling, reordering, and flow asymmetry, so that it can be used on almost any network. KISS is tested in different scenarios, considering traditional client-server protocols, VoIP, and both traditional and new P2P Internet applications. Results are astonishing. The average True Positive percentage is 99.6%, with the worst case equal to 98.1,% while results are almost perfect when dealing with new P2P streaming applications. We presented KISS, a novel classifier explicitly targeting UDP traffic that couples the stochastic description of application protocols with the discrimination power of SVMs. Signatures are extracted from a traffic stream by the means of -like test that allows application protocol format to emerge while ignoring protocol synchronization and semantic rules. A decision process based on SVM is then used to classify the extracted signatures, leading to exceptional performance.. Performance of KISS has been tested in different scenarios, considering both data, VoIP, traditional P2P applications, and novel P2PTV systems. Results are astonishing. The average True Positive percentage is 99.6%, and less than 1% of False Positives are typically detected. Moreover, KISS is very robust to internal parameter setting, and it is efficient considering both memory and computational requirements. Since Port-based classification [1] has become unreliable, a number of different solutions and methodologies have been proposed to classify Internet traffic [4] , [5] , [9] , [17] - [25] . Classification engines can be coarsely divided into three categories, each of them exploiting different ideas. For a good survey, see [8] , while [26] is a complementary work for the survey.. Payload-based techniques [9] , [18] - [20] inspect the content of packets looking for distinctive signatures that allow to recognize a given application. All DPI techniques fall in this class.. Machine-learning-based classification [5] , [21] - [23] , [27] , [28] relies on the rationale that since the nature of the services is extremely diverse (e.g., Web versus VoIP), the corresponding generated traffic is very diverse as well (e.g., short-lived bursts of big packets versus long-lived, constant bitrate flows of small packets). This class of work stems from the characterization and modeling research field, which started from pioneering work [29] . Initial work in this area focused on the offline traffic classification, exploring which flow properties and which classification technique was best suited to discriminate traffic flows according to the different classes of applications [5] , [21] - [23] . More recently, [27] and [28] addressed the problem of "early" classification of individual applications, basing solely on information such as the size and direction (and interpacket gap in the case of [28] ) of the very first packets of each flow: The initial handshake phase of different applications is distinctive and can be used as protocol fingerprint (e.g., SMTP handshake is different from HTTP one).. Finally, behavioral-based classification [4] , [24] , [25] targets the classification of Internet hosts on the sole basis of the transport-layer traffic patterns they generate (e.g., P2P hosts contact many different hosts typically using a single port, whereas a Web server is contacted by different clients with multiple parallel connections).. Our work aims at fine-grained classification of Internet traffic. As such, we consider work targeted to host identification [4] , [24] , [25] or coarse-grained identification [5] , [21] , [22] to not be suited as a comparison for our purpose. Moreover, this work aims at filling a gap in the current Internet classification spectrum, specifically addressing UDP traffic classification. Since UDP is a connectionless protocol, we argue that an approach such as [27] or [28] cannot be applied as no handshake can be reliably identified in this case. Indeed, even the notion of "flow" is fuzzy considering UDP streams.. Works closest to ours are those that belong to the payloadbased class. However, our work is very different from [9] and [18] since the definition of application signatures does not rely on any reverse-engineering of the applications. Instead, our approach is more similar in spirit to [19] and [20] , in which authors automate the extraction of signatures from the application payload. Both [19] and [20] rely on signatures extracted from the beginning of each data stream (more specifically, the first 64-256 B). We remove this assumption so that the classification can start at any point in a flow. This is an important difference since, for example, it opens the door to adopt packet sampling to cope with the ever increasing link data rate.. Another significant difference consists in the technique used to express the payload fingerprint: [19] uses discrete byte encoding, whereas the framework of [20] proposes the use of different models of increasing complexity. Another difference consists in the technique explored to perform the classification. Indeed, in this work we use Support Vector Machines (SVMs) which, to the best of our knowledge, have not yet been deeply tested in the context of Internet traffic classification.

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


Unbiased tournament selection Tournament selection is a popular form of selection which is commonly used with genetic algorithms, genetic programming and evolutionary programming. However, tournament selection introduces a sampling bias into the selection process. We review analytic results and present empirical evidence that shows this bias has a significant impact on search performance. We introduce two new forms of unbiased tournament selection that remove or reduce sampling bias in tournament selection. Selection is an important aspect of evolutionary computation. It dictates what members of the current population affect the next generation. More fit individuals are generally given a higher chance to participate in the recombination process. The motivation is that highly fit members of a population possess good properties that, recombined in the right way, could lead to even better solutions.. The primary concern of all selection schemes is what's known as the loss of diversity. In a generational genetic algorithm (GA), not every individual makes it into the intermediate population. As a result, information encoded in the current population is not transferred into the next generation in its entirety. Loss of diversity has been measured and analyzed for a number of popular selection algorithms( [6] , [3] , [4] , and [7] ).. Of particular interest is tournament selection, which is commonly used with genetic algorithms, genetic programming and evolutionary programming. As pointed out by Poli [8] , the loss of diversity in tournament selection is caused by two factors. He distinguishes between the notions of not being sampled and not being selected. The first holds true about an individual that does not get to participate in a tournament. The second refers to an individual losing a tournament and, thus, not making it into the intermediate population. Poli notes that the objective function is unnecessarily evaluated on individuals that don't affect the next generation and suggests ways to eliminate this computation [8] .. This paper takes a different approach. We propose a selection scheme that eliminates the loss of diversity in tournament selection due to individuals not being sampled. Our algorithm is based on reducing variance in the number of times a particular individual is picked to participate in a tournament. We demonstrate that using our selection algorithm yields better results than the traditional tournament selection when used in a generational genetic algorithm.. We also introduce a new scheduling problem into the literature: Radar Scheduling. The problem is a variation of resource-constrained project scheduling, as defined in [10] . The implementation presented in this papers consists of a greedy scheduler that constructs and evaluates a schedule based on some given permutation of tasks. A GA works with permutations and uses the scheduler to calculate fitness of every solution.. The paper is organized in the following way. In Section 2 we briefly overview some of the popular selection schemes and introduce our novel algorithm. Applications and empirical results are presented in Section 3. Section 4 consists of additional elaboration on the observed behavior. Finally, Section 5 summarizes the work done and describes possible directions to be taken for further analysis. Given the popularity of tournament selection, it is important to understand that there are minor algorithm changes that can improve the performance of a generational genetic algorithm. As has been shown, eliminating the loss of di- Table 6 : Radar Scheduling with POPSIZE=100 and enforced diversity versity due to population members not being sampled leads to better performance. It is also important to note that the effects of failing to sample the most fit individual in the population can be very dramatic when population sizes are small. We introduce two simple forms of tournament selection that eliminate loss of diversity due to a failure to sample. One of these algorithms, parallel unbiased tournament selection, is still embarrassingly parallel and independent in nature. The other algorithm, which we call unbiased tournament selection and which is analyzed empirically in this paper, has been shown to enhance search on a small but representative sample of test problems. We have also pointed out that the use of either tournament selection scheme yielded no significant difference in performance when diversity was enforced.. Future work should further evaluate parallel unbiased tournament selection. We would also like to examine the impact of bias for tournament selection when the selection pressure is less than or greater than 2.0. It would be desirable to have a way of calculating the loss of diversity in both cases.

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