Polar codes for degradable quantum channels Abstract-Channel polarization is a phenomenon in which a particular recursive encoding induces a set of synthesized channels from many instances of a memoryless channel, such that a fraction of the synthesized channels becomes near perfect for data transmission and the other fraction becomes near useless for this task. Mahdavifar and Vardy have recently exploited this phenomenon to construct codes that achieve the symmetric private capacity for private data transmission over a degraded wiretap channel. In the current paper, we build on their work and demonstrate how to construct quantum wiretap polar codes that achieve the symmetric private capacity of a degraded quantum wiretap channel with a classical eavesdropper. Due to the Schumacher-Westmoreland correspondence between quantum privacy and quantum coherence, we can construct quantum polar codes by operating these quantum wiretap polar codes in superposition, much like Devetak's technique for demonstrating the achievability of the coherent information rate for quantum data transmission. Our scheme achieves the symmetric coherent information rate for quantum channels that are degradable with a classical environment. This condition on the environment may seem restrictive, but we show that many quantum channels satisfy this criterion, including amplitude damping channels, photondetected jump channels, dephasing channels, erasure channels, and cloning channels. Our quantum polar coding scheme has the desirable properties of being channel-adapted and symmetric capacity-achieving along with having an efficient encoder, but we have not demonstrated that the decoding is efficient. Also, the scheme may require entanglement assistance, but we show that the rate of entanglement consumption vanishes in the limit of large blocklength if the channel is degradable with classical environment. In a seminal paper on quantum error correction, Shor set out the "goal of [defining] the quantum analog of the Shannon capacity [40] for a quantum channel, and [finding] encoding schemes which approach this capacity" [41] . At the time, it was not really clear how to define the quantum capacity of a quantum channel, but Shor's quantum error correction code [41] gave some clues for constructing more general encoding schemes. Subsequently, several authors contributed increasingly sophisticated quantum error correction codes [10] , [45] , [18] and others established a good definition of and upper bounds on the quantum capacity of a channel [36] , [37] , [4] , [5] , culminating in some high-performing quantum error-correction codes [31] , [30] , [34] , [26] and random-coding based schemes for achieving the coherent information rate [37] of a quantum channel [29] , [42] , [11] . For some channels known as degradable quantum channels [12] , in which the channel to the environment is noisier than the channel to the intended receiver, the random-coding based schemes from Refs. [29] , [42] , [11] achieve their quantum capacity, due to the particular structure of these channels.. In spite of the astounding progress in both quantum error correction [13] and quantum Shannon theory [47] , none of the high performance codes constructed to date are provably capacity achieving, and none of the aforementioned schemes that achieve the capacity are explicit (the proofs instead exploit randomness to establish the existence of a code). Among the schemes that achieve the quantum capacity, perhaps Devetak's [11] provides the most clear recipe to a quantum code designer interested in constructing a capacity-achieving quantum code. His proof takes a cue from a certain security proof of quantum key distribution [43] and the SchumacherWestmoreland correspondence between quantum privacy and quantum coherence [38] , by first establishing the existence of codes that achieve the private capacity of a quantum wiretap channel and then demonstrating how to operate such a code in superposition so that it achieves the quantum capacity. It is also clear that the structure of his codes bears some similarities with Calderbank-Shor-Steane codes [10] , [45] .. Along with the above developments, there have been impressive breakthroughs in classical coding theory and information theory [14] , one of which is Arikan's recent work on polar codes [2] . Arikan's polar codes exploit a phenomenon known as channel polarization, in which a particular recursive encoding induces a set of synthesized channels from the original memoryless noisy channels. The synthesized channels are such that a fraction of them are perfect for data transmission, while the other fraction are completely useless, and the fraction that are perfect is equal to the symmetric capacity of the original channel. The codes are channel adapted, in the sense that Arikan's "polar coding rule" establishes through which of the synthesized channels the sender should transmit data, and this polar coding rule depends on the particular channel being used. The codes are near explicit and have the desirable property that both the encoding and decoding are efficient (the complexity of each is O (N log N ) where N is the blocklength of the code).. Arikan's work might make us wonder whether it would be possible to construct polar codes for transmitting quantum data over general quantum channels, and the development in the classical world most relevant for this task is due to Madhavifar and Vardy [32] . There, they established that a modification of Arikan's original polar coding scheme can achieve the symmetric private capacity of a degraded classical wiretap channel. (In order to make this statement, the sender and receiver actually require access to a small amount of secret key, but the rate of secret key needed vanishes when the code's blocklength becomes large.) Thus, with the Madhavifar-Vardy scheme for polar coding over classical degraded wiretap chan-nels [32] , the Devetak scheme for operating a quantum wiretap code in superposition [11] , and our recent work on polar codes for transmitting classical data over quantum channels [48] , it should be evident that one could put these pieces together in order to construct polar codes for transmitting quantum data over degradable quantum channels.. In this paper, we pursue this direction by constructing polar codes that achieve a symmetric capacity for transmitting quantum data over particular degradable quantum channels. These channels should satisfy the property that encoding classical data in some orthonormal basis at their input leads to commuting states for the environment (essentially, the environment becomes classical), and we clarify later why this is important in our construction. Many degradable channels fall into this class, including amplitude damping channels [16] , photondetected jump channels [1] , erasure channels [19] , dephasing channels [11] , and cloning channels [6] (channels induced by universal cloning machines [17] , [9] ). These noisy channels occur naturally in physical processes, with amplitude damping modeling photon loss or spontaneous emission, the photondetected jump channel modeling the spontaneous decay of atoms with a detected photon emission [1] , the erasure channel being a different model for photon loss [15] , the dephasing channel modeling random phase noise in superconducting systems [7] , and the cloning channel modeling stimulated emission from an atom [33] , [44] , [28] . Our codes are symmetric capacity achieving for all of the above channels, and this follows from analyzing a quantum polar coding rule for these channels.. We summarize briefly how the construction works. First, we consider a quantum wiretap channel with one classical input and two quantum outputs, one for the legitimate receiver (Bob) and the other for the wiretapper (Eve). We demonstrate that these channels polarize in four different ways, based on whether the channels are good or bad for the receiver or the wiretapper. In order to have strong security, we must guarantee that the bad channels for the wiretapper are in fact "really bad" in a precise sense, and it is for this reason that we consider channels with classical environment. (Interestingly, we know of quantum channels where it is not clear to us how to ensure that they become "really bad," and we prove some results in this direction in Appendix A.) The resulting coding scheme is to send the information bits through the channels which are good for Bob and bad for Eve, "frozen" bits through the channels that are bad for both, half of a secret key through channels which are bad for Bob but good for Eve, and randomized bits through the channels good for both. By an analysis similar to that of Madhavifar and Vardy [32] , we can demonstrate that this scheme achieves the symmetric private capacity of a degraded quantum wiretap channel with classical environment, while the rate of secret key required vanishes in the limit of large blocklength.. The main idea for constructing quantum polar codes for degradable quantum channels with classical environment is just to operate the quantum wiretap code in superposition and exploit Arikan's encoding with CNOT gates with respect to some orthonormal basis. This amounts to sending information qubits through the channels good for the receiver Bob and bad for the environment Eve, frozen ancilla qubits through the channels that are bad for both, half of shared entanglement through the channels that are bad for Bob but good for Eve, and superposed ancilla qubits in the state |+ ≡ (|0 + |1 ) / √ 2 through the channels that are good for both. The resulting quantum polar codes are entanglement-assisted [8] , but we can prove that the entanglement consumption rate required vanishes in the limit of large blocklength (in this regard, the codes here are similar to Hsieh et al.'s recent ones [25] ). Operating the quantum successive cancellation decoder from Ref. [48] in a coherent fashion, followed by controlled "decoupling unitaries" allows us to exploit the properties of the quantum wiretap polar code in order to prove that the above quantum polar code performs well (we note that this decoder is similar to Devetak's [11] ). The resulting quantum polar codes achieve the symmetric quantum capacity of degradable channels with classical environment with an encoding circuit that has complexity O (N log N ). The decoding unfortunately remains inefficient, but further efforts may lead to an efficient realization of a decoder.. Recently, Renes et al. have independently constructed quantum polar codes that have both an efficient encoding and decoding, though they achieve the coherent information rate only for Pauli channels [35] . We should clarify the ways in which their scheme is different from ours. First, they restrict their construction to Pauli channels because they are considering the effective classical channels induced in the amplitude (Pauli-Z) and phase (Pauli-X) bases (not all channels, including some of the ones mentioned above, induce classical channels in complementary bases). As a result, they can directly import Arikan's ideas because they are dealing with classical channels in complementary bases. An additional bonus is that they obtain an efficient decoder as well as an efficient encoder, essentially because their decoder is Arikan's successive cancellation decoder implemented as an efficient unitary operation. Their codes require the assistance of shared entanglement, but there are some channels for which they can prove that it is not required.. Our scheme is different from theirs in several ways. First, we have a quantum polar coding rule that is adapted to a given quantum channel. In particular, the rule used for determining the good or bad channels is based on a quantum parameter (fidelity). Also, we demonstrate polarization in terms of two quantum parameters, the fidelity and the Holevo information, by building on our earlier results in Ref. [48] . It is for this reason that our scheme is symmetric capacity-achieving for a wide variety of quantum channels. Also, the first part of our decoder is a coherent version of the quantum successive cancellation decoder from Ref. [48] , rather than one that is based directly on the classical decoder. Since we have not proven that the quantum successive cancellation decoder from Ref. [48] has an efficient implementation, the coherent version of it in this work is certainly not efficient. In spite of the inefficiency of the quantum successive cancellation decoder, this decoder is needed in order to achieve the symmetric quantum capacity of the channels considered here. Finally, all the channels we consider here only require a vanishing rate of entanglement assistance, due to an argument similar to Proposition 22 of Madhavifar and Vardy [32] and the fact that they are degradable with a classical environment.. We structure this paper as follows. Section II reviews our work from Ref. [48] on polar codes for classical-quantum channels. We present in Section III our scheme for private communication over quantum wiretap channels that are degraded with a classical environment. Finally, we demonstrate how to construct quantum polar codes from quantum wiretap polar codes in Section IV. The last section concludes with a summary and some open questions. We constructed polar codes for transmitting classical data privately over a quantum wiretap channel and for transmitting quantum data over a quantum channel. The rates achievable are respectively equal to the symmetric private information and the symmetric coherent information. In both settings, we require that the channel to the environment be effectively classical in order to guarantee that the rates are as claimed. The codes exploit the channel polarization phenomenon, first observed by Arikan in the classical setting [2] and later in Ref. [48] for classical-quantum channels.. Two of the most important problems going forward from here are to determine an efficient quantum decoder and to find other degradable channels outside of the class discussed here for which the symmetric quantum capacity is achievable. For this latter problem, one might need to consider an alternate inequality to that in (10) , in order to determine the "bad channel polarization rate"à la Proposition 4. Such an alternate inequality should have the property that it pushes the process X n closer to zero in some power in order to prove the needed convergence rates [21] . One could also try to determine whether the inequality in (10) might generally hold as n becomes larger. Numerical evidence indicates that the inequality in (10) holds whenever the purity of the two input states is below a certain threshold.. One might question whether we should call the codes developed here and in Ref. [35] "quantum polar codes," if the criterion for a quantum polar code is that it be symmetric capacity-achieving, channel-adapted, and possessing efficient encoders and decoders. The codes constructed here are symmetric capacity-achieving and channel-adapted, but do not have efficient decoders. The codes from Ref. [35] have efficient encoders and decoders, but they are capacity-achieving and channel-adapted only for Pauli channels. Solving the open problems posed here would certainly lead to quantum polar codes that possess all desiderata. ACKNOWLEDGEMENTS MMW acknowledges financial support from the MDEIE (Québec) PSR-SIIRI international collaboration grant. SG was supported by the DARPA Information in a Photon (InPho) program under contract number HR0011-10-C-0159. The authors are grateful to S. Hamed Hassani for helpful discussions concerning Theorem 3 in Ref. [22] and Omar Fawzi and Patrick Hayden for useful discussions.

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


A data-model-fusion prognostic framework for dynamic system state forecasting A novel data-model-fusion prognostic framework is developed in this paper to improve the accuracy of system state long-horizon forecasting. This framework strategically integrates the strengths of the data-driven prognostic method and the model-based particle filtering approach in system state prediction while alleviating their limitations. In the proposed methodology, particle filtering is applied for system state estimation in parallel with parameter identification of the prediction model (with unknown parameters) based on Bayesian learning. Simultaneously, a data-driven predictor is employed to learn the system degradation pattern from history data so as to predict system evolution (or future measurements). An innovative feature of the proposed fusion prognostic framework is that the predicted measurements (with uncertainties) from the data-driven predictor will be properly managed and utilized by the particle filtering to further update the prediction model parameters, thereby enabling markedly better prognosis as well as improved forecasting transparency. As an application example, the developed fusion prognostic framework is employed to predict the remaining useful life of lithium ion batteries through electrochemical impedance spectroscopy tests. The investigation results demonstrate that the proposed fusion prognostic framework is an effective forecasting tool that can integrate the strengths of both the data-driven method and the particle filtering approach to achieve more accurate state forecasting. Condition-based maintenance is a program that recommends maintenance decisions based on the information collected through system condition monitoring (or system state estimation) and equipment failure prognostics (or system state forecasting), in which prognostics still remains as the least mature element in both research and real-world applications (Jardine et al., 2006) . Prognostics entails the use of the current and previous system states (or observations) to predict the future states of a dynamic system. Reliable forecast information can be used to schedule repairs and maintenance in advance and provide an alarm before faults reach critical levels so as to prevent machinery performance degradation, malfunction, or even catastrophic failures (Liu et al., 2009 ).. In general, prognostics can be conducted using either datadriven methods or model-based approaches. Data-driven methods use pattern recognition and machine learning to detect changes in system states (Yagiz et al., 2009; Gupta and Ray, 2007) . The classical data-driven methods for nonlinear system prediction include the use of stochastic models such as the autoregressive (AR) model, the threshold AR model (Tong and Lim, 1980) , the bilinear model (Subba, 1981) , the projection pursuit (Friedman and Stuetzle, 1981) , the multivariate adaptive regression splines (Friedman, 1991) , and the Volterra series expansion (Brillinger, 1970) . Since the last decade, more research interests in data-driven system state forecasting have been focused on the use of flexible models such as various types of neural networks (NNs) (Atiya et al., 1999; Liang and Liang, 2006) and neural fuzzy (NF) systems (Husmeier, 1999; Korbicz, 2004; Jang, 1993) . Data-driven methods rely on past patterns of the degradation of similar systems to project future system states; their forecasting accuracy depends on not only the quantity but also the quality of system history data, which could be a challenging task in many real applications (Liu et al., 2009; Wang and Vrbanek, 2008) . Another principal disadvantage of data-driven methods is that the prognostic reasoning process is usually opaque to users (Tse and Atherton, 1999) ; consequently, they are not suitable for some advanced applications where forecast reasoning transparency is required (e.g., credit card cheating, earthquake and stock market prediction).. Model-based approaches typically involve building models (or mathematical functions) to describe the physics of the system states and failure modes; they incorporate physical understanding of the system into the estimation of system state and/or remaining useful life (RUL) (Adams, 2002; Luo et al., 2003; Chelidze and Cusumano, 2004) . Model-based approaches, however, may not be suitable for many industrial applications where the physical parameters and fault modes may vary under different operation conditions (Pecht and Jaai, 2010) . On one hand, it is usually difficult to tune the derived models in situ to accommodate time-varying system dynamics. On the other hand, modelbased approaches cannot be used for complex systems whose internal state variables are inaccessible (or hard) to direct measurement using general sensors. In this case, inference has to be made from indirect measurements using techniques such as particle filtering (PF). The PF-based approaches have been used for prognostic applications (Saha et al., 2009) , in which the PF is employed to update the nonlinear prediction model and the identified model is applied for system state forecasting. However, a limitation associated with the classical PF-based predictors is that the prediction model parameters cannot be updated during the prognostic period since no new measurements are available. The prediction accuracy could be low in many applications because the identified model during the state estimation period may not be accurate and robust.. To address the aforementioned challenges, a data-modelfusion framework is proposed in this work for system state prognostics. The developed framework aims to integrate the strengths of the data-driven prognostic method and the modelbased PF approach for a more reliable system state forecasting. The proposed fusion framework is new in the following aspects: (1) the prediction uncertainties from the data-driven predictor can be properly managed and utilized through the fusion framework so as to further update the prediction model parameters; (2) the fusion prognostic framework can overcome the aforementioned limitations of both the data-driven method and the modelbased PF approach so as to make prediction models more interpretable and transparent; (3) as an application example, the developed fusion prognostic framework is implemented for the RUL prediction of lithium-ion batteries. This paper is organized as follows. The proposed fusion prognostic framework is described in Section 2. The effectiveness of this fusion framework is demonstrated in Section 3 via an application in battery RUL prediction. A summary of important observations and conclusive remarks are given in Section 4. A data-model-fusion prognostic framework has been developed in this work for system state forecasting. This fusion framework is able to integrate the strengths of both the datadriven prognostic method and the model-based particle filtering in system state prediction while alleviating their respective limitations. A unique feature of the proposed fusion framework is to improve the transparency of the data-driven method in the sense that its prediction uncertainty can be properly managed and utilized by a particle filtering based learning process. As a result, the nonlinear prediction model parameters can be updated in a continuous and recursive manner so as to further improve the prediction accuracy. The effectiveness of the developed fusion prognostic framework has been demonstrated through an application example in predicting the remaining useful life of lithium ion batteries. The test results have indicated that the proposed fusion prognostic framework can effectively track system states by incorporating the predicted measurements from the datadriven predictor so as to adaptively update the prediction model parameters; thereby it outperforms the related data-driven predictors and the classical particle filtering based approaches in system state forecasting. As a final note, it is also worth mentioning the following considerations: (1) the goal of this paper is to propose a novel data-model fusion framework for prognostic applications. This concept can be analogous to sensor fusion using the extended Kalman filter in the field of fault diagnostics. In implementation, this fusion prognostic framework involves two principal components: a data driven prognostic method by which the uncertainty of the associated data driven forecasting can be estimated, and a particle filtering based system degradation model by which the system aging trend can be interpreted. For different applications, these two components will differ and must be determined. (2) In this work, three data-driven prognostic methods (i.e., the NN predictor, the NF predictor, and the RNF predictor) and a generic particle filtering have been tested. Testing results have shown that the proposed fusion framework outperforms each of these prognostic methods. Actually, it is safe to say that the proposed fusion framework could work well with virtually any data-driven prognostic methods as long as the prediction uncertainty can be estimated. (3) The authors have applied the most commonly-used metric (i.e., the aÀl prognostic metric where a is the prediction error andl is the remaining time from the earliest prediction to Fig. 9 . The battery RULs predicted by the proposed fusion prognostic framework (dots, in red), the classical PF-based predictor (triangles, in magenta), and the RNF predictor (squares, in blue); the shaded green area is the 90% accuracy cone (a¼0.1, l¼0.5). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) the actual end-of-life of a system) in failure prediction to quantify the prognostic performance of the proposed data-model fusion prognostic framework; (4) The motivation of this work is elaborated in Section 2 in which the limitations of two principal components of the proposed fusion prognostic framework are discussed. This framework aims to integrate the advantages of both the data-driven predictor and the PF approach while alleviating their respective limitations, so as to develop a more reliable system state forecasting paradigm. (5) Battery prediction is employed in this work as an application example to demonstrate the effectiveness of the proposed fusion framework. The authors use this example instead of others because they have been working on battery RUL prediction for a few years and both the experimental data and the physical prediction model for battery RUL prediction are readily available. It is also expected that the readers or other researchers will apply this framework in the areas of their expertise to further investigate its effectiveness. (6) Compared to fault diagnostic, failure prognostic is a much less mature and more challenging research area. One of the main reasons is the lack of sufficient experimental/field data to support fundamental studies for developing proper prognostic models. Taking the lithium ion battery RUL prediction as an example, it usually takes more than a year (Fig. 4) to age a battery in a lab environment. (7) A collaborative research is currently underway among Carleton University, Nanjing University, and National Research Council of Canada to extend this prognostic framework to failure prognostics of auxiliary power units (APUs) of aircrafts. However, considering the complexity of APUs (thus the associated data base), the research work, including the development of both the physical prognostic model, data-driven prognostic model, will take years to finish and the research results will be published in the follow-up publications. (8) The in-depth statistical analysis (as suggested by one of the reviewers) of the proposed prognostic method deserve further investigation and will be conducted when the testing results from the APUs are obtained. (9) The performance comparison of the proposed fusion framework with other related prognostic methods, e.g., Gaussian process regression (Goebel et al., 2008) , independent component analysis (Lee et al., 2004) , and Gaussian mixture model (Yu and Qin, 2008, 2009) , will remain as topics for future studies. Fig. 11 . The battery RULs predicted by the proposed fusion prognostic framework (dots, in red), the classical PF-based predictor (triangles, in magenta), and the NN predictor (squares, in blue); the shaded green area is the 90% accuracy cone (a¼ 0.1, l¼0.5). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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


Typestate verification: Abstraction techniques and complexity results Abstract. We consider the problem of typestate verification for shallow programs; i.e., programs where pointers from program variables to heap-allocated objects are allowed, but where heap-allocated objects may not themselves contain pointers. We prove a number of results relating the complexity of verification to the nature of the finite state machine used to specify the property. Some properties are shown to be intractable, but others which appear to be quite similar admit polynomial-time verification algorithms. While there has been much progress on many aspects of automated program verification, we are not aware of any previous work relating the difficulty of typestate verification to properties of the finite state automaton. Our results serve to provide insight into the inherent complexity of important classes of verification problems. In addition, the program abstractions used for the polynomial-time verification algorithms may be of independent interest. The desire for more reliable software has led to increasing interest in extended static checking: statically verifying whether a program satisfies certain desirable properties. A technique that has received particular attention is that of finite state or typestate verification (e.g., see [24, 23, 19, 5, 7, 3, 8, 12, 11, 16, 1] ). In this model, objects of a given type may exist in one of several states; the operations permitted on an object depend on the state of the object, and the operations may potentially alter the state of the object. The goal of typestate verification is to statically determine if the execution of a given program may cause an operation to be performed on an object in a state where the operation is not permitted.. Typestate verification can thus be used to check that objects satisfy certain kinds of temporal properties; for example, that an object is not used before it is initialized, or that a file is not used after it is closed. The properties we will consider in this paper will be specified using regular expressions or finite state automata which define the set of valid sequences of operations that can be performed on a single object.. Our goal in this paper is to develop an initial understanding of how the difficulty of performing typestate verification relates to the nature of the property being verified. While there has been much progress on many aspects of automated program verification, we are not aware of any previous work relating the difficulty of typestate verification to properties of the finite state automaton. This work is part of a broader effort to develop efficient program verification techniques that are tailored to the property being verified [21] . A logical formula can usually be simplified into a number of equivalent forms. Hence, a weakestprecondition can often be expressed in many ways. The form we chose to use in expressing weakest-preconditions above is critical to deriving a polynomial time verification algorithm. As an example, consider the read * ; close example. The following is an alternative, correct, weakestprecondition equation, which says that an object in the err state is possible after x.close() iff either x points to an object in state q2 or an object exists in the err state before the statement:. The actual formulation we used. actually contains some redundancy. In particular, {x}, {err, q2} is equivalent to {x}, {err} ∨ {x}, {q2} . But the disjunct {x}, {err} is redundant because it implies Error, another disjunct in our formula.. However, equation 2 is preferable to equation 1. In particular, we have seen that we can determine in polynomial time if {x}, {err, q2} is possible at any program point. However, one can show that determining if {x}, {q2} is possible at a program point is PSPACE-hard, adapting the proof we present in Section 4. Thus, unless PSPACE = P, a distributively WP-closed set containing {x}, {q2} of polynomial size does not exist. There has been a significant resurgence of interest in partial verification recently. Many, though not all, of the problems considered in the literature can be expressed as typestate verification problems.. While significant progress has been made in several dimensions, developing verification techniques that are precise enough and scale up to handle industrial-size applications is still a challenge and is the motivation for the complexity results of the form presented in this paper.. One of the challenges in typestate verification is dealing with aliasing. Some approaches avoid the issue: e.g., the original work on typestate verification [24, 23] did not allow any aliasing, and more recent work on typestate verification based on linear types [7] also restrict aliasing severely. Some other approaches (e.g. [6] ) do the alias analysis and typestate verification separately: an initial phase performs a conservative alias analysis for the program, and a subsequent phase uses the information from the alias analysis to do typestate verification. This can lead to imprecise results, as illustrated by the examples in Fig. 1 .. A second challenge in typestate verification is dealing with infeasible paths in the program. Das et al. [6] present efficient algorithms that are path-sensitive (i.e., eliminate certain infeasible paths from consideration during analysis) but do not track aliasing precisely (as explained above). In contrast, our algorithms track (one-level) aliasing precisely, but are not path-sensitive. It might be worthwhile trying to combine these techniques.. Several recent verification approaches [2, 15] , based on predicate abstraction [14] , avoid imprecision (e.g., due to aliasing or infeasible paths) by iteratively refining the abstractions as necessary, but are fundamentally exponential algorithms. These techniques use symbolic and theorem-proving techniques (during verification) to identify a set P of "relevant" predicates, and then use the powerset lattice 2 P →{true,f alse} for abstraction, and then model check the resulting finite state system (and usually iterate with increasingly larger sets of predicates until a satisfactory result is obtained). The worst-case complexity of a single iteration is exponential in the number of predicates. In contrast, for our upper bounds, we describe the set of predicates Q to be used a priori (and do not rely on expensive predicate discovery at verification time). More importantly, our verification algorithms utilize the function-space lattice Q → {f alse, maybe} for abstraction, with a worst-case complexity that is linear in the number of predicates. Our selection of predicates ensures that there is no loss of precision in using the smaller function-space lattice, i.e., we ensure that our abstraction is complete (e.g., see [13] ). Our approach yields polynomial time verification algorithms for various problems, while none of the predicate abstraction based techniques mentioned above can guarantee a polynomial time worst-case complexity for the same problems (even though they may perform verification with a smaller number of predicates than our algorithm).. Our lower bound results follow the tradition set by earlier complexity results due to Landi and Ryder [17] and Muth and Debray [18] .

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


A Case-Based Approach for Coordinated Action Selection in Robot Soccer Designing coordinated robot behaviors in uncertain, dynamic, real-time, adversarial environments, such as in robot soccer, is very challenging. In this work we present a case-based reasoning approach for cooperative action selection, which relies on the storage, retrieval, and adaptation of example cases. We focus on cases of coordinated attacking passes between robots in the presence of the defending opponent robots. We present the case representation explicitly distinguishing between controllable and uncontrollable indexing features, corresponding to the positions of the team members and opponent robots, respectively. We use the symmetric properties of the domain to automatically augment the case library. We introduce a retrieval technique that weights the similarity of a situation in terms of the continuous ball positional features, the uncontrollable features, and the cost of moving the robots from the current situation to match the case controllable features. The case adaptation includes a best match between the positions of the robots in the past case and in the new situation. The robots are assigned an adapted position to which they move to maximize the match to the retrieved case. Case retrieval and reuse are achieved within the distributed team of robots through communication and sharing of own internal states and actions. We evaluate our approach, both in simulation and with real robots, in laboratory scenarios with two attacking robots versus two defending robots as well as versus a defender and a goalie. We show that we achieve the desired coordinated passing behavior, and also outperform a reactive action selection approach. In order for a robot to perform an apparently simple task, such as actuating a ball towards a goal point, the robot needs multiple capabilities, including object detection, perception of the environment, building of an internal world model, making decisions when planning the task, navigation while avoiding obstacles, execution of planned actions, and recovering from failure. The complexity of each individual ability, and therefore the overall robot's behavior design, is related to the complexity of the environment where the robot carries out the task: the higher the complexity of the environment, the more challenging the robot's behavior design. Robot soccer is a particularly complex environment, particularly due to its dynamic nature resulting from the presence of multiple teammate and opponent robots.. In general in multi-robot domains, and robot soccer in particular, collaboration is desired so that the group of robots work together to achieve a common goal. It is not only important to have the agents collaborate, but also to do it in a coordinated manner so that the task can be organized to obtain effective results. A wide variety of methods has been investigated to address multiagent coordination, including task division with role assignment (Stone and Veloso, 1999; Vail and Veloso, 2003; Gerkey and Mataric, 2004) , and establishment of mutual beliefs of the intentions to act (Grosz and Kraus, 1996) . Communication among agents underlies these approaches for collaboration.. In this work, we are particularly interested in the action selection and coordination for joint multi-robot tasks, motivated by a prototype environment of robot soccer. We have successfully applied Case-Based Reasoning (CBR) techniques to model the action selection of a team of robots within the robot soccer domain (Ros et al., 2006; . However, our previous approach did not address the dynamic intentional aspect of the environment, in particular, in robot soccer, the presence of adversaries. Many efforts aim at modeling the opponents (Wendler and Bach, 2004; Ahmadi et al., 2003; Steffens, 2004; Huang et al., 2003; Miene et al., 2004; Marling et al., 2003) , in particular when the perception is centralized (Riley and Veloso, 2002) . Instead, we address here a robot soccer framework in which the robots are fully distributed, without global perception nor global control, and can communicate.. We follow a CBR approach where cases are recorded and model the state of the world at a given time and prescribe a successful action (Kolodner, 1993; Lopez de Mantaras et al., 2006; Riesbeck and Schank, 1989; Veloso, 1994) . A case can be seen as a recipe that describes the state of the environment (problem description) and the actions to perform in that state (solution description). Given a new situation, the most similar past case is retrieved and its solution is reused after some adaptation process to match the current situation. We model the case solution as a set of sequences of actions, which indicate what actions each robot should perform (Veloso, 1994) . Our case-based approach is novel in the sense that our cases represent a multi-robot situation where the robots are distributed in terms of perception, reasoning, and action, and can communicate. Our case-based retrieval and reuse phases are therefore based on messages exchanged among the robots about their internal states, in terms of beliefs and intentions.. Our case representation ensures that the solution description in the cases indicates the actions the robots should perform; that the retrieval process allocates robots to actions; and finally, with the coordination mechanism, that the robots share their individual intentions to act. Our approach allows for the representation of challenging rich multi-robot actions, such as passes in our robot soccer domain, which require well synchronized positioning and actions.. Previously, our retrieval process consisted of obtaining the most similar case based on two measures: similarity between the problem to solve and a given case, and the cost of adapting the problem to that case. Opponents were static, and therefore, simply modeled as obstacles the robots had to avoid . We extend the retrieval algorithm to include an applicability measure that determines if the reuse of a given case is feasible or not, as a function of the opponent and despite its static similarity degree. Furthermore we view the position of the opponents and the ball in a case description as uncontrollable features, while the positions of the teammate robots are viewed as controllable features, in the sense that robots in the current situation can move to match the positions in the past cases.. In terms of the reuse step, before the robots start executing the assigned actions according to the case, they pre-position themselves in the initial positions of the retrieved case. Instead of following an independent positioning strategy , we propose an alternative positioning strategy that reduces the time spent on satisfying the initial conditions. We show empirical results that confirm the effectiveness of our collaborative approach compared to an individualistic approach. The scenarios include two moving opponents and two attacking robots.. The article is organized as follows. Section 2 presents our particular robot soccer domain and describes the case representation. Section 3 introduces the case retrieval in terms of matching the situation. Section 4 focuses on the multi-robot architecture for case retrieval and case reuse in our distributed multi-robot system. Section 5 presents the empirical evaluation. We review related work within the robot soccer domain in Section 6. Finally, Section 7 draws the conclusions and discusses future work. We can classify the work reviewed based on (i ) the problem the work is addressed and (ii ) the league where it is applied. The problems we have focused our attention on are: action selection, opponent modeling or state recognition (team formation recognition), role assignment, positioning (locations on the field where the robots should move), and skills (learning low level actions such as kick the ball, walk, etc.). The available leagues are: Simulation League, Small-Size League, Middle-Size League and Four-Legged League.. Thus, within this classification, our work covers the action selection, role assignment, and positioning problem domains within the Four-Legged League.. As we describe through this paper, the CBR system retrieves and adapts a case specifying the actions the robots must perform (action selection problem). Moreover, it indicates which actions each robot should perform and their initial positions. Hence, the role assignment and positioning problems are solved through the case reuse. Although two works have been presented in the action selection domain within this league (Karol et al., 2004; Ahmadi and Stone, 2008) , we introduce a complete framework addressing the decision-making of a multi-robot system, where a set of actions for each robot is selected (not only two possible actions as in (Ahmadi and Stone, 2008) ), and the subsequent execution of these actions (the work presented in (Karol et al., 2004 ) is preliminary and only refers to the first stage, the decision-making). Furthermore, we include the cooperative aspect in the task execution through explicit passes between robots, not only to enhance collaboration among robots, but also to face the adversarial component of the domain. This latter component has only been explicitly addressed in (Ahmadi and Stone, 2008) and (Kim et al., 1997) .. Finally, to evaluate our approach, we have performed experiments consisting of two vs. two scenarios both in simulation and with real robots. Two attackers collaboratively play against two non-random opponents and making use of explicit passes, when possible. It is important to remark that experimenting with real robots with more than one robot per team is not common due to the complexity of the task. Indeed, the experiments in the reviewed literature dealing with real robots were performed with the following limitations: (i ) with a single robot (Kleiner et al., 2003; Park et al., 2006) ; (ii ) without opponents (Lee et al., 2005) ; (iii ) in 1 vs. 1 scenarios (Wu and Lee, 2004; Kim et al., 1997) ; (iv ) against one static opponent (Kleiner et al., 2003); or (v ) against two random opponents (opponents just moved from one position to another randomly) (Ahmadi and Stone, 2008) . In this work we have presented a Case-Based Reasoning approach for action selection and coordination in joint multi-robot tasks in domains where imprecision, uncertainty, unpredictability and real-time response are present. Robot soccer is an example of such domains, and it is the one in which this work is applied. Thus, we must also consider one more ingredient when attacking the problem: the adversarial component.. We have successfully extended our previous work with new mechanisms to handle this latter component. More precisely, we have introduced the applicability measure, which is composed of two functions: the free path function, to guarantee that there are no opponents intercepting the ball's path; and the opponent similarity, to evaluate the relevance of the existing opponents on the field. Given a new problem to solve, the retrieval process first evaluates all cases filtering those that are not useful to solve the current problem. Next, the candidate cases are ranked based on a set of criteria. After a case is retrieved, the adaptation phase takes place. The robots move to a more appropriate position to directly reuse the case solution using a new positioning strategy to prevent the robots from being inactive too much time. Finally, the robots reuse its solution executing the sequences of actions specified in the case. To this end, a multi-robot architecture and a coordination mechanism have been presented.. To evaluate the performance of our approach, we have compared it with a reactive approach, both in simulation and with real robots. We have tested four scenarios where two attackers (either using the CBR or the reactive approach) played against two moving opponents, an important step forward with respect to our previous work. In simulation we have considered two different defending configurations: a defender and a goalie (DG), and two defenders (2D). We have selected the DG configuration in the real robots experiments mainly because the trials take a much shorter time. The results have shown that generally the CBR approach outperforms the reactive one both in simulation and with real robots and against both defending configurations, not only having more opportunities to score, but also reducing the number of balls out of the field, avoiding the opponents and moving the ball quicker towards the attacking goal. Moreover, because of the use of passes between teammates, the defenders playing against the CBR attackers had less opportunities to steal or touch the ball than when playing against the reactive attackers. This clearly shows that cooperation (having passes in the soccer domain) is a good strategy in this type of adversarial domains. To achieve cooperation it is essential to know who does what. Case-based reasoning allows us to easily include this information in the case description, so all agents are aware of the actions they have to perform. Jointly with a coordination mechanism, the multi-agent system achieves a cooperative behavior.. As future work, we are interested in combining both deliberative and reactive approaches since as we could observe from the experiments, in some situations, acting reactively fast is also important. Hence, by combining both approaches, we can benefit from their advantages.. With respect to the CBR system, we believe that the time and score of the game should be taken into account in the decision making process. We could make use of these features to model the team strategy, i.e. more conservative or more aggressive. Moreover, we could combine them with the fuzzy representation of the applicability functions. Thus, the trajectories and opponents' scopes could be dynamically enlarged or shrunk based on the strategy the team should play with. The team strategy could also be modified during a game by changing the sorting criteria of the retrieval process.. We could also consider including a more sophisticated negotiation process in the multi-robot architecture for deciding which case to reuse. Instead of having a single robot selecting the retrieved case, all retrievers could propose their solutions, and by means of negotiation techniques, agree on which is the most suitable one given the state of the game.. Finally, including automatic case acquisition and revision in the current approach is fundamental if we expect the system to automatically improve its performance as well as to adapt and to learn new situations encountered through time. However, because of the complexity of the domain, the automatic acquisition of cases is very challenging. To this end at least the following questions should be analyzed: (i ) when to consider that a potential case has started ?, i.e. identify the initial situation; (ii ) when does it finish?, i.e. which are the actions to store as part of the solution of a case; and, (iii ) how to evaluate the outcome of the performed actions to determine if the case is useful or not for achieving the robot's goals (i.e. positive or negative feedback) ? This is known as the credit assignment problem. In general, the results obtained both in simulation and with the real robots, confirm that the Case-Based Reasoning approach is better than the reactive approach, not only on placing a higher percentage of balls close to the opponent's goal, but also achieving a lower percentage of out balls. More precisely, the results obtained in the third scenario with the real robots confirms the simulation results. In the fourth scenario, once again the average of balls out is higher for the reactive approach, which confirms that the defense had more chances to steal the ball.. However, in the last scenario the reactive approach achieved a higher percentage of balls to goal compared to the CBR approach. We must point out that comparing the ideal executions of both approaches (Figures 15 and 16 ) we can easily observe that the reactive approach is more efficient in moving faster the ball towards the attacking goal, i.e. with two kicks the robots can score. On the contrary, the attackers with the CBR approach need at least three kicks to complete the same task. Hence, the chances for the goalie to move towards a better position to block the ball also increase, as confirmed in the percentage of blocked balls by the goalie (60% for the CBR approach vs. 33% for the reactive approach). These results also support the finding that the CBR approach had more chances to get closer to the attacking goal, i.e. succeeded in avoiding the defender in the first step (Figure 15b ). When playing with the reactive approach, the attacker's first action was blocked most of the times (Figure 16a ). Otherwise, as the ideal sequences shows, the attackers using the reactive approach would have had even more opportunities to score, considerably increasing the percentage of to goal balls.. We must analyze one last aspect. Comparing the results obtained for the fourth scenario in simulation and with the real robots, we can see that in simulation, the CBR approach outperformed the reactive approach. However, we must always have in mind that the high uncertainty in the perception for the robots is not considered in the simulated environment. If we take a look again at Figure 15c , we can observe that after reusing the first case, the ball stops close to the penalty area. Although the goalie is not allowed to leave its home region (the penalty area), he may believe that the ball is within its home region due to the high uncertainty in the robot's world model and, therefore, may try to grab it anyway. In the simulated environment this situation occurs, unless the ball is actually within the penalty area. Hence, while in the real world the trial could end with the goalie clearing the ball, i.e. block ball in the statistics, in simulation the attacker would probably try to score, i.e. a to goal ball.. Regarding the defenders' performance, we have confirmed that using the CBR approach is a better strategy to avoid the defenders stealing the ball because of the explicit passes between teammates. The reactive strategy almost doubles the chances for the defender to steal the ball compared to the CBR approach, even playing shorter periods of time.. With respect to the attackers, when using the CBR approach the number of times that the default (reactive) behavior was triggered was not significant and, therefore, we do not include these measures in the results. These situations only occurred when the costs of all the available cases exceeded the cost threshold, but not because there were no cases defined. However, when the attackers moved towards the ball using the reactive approach, they reduced their distances to the ball and, consequently, some of the available cases that had previously been filtered out became applicable.. Finally, and to conclude, the fourth scenario verifies that in this kind of domains (high uncertainty, highly dynamic and real time response requirements) it is useful to have a reactive strategy rather than a deliberative strategy to solve critical situations. When the players are in front of the goal, there is no need of reasoning about interesting plays. Instead, trying to score as fast as possible is the best strategy. Thus, acting fast is crucial. However, a deliberative strategy that includes collaboration outperforms a reactive one for the remaining situations, i.e. in the middle, back, sides and corners of the field where the opponents have more opportunities to steal the ball. We first briefly describe the different RoboCup Soccer leagues because they are subsequently referred to in this section that reviews related work. Then, we focus on CBR techniques that have been applied to the RoboCup domain and mostly related to the action selection problem. Next, we take a look at works where other approaches besides CBR have been previously used in the same context. Finally, we situate our work within the reviewed literature summarizing the main features of our CBR approach. Two types of experiments were performed: simulated experiments and real robots experiments in two vs. two scenarios. We next roughly describe the case base used during the experimentation, the behaviors of the robots, and the evaluation measures.

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Power-rate synchronization of coupled genetic oscillators with unbounded time-varying delay Abstract In this paper, a new synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay is investigated. The dynamical system under consideration consists of an array of linearly coupled identical genetic oscillators with each oscillators having unbounded time-delays. A new concept called power-rate synchronization, which is different from both the asymptotical synchronization and the exponential synchronization, is put forward to facilitate handling the unbounded time-varying delays. By using a combination of the Lyapunov functional method, matrix inequality techniques and properties of Kronecker product, we derive several sufficient conditions that ensure the coupled genetic oscillators to be power-rate synchronized. The criteria obtained in this paper are in the form of matrix inequalities.Illustrative example is presented to show the effectiveness of the obtained results. It is well known that gene expression is a complex process regulated at several stages in the synthesis of proteins (Lewin 1999) . DNA microarray technology has made it possible to measure gene expression levels on a genomic scale, and has therefore been widely applied to gene transcription analysis (Fraser et al. 2004) . With the available huge amount of microarray data, researchers have been trying to understand how biological activities are governed by the connectivity of genes and proteins, which is described in terms of genetic regulatory networks (GRNs). A gene network consists of a group of genes that interact among themselves in order to synthesize certain products, i.e., proteins. The types and amount of proteins produced by a gene network have a fundamental effect on the development of the gene network itself, and on the biological systems with which the network interacts. Since genetic networks are biochemical dynamical systems, mathematical modelling of genetic networks as dynamical system models provides a powerful tool for studying gene regulation processes in living organisms . The study of genetic regulatory networks have become an attractive area and received great attention over past decade (Li et al. 2006a, b; Ren and Cao 2008; Wang et al. 2010; Xiao et al. 2014; Cao and Ren 2008; Ren et al. 2015) .. Synchronization is an important dynamical behavior that has been intensively investigated in the last decade (Wu and Chua 1995; Li and Chen 2004; Lu and Chen 2004; Wang and Cao 2006; Huang et al. 2008; Qiu and Cao 2009) . There are many benefits of having synchronization in some applications, such as secure communication, human heartbeat regulation, chemical reaction, power systems protection, ecological systems and so on. Recently, it has been discovered that the phase synchronization can also exist in many biologically plausible systems and such synchronization is a main mechanism to fulfil cellular communication. Therefore, it is not surprising that the synchronization problem of coupled genetic oscillators has begun to attract a great deal of research attention due to its potential applications in GRNs. For example, in McMillen et al. (2002) , Kuznetsov et al. (2004) , and Wang and Chen (2005) , the synchronization was studied in biological networks of identical genetic oscillators and, in Garcia-Ojalvo et al. (2004) , Gonze et al. (2005) and Li et al. (2006b) , the synchronization was investigated for coupled nonidentical genetic oscillators. In Li et al. (2007) , based on a system biology approach, the authors provided a general theoretical result on the synchronization of genetic oscillators with stochastic perturbations.. It is now well known that the slow processes of transcription, translation, and translocation or the finite switching speed of amplifiers will inevitably cause time delays, which should be taken into account in the biological systems or artificial genetic networks in order to have more accurate models . The effects of transcriptional delays should be assessed in the dynamics of genetic networks whose time scales are short and transcription is regulated by feedback . The time-delay can be of a discrete nature, which assumes each macromolecule takes the same length of time to translocate from its place of synthesis to the location where it exerts an effect. The time-delay can also be time-varying where the delay changes as time changes. In the context of GRNs, both discrete time-delays and time-varying delays have recently been considered in Cao and Ren (2008) , Monk (2003) , Grammaticos et al. (2006) and .. It should be pointed out that, in most existing literature considering delays in GRNs, only the stability has been considered, and the important synchronization behavior has not received much attention yet. Furthermore, almost all relevant research works have assumed that the time delays involved in GRNs are bounded. This is, unfortunately, not always the case in practice. If movement of mRNA from a transcription site to translation sites is an active process with a significant range of transport times for individual molecules, unbounded delay would be the proper modeling framework. For example, it has been shown in Smolen et al. (2000) that the time-delay can occur in a distributed hence unbounded way. Therefore, it is of great interest in both theory and applications to investigate the synchronization problem of the genetic regulatory networks with unbounded delays. To the best of the authors' knowledge, up to now, little effort has been made towards such a challenging problem, which motivates the present study.. In this paper, we investigate a new synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay. The dynamical system under consideration consists of an array of linearly coupled identical genetic oscillators with each oscillators having unbounded time-delays. It is worth noting that, in Chen and Wang (2007) , pioneering research has been carried out for the power-rate stability of dynamical systems with unbounded time-varying delays and neat results have been reported. Motivated by the excellent work of Chen and Wang (2007) , we put forward a new concept called power-rate synchronization in order to cater the unbounded time-delays in GRNs. Notice that the power-rate synchronization is significantly different from the traditional asymptotical synchronization and the exponential synchronization. By resorting to the Lyapunov functional method and properties of Kronecker product, we develop a new matrix inequality approach to derive several sufficient conditions under which the coupled genetic oscillators are synchronized in the power-rate sense. Illustrative example is presented to show the effectiveness of the obtained results.. The rest of this paper is organized as follows. In ''Problem formulation and preliminaries'' section, problem formulation and preliminaries are given. In ''Power-rate synchronization'' section, sufficient criteria are derived for the power-rate synchronization of coupled genetic oscillators with unbounded time-varying delay. In ''Illustrative example'' section, an example is given to show the effectiveness of the proposed results. Finally, conclusions are drawn in the last section. In this paper, we have investigated the synchronization problem on the genetic oscillators with unbounded timevarying delays. Some criteria are given to ensure the power-rate synchronization of the coupled genetic oscillators. The obtained conditions are derived in terms of matrix inequalities. Synchronization with some power convergence rate is first proposed on genetic oscillators. From Chen and Wang (2007) and the concept of synchronization, we know that there indeed exist some dynamical systems which are power-rate synchronized but not exponentially synchronized. We have employed a simulation example to illustrate the effectiveness of the proposed results.

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


Some lattice attacks on DSA and ECDSA Abstract In this paper, using the LLL reduction method and computing the integral points of two classes of conics, we develop attacks on DSA and ECDSA in case where the secret and the ephemeral key of a signed message or theirs modular inverses are sufficiently small and in case where the ephemeral keys or theirs modular inverses of two signed messages are sufficiently small. In this paper, combining lattice reduction techniques with algorithms for computing the integral solutions of the equations bx + cy + dx y = 0 and b + cy + dx y = 0, we develop some rigorous attacks on DSA and ECDSA. If one signature is available having secret key a and ephemeral key k and the numbers in at least one of the sets {a, k −1 }, {k, a −1 }, {a −1 , k −1 } satisfy a certain inequality, then a can be computed in practice. The same happens, if two signatures are available with ephemeral keys k 1 , k 2 and the numbers in at least one of the sets {k 1 , k The above attacks can also be applied on other schemes where the secret and the ephemeral keys are solutions of a modular bivariate linear equation as in DSA or of a modular bivariate equation of second degree. For instance, such schemes are Schnorr' signature, Heyst-Pedersen signature, etc [14, 19] .. Our attacks on DSA require O(q (log p) 3 ) bit operations and on ECDSA need O(q ) bit operations and O(q ) elliptic curve group operations. Furthermore, these two attacks need the factorization of one or two integers < q. Note that these numbers are random and not chosen in such a way that their factorization is difficult. In practice the size of q is 160 and so, the factorization of the above numbers is quite easy with the current algorithms. Therefore, our attacks can be quite efficient.

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Monochromatic Clique Decompositions of Graphs Let G be a graph whose edges are colored with k colors, and H = (H 1 , . . . , H k ) be a k-tuple of graphs. A monochromatic Hdecomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic copy of H i in color i, for some 1 ≤ i ≤ k. Let φ k (n, H) be the smallest number φ, such that, for every order-n graph and every k-edge-coloring, there is a monochromatic H-decomposition with at most φ elements. Extending the previous results of Liu and Sousa [Monochromatic K r -decompositions of graphs, J Graph Theory 76 (2014), 89-100], we solve this problem when each graph in H is a clique and n ≥ n 0 (H) is sufficiently large. All graphs in this article are finite, undirected, and simple. For standard graph-theoretic terminology the reader is referred to [3] .. Given two graphs G and H, an H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a subgraph isomorphic to H. Let φ(G, H ) be the smallest possible number of parts in an H-decomposition of G. It is easy to see that, if H is nonempty, we have φ(G, H ) = e(G) − ν H (G)(e(H ) − 1), where ν H (G) is the maximum number of pairwise edge-disjoint copies of H that can be packed into G. Dor and Tarsi [4] showed that if H has a component with at least three edges then it is NP-complete to determine if a graph G admits a partition into copies of H. Thus, it is NP-hard to compute the function φ(G, H ) for such H. Nonetheless, many exact results were proved about the extremal function. which is the smallest number such that any graph G of order n admits an H-decomposition with at most φ(n, H ) elements.. This function was first studied, in 1966, by Erdős et al. [6] , who proved that φ(n, K 3 ) = t 2 (n), where K s denotes the complete graph (clique) of order s, and t r−1 (n) denotes the number of edges in the Turán graph T r−1 (n), which is the unique (r − 1)-partite graph on n vertices that has the maximum number of edges. A decade later, Bollobás [2] proved that φ(n, K r ) = t r−1 (n), for all n ≥ r ≥ 3.. Recently, Pikhurko and Sousa [13] studied φ(n, H ) for arbitrary graphs H. Their result is the following. Theorem 1.1 [13] . Let H be any fixed graph of chromatic number r ≥ 3. Then, φ(n, H ) = t r−1 (n) + o(n 2 ).. Let ex(n, H ) denote the maximum number of edges in a graph on n vertices not containing H as a subgraph. The result of Turán [20] states that T r−1 (n) is the unique extremal graph for ex(n, K r ). The function ex(n, H ) is usually called the Turán function for H. Pikhurko and Sousa [13] also made the following conjecture. Conjecture 1.2 [13] . For any graph H of chromatic number r ≥ 3, there exists n 0 = n 0 (H ) such that φ(n, H ) = ex(n, H ) for all n ≥ n 0 .. A graph H is edge-critical if there exists an edge e ∈ E(H ) such that χ (H ) > χ(H − e), where χ (H ) denotes the chromatic number of H. For r ≥ 4, a clique-extension of order r is a connected graph that consists of a K r−1 plus another vertex, say v, adjacent to at most r − 2 vertices of K r−1 . Conjecture 1.2 has been verified by Sousa for some edge-critical graphs, namely, clique-extensions of order r ≥ 4 (n ≥ r) [18] and the cycles of length 5 (n ≥ 6) and 7 (n ≥ 10) [17, 19] . Later,Özkahya and Person [12] verified the conjecture for all edge-critical graphs with chromatic number r ≥ 3. Their result is the following. and Sousa in Theorem 1.1. In fact, they proved that the error term o(n 2 ) can be replaced by O(n 2−α ) for some α > 0. Furthermore, they also showed that this error term has the correct order of magnitude. Their result is indeed an extension of Theorem 1.3 since the error term O(n 2−α ) that they obtained vanishes for every edge-critical graph H.. Motivated by the recent work about H-decompositions of graphs, a natural problem to consider is the Ramsey (or colored) version of this problem. More precisely, let G be a graph on n vertices whose edges are colored with k colors, for some k ≥ 2 and let H = (H 1 , . . . , H k ) be a k-tuple of fixed graphs, where repetition is allowed. A monochromatic H-decomposition of G is a partition of its edge set such that each part is either a single edge, or forms a monochromatic copy of H i in color i, for some 1 ≤ i ≤ k. Let φ k (G, H) be the smallest number, such that, for any k-edge-coloring of G, there exists a monochromatic H-decomposition of G with at most φ k (G, H) elements. Our goal is to study the function. which is the smallest number φ such that, any k-edge-colored graph of order n admits a monochromatic H-decomposition with at most φ elements. In the case when. The function φ k (n, K r ), for k ≥ 2 and r ≥ 3, has been studied by Liu and Sousa [11] , who obtained results involving the Ramsey numbers and the Turán numbers. Recall that for k ≥ 2 and integers r 1 , . . . , r k ≥ 3, the Ramsey number for K r 1 , . . . , K r k , denoted by R(r 1 , . . . , r k ), is the smallest value of s, such that, for every k-edge-coloring of K s , there exists a monochromatic K r i in color i, for some 1 ≤ i ≤ k. For the case when r 1 = · · · = r k = r, for some r ≥ 3, we simply write R k (r) = R(r 1 , . . . , r k ). Since R(r 1 , . . . , r k ) does not change under any permutation of r 1 , . . . , r k , without loss of generality, we assume throughout that 3 ≤ r 1 ≤ · · · ≤ r k . The Ramsey numbers are notoriously difficult to calculate, even though, it is known that their values are finite [15] . To this date, the values of R(3, r 2 ) have been determined exactly only for 3 ≤ r 2 ≤ 9, and these are shown in the following table [14] . The remaining Ramsey numbers that are known exactly are R(4, 4) = 18, R(4, 5) = 25, and R(3, 3, 3) = 17. The gap between the lower bound and the upper bound for other Ramsey numbers is generally quite large.. For the case R(3, 3) = 6, it is easy to see that the only 2-edge-coloring of K 5 not containing a monochromatic K 3 is the one where each color induces a cycle of length 5. From this 2-edge-coloring, observe that we may take a "blow-up" to obtain a 2edge-coloring of the Turán graph T 5 (n), and easily deduce that φ 2 (n, K 3 ) ≥ t 5 (n). See Figure 1 .. This example was the motivation for Liu and Sousa [11] to study K r -monochromatic decompositions of graphs, for r ≥ 3 and k ≥ 2. They have recently proved the following result. [11] .

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From Algorithms to (Sub-)Symbolic Inferences in Multi-Agent Systems Extending metaphorically the Moisilean idea of "nuanced-reasoning logic" and adapting it to the e-world age of Information Technology (IT), the paper aims at showing that new logics, already useful in modern software engineering, become necessary mainly for Multi-Agent Systems (MAS), despite obvious adversities. The first sections are typical for a position paper, defending such logics from an anthropocentric perspective. Through this sieve, Section 4 outlines the features asked for by the paradigm of computing as intelligent interaction, based on "nuances of nuanced-reasoning", that should be reflected by agent logics. To keep the approach credible, Section 5 illustrates how quantifiable synergy can be reached -even in advanced challenging domains, such as stigmergic coordination -by injecting symbolic reasoning in systems based on sub-symbolic "emergent synthesis". Since for future work too the preferred logics are doxastic, the conclusions could be structured in line with the well-known agent architecture: Beliefs, Desires, Intentions.

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