DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

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focusing on the computational complexity of individual orbits. By not requiring a Hamiltonian, computational mechanics can be applied in a wide range of contexts, including those where an energy function for the system may not manifest like for the supply chains. Notions of Complexity, Emergence and Self-Organization have also been formalized and quantified in terms of various information measures (Shalizi 2000). Given a time series, the (unknowable) exact states of an observed system are translated into sequence of symbols via a measurement channel (Crutchfield 1992). Two histories (i.e., two series of past data) carry equivalent information if they lead to the same (conditional) probability distribution in the future (i.e., if it makes no difference whether one or the other data-series is observed). Under these circumstances, i.e., the effects of the two series being indistinguishable, they can be lumped together. This procedure identifies causal states, and also identifies the structure of connections or succession in causal states, and creates what is known as an “epsilonmachine”. The ε -machines form a special class of Deterministic Finite State Automata (DFSA) with transitions labeled with conditional probabilities and hence can also be viewed as Markov chains. However, finite-memory machines likeε -machines may fail to admit a finite size model implying that the number of casual states could turn out to be infinite. In this case, a more powerful model than DFSA needs to be used. One proceeds by trying to use the next most powerful model in the hierarchy of machines known as the casual hierarchy (Crutchfield 1994), in analogy with the Chomsky hierarchy of formal languages. While “ε -machine reconstruction” refers to the process of constructing the machine given an assumed model class, “hierarchical machine reconstruction” describes a process of innovation to create a new model class. It detects regularities in a series of increasingly accurate models. The inductive jump to a higher computational level occurs by taking those regularities as the new representation. ε -machines reflect a balanced utilization of deterministic and random information processing and this is discovered automatically during ε -machine reconstruction. These machines are unique and optimal in the sense that they have maximal predictive power and minimum model size (hence satisfy Principle of Occam Razor i.e. causes should not be multiplied beyond necessity). ε -machine provides a minimal description of the pattern or regularities in a system in the sense that the pattern is the algebraic structure determined by the causal states and their transitions. ε -machines are also minimally stochastic. Hence computational mechanics acts as a method for automatic pattern discovery. ε -machine is the organization of the process, or at least of the part of it which is relevant to our measurements. The ε -machine being a model of the observed time series from a system can be used to define and calculate macroscopic or global properties that reflect the characteristic average information processing capabilities of the system. Some of these include Entropy rate, Excess entropy and Statistical Complexity (Feldman and Crutchfield 1998) and (Crutchfield and Feldman 2001). The entropy density indicates how predictable the system is. Excess entropy on other hand provides a measure of the apparent memory stored in a spatial configuration and represents how hard it is the prediction.ε -machine reconstruction leads to a natural measure of the statistical complexity of a process, namely the amount of information needed to specify the state of the ε -machine i.e. the Shannon Entropy. Statistical Complexity is distinct and dual from information theoretic entropies and dimension (Crutchfield and Young 1989). The existence of chaos shows that there is rich variety of unpredictability that spans the two extremes: periodic and random behavior. This behavior between two extremes while of intermediate information content is more complex in that the most concise description (modeling) is an amalgam of regular and stochastic processes. Information theoretic description of this spectrum in terms of dynamical entropies measures raw diversity of temporal patterns. The dynamical entropies however do not measure directly the computational effort required in modeling the complex behavior, which is what statistical complexity captures.
Computational mechanics sets limits on how well processes can be predicted and shows how at least in principle, those limits can be attained. ε -machines are what any prediction method would build, if only they could. Similar to ε -machine reconstruction, techniques exists which can be used to discover casual architecture in memory less transducers, transducers with memory and spatially extended systems (Shalizi 2000). Computational mechanics can be used for modeling and prediction in supply chains in the following way: • In systems like supply chain, it is difficult to define analogs of various thermodynamic quantities like energy, temperature, pressure etc as we can do for physical systems. Each component in the network has a cognition, which is absent in physical systems; say a molecule of a gas. Due to such difficulties statistical mechanics cannot be applied directly to build prediction models for supply chains. As discussed previously by not requiring a Hamiltonian (the energy like function), computational mechanics is still applicable in case of supply chains. • ε -machines can be built to discover patterns in behavior of various quantities in supply chains like the inventory levels, demand fluctuations, etc. • ε -machines can be used for prediction through a process known as “synchronization” (Crutchfield and Feldman 2003). • ε -machines can be used to calculate various global properties like entropy rate, excess entropy and statistical complexity, that reflect how the system stores and processes information. The significance of these quantities has been discussed earlier. • We can also quantify notions of Complexity, Emergence and Self-Organization in terms of various information measures derived from ε -machines. By evaluating such quantities we can compare complexity of different supply chains and quantify the extent to which the network is showing emergence. We can also infer when a supply chain is undergoing self-organization and to what extent. Such quantification can help us to compare precisely what policies or cognitive capabilities possessed by individual agents can lead to different degrees of emergence and self-organization. Hence we can decide to what extent we desire to enforce the control and to what extent we want to let the network emerge. 7. Network Dynamics The ubiquity of networks in the social, biological and physical sciences and in technology leads naturally to an important set of common problems, which are being currently studied under the rubric of “Network Dynamics” (Strogatz 2001). Structure always affects function and it is important to consider dynamical and structural complexity together in the study of networks. For instance, the topology of social networks affects the spread of information and disease, and the topology of the power grid affects the robustness and stability of power transmission. The different problem areas in network dynamics are discussed below. One area of research in this field has been primarily concerned with the dynamical complexity in regular networks without regard to other network topologies. While the collective behavior depends on the details of the network, some generalization can still be drawn (Strogatz 2001). For instance, if the dynamical system at each node has stable fixed points and no other attractor, the network tends to lock into a static fixed pattern. If the nodes have competing interactions, network may become frustrated and display enormous number of locally stable equilibria. In the intermediate case where each node has a stable limit cycle, synchronization and patterns like traveling waves can be observed. For non-identical oscillators temporal analogue of phase transition can be seen with the control parameter as the coupling coefficient. At the opposite extreme if each node has identical chaotic attractor, the network can synchronize their erratic fluctuations. For a wide range of network topologies, synchronized chaos requires that the coupling be neither too weak nor too strong; otherwise spatial instabilities are triggered. Related
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Ultra*Log PSU/IAI Final Report for
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Contents Contents .................
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Executive Summary Ultra*Log is a De
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2.3 Gnanasambandam, N., Lee, S., Ku
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6 Characterization and analysis of
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timizing simultaneously the link de
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where ∆ 1 (j) ≥ 0 and ∆ 2 (i,
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Table 2. GA Results Agent N A D max
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connected component, in which a pat
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Random Small-world Scale-free with
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Growth mechanisms Start with a smal
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Table 2. The proposed network’s c
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tems. Proceedings of the Second Joi
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Proceedings of the 1st Open Cougaar
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1 SITUATION IDENTIFICATION USING DY
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3 3.2 Behavior In SSC society an ag
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5 All the behavior parameters may n
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Estimating Global Stress Environmen
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chaotic deterministic time series.
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100% 1 95% 2 14 15 64% TAO 4 64% 62
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interactions as a dynamical system
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ehavior states under varying system
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Extensive testing and validation of
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5 Conclusions and Future Research T
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2 to function critically even under
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4 points into a corresponding inner
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6 stresses well. The Warehouse 1 ag
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Table 1: Notation Symbol Descriptio
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4 Conclusions and Future Work 4.1 C
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O, U Stresses Physical/Infrastructu
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Manuscript for IEEE Transactions on
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2 discuss problem domain and in Sec
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4 t Si t ∫ + ( ) i ) t When RA i
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6 S UB i ∑ = P LI / LI . (16) i p
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8 policies while underutilizing in
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architecture based on both the Grid
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to machines (network topology) and
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component is a member of one of the
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denotes the immediate predecessors
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limit of large number of tasks. If
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Min-min heuristic algorithm Step 1:
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We set up eight different experimen
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0. 4 8057 8237 0.5 6439 6635 0.6 53
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pp. 191-200. [3] I. Foster, C. Kess
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Technology, Cambridge, MA, 1995. [2
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Manuscript for IEEE TRANSACTIONS ON
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Page 176 and 177: Acknowledgements The work described
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Page 202 and 203: Min, H. and Zhou, G., 2002, Supply
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