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Abstract-Recently, fully connected recurrent neural networks have been proven to be computationally rich-at least as powerful as Turing machines. This work focuses on another network which is popular in control applications and has been found to be very effective at learning a variety of problems. These networks are based upon Nonlinear AutoRegressive models with eXogenous Inputs (NARX models), and are therefore called NARX networks. As opposed to other recurrent networks, NARX networks have a limited feedback which comes only from the output neuron rather than from hidden states. They are formalized by y(t) = 9(u(t 0 n u );111; u(t 0 1);u(t);y(t 0 n y );111 ; y(t 0 1)) where u(t) and y(t) represent input and output of the network at time t, nu and ny are the input and output order, and the function 9 is the mapping performed by a Multilayer Perceptron.We constructively prove that the NARX networks with a finite number of parameters are computationally as strong as fully connected recurrent networks and thus Turing machines. We conclude that in theory one can use the NARX models, rather than conventional recurrent networks without any computational loss even though their feedback is limited. Furthermore, these results raise the issue of what amount of feedback or recurrence is necessary for any network to be Turing equivalent and what restrictions on feedback limit computational power.

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Abstract. We describe a software self-checking mechanism designed to improve the tamper resistance of large programs. The mechanism consists of a number of testers that redundantly test for changes in the executable code as it is running and report modifications. The mechanism is built to be compatible with copy-specific static watermarking and other tamper-resistance techniques. The mechanism includes several innovations to make it stealthy and more robust.

It has previously been shown that gradient-descent learning algorithms for recurrent neural networks can perform poorly on tasks that involve long-term dependencies, i.e. those problems for which the desired output depends on inputs presented at times far in the past. We show that the long-term dependencies problem is lessened for a class of architectures called nonlinear autoregressive models with exogenous (NARX) recurrent neural networks, which have powerful representational capabilities. We have previously reported that gradient descent learning can be more effective in NARX networks than in recurrent neural network architectures that have "hidden states" on problems including grammatical inference and nonlinear system identification. Typically, the network converges much faster and generalizes better than other networks. The results in this paper are consistent with this phenomenon. We present some experimental results which show that NARX networks can often retain information for two to three times as long as conventional recurrent neural networks. We show that although NARX networks do not circumvent the problem of long-term dependencies, they can greatly improve performance on long-term dependency problems. We also describe in detail some of the assumptions regarding what it means to latch information robustly and suggest possible ways to loosen these assumptions.