The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

We explore the simulation and computational capabilities of hybrid and continuous dynamical systems. The continuous dynamical systems considered are ordinary differential equations (ODEs). For hybrid systems we concentrate on models that combine ODEs and discrete dynamics (e.g., finite automata). We review and compare four such models from the literature. Notions of simulation of a discrete dynamical system by a continuous one are developed. We show that hybrid systems whose equations can describe a precise binary timing pulse (exact clock) can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of R n . The simulations require continuous ODEs in R 2n with the exact clock as input. All four hybrid systems models studied here can implement exact clocks. We also prove that any discrete dynamical system in Z n can be simulated by continuous ODEs in R 2n+1 . We use this to show that smooth ODEs in R 3 can simulate arbitrary Turing machines, and henc...

Previous work on learning regular languages from exemplary training sequences showed that Long Short- Term Memory (LSTM) outperforms traditional recurrent neural networks (RNNs). Here we demonstrate LSTM's superior performance on context free language (CFL) benchmarks for recurrent neural networks (RNNs), and show that it works even better than previous hardwired or highly specialized architectures.

In [18] we have shown how to construct a 3-layered recurrent neural network that computes the fixed point of the meaning function TP of a given propositional logic program P, which corresponds to the computation of the semantics of P. In this article we consider the first order case. We define a notion of approximation for interpretations and prove that there exists a 3-layered feed forward neural network that approximates the calculation of TP for a given first order acyclic logic program P with an injective level mapping arbitrarily well. Extending the feed forward network by recurrent connections we obtain a recurrent neural network whose iteration approximates the fixed point of TP. This result is proven by taking advantage of the fact that for acyclic logic programs the function TP is a contraction mapping on a complete metric space defined by the interpretations of the program. Mapping this space to the metric space IR with Euclidean distance, a real valued function fP can be defined which corresponds to TP and is continuous as well as a contraction. Consequently it can be approximated by an appropriately chosen class of feed forward neural networks.

This paper shows that the weights of continuous-time feedback neural networks are uniquely identifiable from input/output measurements. Under very weak genericity assumptions, the following is true: Assume given two nets, whose neurons all have the same nonlinear activation function oe; if the two nets have equal behaviors as "black boxes" then necessarily they must have the same number of neurons and ---except at most for sign reversals at each node--- the same weights. Moreover, even if the activations are not a priori known to coincide, they are shown to be also essentially determined from the external measurements. Key words: Neural networks, identification from input/output data, control systems 1 Introduction Many recent papers have explored the computational and dynamical properties of systems of interconnected "neurons." For instance, Hopfield ([7]), Cowan ([4]), and Grossberg and his school (see e.g. [3]), have all studied devices that can be modelled by sets of nonlinear dif...

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) = \Psi i u(t \Gamma nu ); : : : ; u(t \Gamma 1); u(t); y(t \Gamma ny ); : : : ; y(t \Gamma 1) j ; 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 \Psi is the mapping performed by a Multilayer Perceptron. We constructively prove that the NARX networks with a finite number of parameters are computation...

It is often difficult to predict the optimal neural network size for a particular application. Constructive or destructive methods that add or subtract neurons, layers, connections, etc. might offer a solution to this problem. We prove that one method, Recurrent Cascade Correlation, due to its topology, has fundamental limitations in representation and thus in its learning capabilities. It cannot represent with monotone (i.e. sigmoid) and hard-threshold activation functions certain finite state automata. We give a "preliminary" approach on how to get around these limitations by devising a simple constructive training method that adds neurons during training while still preserving the powerful fully-recurrent structure. We illustrate this approach by simulations which learn many examples of regular grammars that the Recurrent Cascade Correlation method is unable to learn. 1 Introduction Choosing the architecture of a neural network for a particular problem usually requires some prior k...

The temporal distance between events conveys information essential for numerous sequential tasks such as motor control and rhythm detection. While Hidden Markov Models tend to ignore this information, recurrent neural networks (RNNs) can in principle learn to make use of it. We focus on Long Short-Term Memory (LSTM) because it has been shown to outperform other RNNs on tasks involving long time lags. We find that LSTM augmented by "peephole connections" from its internal cells to its multiplicative gates can learn the fine distinction between sequences of spikes spaced either 50 or 49 time steps apart without the help of any short training exemplars. Without external resets or teacher forcing, our LSTM variant also learns to generate stable streams of precisely timed spikes and other highly nonlinear periodic patterns. This makes LSTM a promising approach for tasks that require the accurate measurement or generation of time intervals.

95 CDC- Keywords: complexity, controllability, nonlinear Extended Summary for Invited Session entitled Computational Complexity Issues in Control 1. Introduction It is obvious that many control problems are in general easier to solve for linear systems than for arbitrary, not necessarily linear, ones. An interesting and worthy area of research deals with the attempt to make mathematically precise the increases in difficulty that may arise when passing to the nonlinear case. By obtaining such precise statements, one gains an understanding of which analysis and/or design problems may be expected to be intractable. For instance, even for apparently mildly nonlinear systems it becomes impossible to check if a state ever reaches the origin. More interestingly perhaps, one also can then explain in what sense some variants of problems are easier than others for nonlinear systems. An example of this later aspect is given by comparing the characterization of the accessibility property (being ...

Recurrent neural networks readily process, recognize and generate temporal sequences. By encoding grammatical strings as temporal sequences, recurrent neural networks can be trained to behave like deterministic sequential finite-state automata. Algorithms have been developed for extracting grammatical rules from trained networks. Using a simple method for inserting prior knowledge (or rules) into recurrent neural networks, we show that recurrent neural networks are able to perform rule revision. Rule revision is performed by comparing the inserted rules with the rules in the finite-state automata extracted from trained networks. The results from training a recurrent neural network to recognize a known non-trivial, randomly generated regular grammar show that not only do the networks preserve correct rules but that they are able to correct through training inserted rules which were initially incorrect. (By incorrect, we mean that the rules were not the ones in the randomly generated gra...