A Survey on Continuous Time Computations

TL;DR

This survey reviews theories of continuous time computation, highlighting their role in understanding the computational capabilities and complexities of continuous dynamical systems and analog models.

Contribution

It provides a comprehensive overview of existing models, summarizes key results, and references relevant literature in the field of continuous time computation.

Findings

Summarizes various models of continuous time computation

Highlights the computational power of continuous dynamical systems

Discusses the complexity and hardness results related to continuous models

Abstract

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature.

Figures

Equations

$$p\left(t,y,y^{\prime },...,y^{(n)}\right)=0,$$

$$C_i{u}_i^{\prime }(t)=\sum _j{W}_{i,j}{V}_j-u_i/R_i+I_i,$$

$$E=-\frac{1}{2}\sum _i\sum _j{W}_{i,j}{V}_i{V}_j+\sum _i\frac{1}{R_i}{\int }_0^{V_i}{\sigma }^{-1}(V)dV+\sum _i{I}_i{V}_i.$$

$$P_v(t)=0+\sum _u\sum _{s\in F_u,s<t}{w}_{u,v}(s){ϵ}_{u,v}(t-s).$$

$$y^{\prime }=f(y),y(t_0)=x$$

$$\varphi (0,x)=x$$

$$\varphi (t,\varphi (s,x))=\varphi (t+s,x).$$

$$y_{t+1}=f(y_t),y_0=x.$$

$$f_{n}=\left\{\begin{array}[]{lll}2^{-m}&\textrm{if}&\textrm{$\mathcal{M}$ stops in $m$ steps on input $n$}\\ 0&\textrm{if}&\textrm{$\mathcal{M}$ does not stop on input $n$}\\ \end{array}\right.$$

$$g(x)=f_{\lfloor x+1/2\rfloor }{e}^{-{\tan }^2\pi x},$$

$$\left\{\begin{array}[]{lll}y^{\prime}_{1}&=&g(y_{2})-1\\ y^{\prime}_{2}&=&0\\ \end{array}\right.$$

$$||y(T)-y_N^{\ast }||\le \frac{h}{\lambda }\left[\frac{R}{2}+\frac{\sigma }{h2}\right](e^T\lambda -1),$$

$$Probability(x_{i+1}\in B)={\int }_{q\in B}z(f(x_i,a_i),q)d\mu ,$$

Full text

1 Introduction

Continuous time systems arise as soon as one attempts to model systems that evolve over a continuous space with a continuous time. They can even emerge as natural descriptions of discrete time or space systems. Utilizing continuous time systems is a common approach in fields such as biology, physics or chemistry, when a huge population of agents (molecules, individuals, …) is abstracted into real quantities such as proportions or thermodynamic data [Hirsch et al., 2003], [Murray, 2002].

There are several approaches that have led to theories on continuous time computations. We will explore in greater depth two primary approaches. One, which we call inspired by continuous time analog machines, has its roots in models of natural or artificial analog machinery. The other, we refer to as inspired by continuous time system theories, is broader in scope. It comes from research on continuous time systems theory from a computational perspective. Hybrid systems and automata theory, for example, are two sources.

A wide range of problems related to theories of continuous time computations are encompassed by these two approaches. They originate in fields as diverse as verification (see e.g. [Asarin et al., 1995]), control theory (see e.g. [Branicky, 1995b]), VLSI design (see e.g. [Mills, 1995], [Mills et al., 2005]), neural networks (see for example [Orponen, 1997]) and recursion theory on the reals (see e.g. [Moore, 1996]).

At its beginning, continuous time computation theory was mainly concerned with analog machines. Determining which systems can actually be considered as computational models is a very intriguing question. This relates to the philosophical discussion about what is a programmable machine, which is beyond the scope of this chapter. Nonetheless, there are some early examples of built analog devices that are generally accepted as programmable machines. They include Bush’s landmark 1931 Differential Analyzer [Bush, 1931], as well as Bill Phillips’ Finance Phalograph, Hermann’s 1814 Planimeter, Pascal’s 1642 Pascaline, or even the 87 b.c. Antikythera mechanism: see [Coward, 2006]. Continuous time computational models also include neural networks and systems that can be built using electronic analog devices. Since continuous time systems are conducive to modeling huge populations, one might speculate that they will have a prominent role in analyzing massively parallel systems such as the Internet [Papadimitriou, 2001].

The first true model of a universal continuous time machine was proposed by Shannon [Shannon, 1941], who introduced it as a model of the Differential Analyzer. During the 1950s and 60s an extensive body of literature was published about the programming of such machines111See for example the very instructive Doug Coward ’s web Analog Computer Museum [Coward, 2006] and its bibliography. This literature reveals a quite forgotten art of programming continuous time and hybrid (digital-analog) machines, which level of sophistication is close to today’s engineering programming.. There were also a number of significant publications on how to use analog devices to solve discrete or continuous problems: see e.g. [Vergis et al., 1986] and the references therein. However, most of this early literature is now only marginally relevant given the ways in which our current understanding of computability and complexity theory have developed.

The research on artificial neural networks, despite the fact that it mainly focused on discrete time analog models, has motivated a change of perspective due to its many shared concepts and goals with today’s standard computability and complexity theory [Orponen, 1997], [Orponen, 1994]. Another line of development of continuous time computation theory has been motivated by hybrid systems, particularly by questions related to the hardness of their verification and control: see for example [Branicky, 1995b] and [Asarin et al., 1995].

In recent years there has also been a surge of interest in alternatives to classical digital models other than continuous time systems. Those alternatives include discrete-time analog-space models like artificial neural networks [Orponen, 1997], optical models [Woods and Naughton, 2005], signal machines [Durand-Lose, 2005] and the Blum Shub and Smale model [Blum et al., 1998]. More generally there have also been many recent developments in non-classical and more-or-less realistic or futuristic models such as exotic cellular automata models [Grigorieff and Margenstern, 2004], molecular or natural computations [Head, 1987], [Adleman, 1994], [Lipton, 1995], [Păun, 2002], black hole computations [Hogarth, 1992], or quantum computations [Deutsch, 1985], [Gruska, 1997], [Shor, 1994], [Kieu, 2004]. Some of these contributions are detailed in this volume.

The computational power of discrete time models are fairly well known and understood thanks in large part to the Church-Turing thesis. The Church-Turing thesis states that all reasonable and sufficiently powerful models are equivalent. For continuous time computation, the situation is far from being so clear, and there has not been a significant effort toward unifying concepts. Nonetheless, some recent results establish the equivalence between apparently distinct models [Gra a and Costa, 2003], [Gra a, 2004], [Graça et al., 2005], and [Bournez et al., 2007], which give us hope that a unified theory of continuous time computation may not be too far in the future.

This text can be considered an up-to-date version of Orponen’s 1997 survey [Orponen, 1997]. Orponen states at the end of his introduction of that the effects of imprecision and noise in analog computations are still far from being understood and that a robust complexity theory of continuous time models has yet to be developed. Although this evaluation remains largely accurate with regard to imprecision and noise, we will see in the present survey that in the intervening decade much progress in understanding the computability and even the complexity of continuous time computations has been made.

This chapter is organized as follows. In Section 2, we review the most relevant continuous time models. In sections 3 and 4 we discuss, respectively, computability and complexity issues in continuous time computations. In these sections we focus mainly on continuous time dynamical systems. In Section 5 we address the effect of imprecision and noise in analog computations. Finally, in Section 6, we conclude with some general insights and directions of further research in the field of continuous time computation.

2 Continuous Time Models

With a historical perspective in mind, we outline in this section several of the major classes of continuous time models that motivated interest in this field These models also illustrate concepts like continuous dynamics and input/output.

2.1 Models inspired by analog machines

2.1.1 GPAC and other circuit models

Probably, the best known universal continuous time machine is the Differential Analyzer, built at MIT under the supervision of Vannevar Bush [Bush, 1931] for the first time in 1931. The idea of assembling integrator devices to solve differential equations dates back to Lord Kelvin in 1876 [Thomson, 1876]. Mechanical222And even MECANO machines: see [Bowles, 1996]., and later on electronic, differential analyzers were used to solve various kinds of differential equations primarily related to problems in the field of engineering: see for e.g. [Bowles, 1996], or more generally [Williams, 1996] for historical accounts. By the 1960s differential analysers were progressively discarded in favor of digital technology.

The first theoretical study of the computational capabilities of continuous time universal machines was published by Shannon. In [Shannon, 1941], he proposed what is now referred to as the General Purpose Analog Computer (GPAC) as a theoretical model of Vannevar Bush’s differential analyzer. The model, later refined in the series of papers [Pour-El, 1974], [Lipshitz and Rubel, 1987], [Gra a and Costa, 2003], [Gra a, 2004], consists of families of circuits built with the basic units presented in Figure 1. There are some restrictions to the kinds of interconnectivity that are allowed to avoid undesirable behavior: e.g. non-unique outputs. For further details and discussions, refer to [Gra a and Costa, 2003] and [Gra a, 2002].

Shannon, in his original paper, already mentions that the GPAC generates polynomials, the exponential function, the usual trigonometric functions, and their inverses (see Figure 2). More generally, he claimed in [Shannon, 1941] that a function can be generated by a GPAC iff it is differentially algebraic. i.e. it satisfies some algebraic differential equation of the form

[TABLE]

where $p$ is a non-zero polynomial in all its variables. As a corollary, and noting that the Gamma function $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}dt$ or the Riemann’s Zeta function $\zeta(x)=\sum_{k=0}^{\infty}\frac{1}{k^{x}}$ are not d.a. [Rubel, 1989], it follows that the Gamma and the Zeta functions are examples of functions that can not be generated by a GPAC.

However, Shannon’s proof relating functions generated by GPACs with differentially algebraic functions was incomplete (as pointed out and partially corrected by [Pour-El, 1974], [Lipshitz and Rubel, 1987]). However, for the more robust class of GPACs defined in [Gra a and Costa, 2003], the following stronger property holds: a scalar function $f:\mathbb{R}\to\mathbb{R}$ is generated by a GPAC iff it is a component of the solution of a system $y^{\prime}=p(t,y)$, where $p$ is a vector of polynomials. A function $f:\mathbb{R}\to\mathbb{R}^{k}$ is generated by a GPAC iff all of its components are.

The $\Gamma$ function is indeed GPAC computable, if a notion of computation inspired from recursive analysis is considered [Gra a, 2004]. GPAC computable functions in this sense correspond precisely to computable functions over the reals [Bournez et al., 2007].

Rubel proposed [Rubel, 1993] an extension of Shannon’s original GPAC. In Rubel’s model, the Extended Analog Computer (EAC), operations to solve boundary value problems or to take certain infinite limits were added. We refer to [Mills, 1995] and [Mills et al., 2005] for descriptions of actual working implementations of Rubel’s EAC.

More broadly, a discussion of circuits made of general basic units has been presented recently in [Tucker and Zucker, 2007]. Equational specifications of such circuits, as well as their semantics, are given by fixed points of operators over the space of continuous streams. Under suitable hypotheses, this operator is contracting and an extension of Banach fixed point theorem for metric spaces guarantees existence and unicity of the fixed point. Moreover, that fixed point can also be proved to be continuous and concretely computable whenever the basic modules also have those properties.

2.1.2 Hopfield network models

Another well known continuous time model is the “neural network” model proposed by John Hopfield in 1984 in [Hopfield, 1984]. These networks can be implemented in electrical [Hopfield, 1984] or optical hardware [Stoll and Lee, 1988].

A symmetric Hopfield network is made of a finite number, say $n$, of simple computational units, or neurons. The architecture of the network is given by some (non oriented) graph whose nodes are the neurons, and whose edges are labeled by some weights, the synaptic weights. The graph can be assumed to be complete by replacing the absence of a connection between two nodes by an edge whose weight is null.

The state of each neuron $i$ at time $t$ is given by some real value $u_{i}(t)$. Starting from some given initial state $\vec{u}_{0}\in\mathbb{R}^{n}$, the global dynamic of the network is defined by a system of differential equations

[TABLE]

where $V_{i}=\sigma(u_{i})$, $\sigma$ is some saturating function such as $\sigma(u)=\alpha\tan u+\beta$, $W_{i,j}=W_{j,i}$ is the weight of the edge between $i$ and $j$, $C_{i},I_{i},R_{i}$ are some constants [Hopfield, 1984].

Hopfield proved in [Hopfield, 1984], by a Lyapunov-function argument, that such systems are globally asymptotically stable, i.e. from any initial state, the system relaxes toward some stable equilibrium state. Indeed, consider for example the energy function [Hopfield, 1984]

[TABLE]

The function $E$ is bounded and its derivative is negative. Hence the time evolution of the whole system is a motion in a state space that seeks out (possibly local) minima of $E$.

This convergence behavior has been used by Hopfield to explore various applications such as associative memory, or to solve combinatorial optimization problems [Hopfield, 1984], [Hopfield and Tank, 1985].

An exponential lower bound on the convergence time of continuous time Hopfield networks has been related to their dimension in [Šíma and Orponen, 2003a]. Such continuous time symmetric networks can be proved to simulate any finite binary-state discrete-time recurrent neural network [Orponen and Šíma, 2000], [Šíma and Orponen, 2003b].

2.1.3 Networks of spiking neurons

If one classifies, following [Maass, 1997b], neural network models according to their activation functions and dynamics, three different generations can be distinguished. The first generation, with discontinuous activation functions, includes multilayer perceptrons, Hopfield networks, and Boltzmann machines (see for example [Abdi, 1994] for an introduction to all mentioned Neural Networks models). The output of this generation of networks is digital. The second generation of networks use continuous activation functions instead of step or threshold functions to compute the output signals. These include feedforward and recurrent sigmoidal neural network, radial basis functions networks, and continuous time Hopfield networks. Their input and output is analog. The third generation of networks is based on spiking neurons and encodes variables in time differences between pulses. This generation exhibits continuous time dynamics and is the most biologically realistic [Maass and Bishop, 1998].

There are several mathematical models of spiking neurons of which we will focus on one, whose computational properties have been investigated in depth. The Spiking Neural Network model is represented by a finite directed graph. To each node $v$ (neuron) of the graph is associated a threshold function $\theta_{v}:\mathbb{R^{+}}\to\mathbb{R}\cup\{\infty\}$, and to each edge $(u,v)$ (synapse) is associated a response-function $\epsilon_{u,v}:\mathbb{R^{+}}\to\mathbb{R}$ and a weight-function $w_{u,v}$.

For a non-input neuron $v$, one defines its set $F_{v}$ of firing times recursively. The first element of $F_{v}$ is $\inf\{t|P_{v}(t)\geq\theta_{v}(0)\}$, and for any $s\in F_{v}$, the next larger element of $F_{v}$ is $\inf\{t|t>s\mbox{ and }P_{v}(t)\geq\theta_{v}(t-s)\}$ where

[TABLE]

The [math] above can be replaced by some bias function. We use it here to guarantee that $P_{v}$ is well-defined even if $F_{u}=\emptyset$ for all $u$ with $w_{u,v}\neq 0$. To approximate biological realism, restrictions are placed on the allowed response-functions and bias-functions of these models: see [Maass, 1997b], [Maass, 1999], [Natschläger and Maass, 2002], or [Maass, 2002], [Maass, 2003], for discussions on the model. In particular, rapidly fading memory is a biological constraint which prevents chaotic behavior in networks with a continuous time dynamic. Recently, the use of feedback to overcome the limitations of such a constraint was analized in [Maass et al., 2007].

The study of the computational power of several variants of spiking neural networks was initiated in [Maass, 1996a]. Noisy extensions of the model have been considered [Maass, 1996b], [Maass, 1997a], [Maass and Natschläger, 2000]. A survey of complexity results can be found in [Šíma and Orponen, 2003c]. Restrictions that are easier to implement in hardware versions have also been investigated in [Maass and Ruf, 1999].

2.1.4 $\mathbb{R}$-recursive functions

Moore proposed a theory of recursive functions on the reals in [Moore, 1996], which is defined in analogy with classical recursion theory and corresponds to a conceptual analog computer operating in continuous time. As we will see, this continuous time model has in particular the capability of solving differential equations, similar to an idealized analog integrator of the GPAC. In fact, the theory of $\mathbb{R}$-recursive functions can be seen as an extension of Shannon’s theory for the GPAC. A general discussion of the motivations behind $\mathbb{R}$-recursion theory can be found in [Mycka and Costa, 2005].

A function algebra $[B_{1},B_{2},...;O_{1},O_{2},...]$ is the smallest set containing basic functions $\{B_{1},B_{2},...\}$ and closed under certain operations $\{O_{1},O_{2},...\}$, which take one or more functions in the class and create new ones. Although function algebras have been defined in the context of recursion theory on the integers, and widely used to characterize computability and complexity classes [Clote, 1998], they are equally suitable to define classes of real valued recursive functions.

The $\mathbb{R}$-recursive functions were first defined in [Moore, 1996]. These are given by the function algebra $\mathcal{M}=[0,1,U;\mathsf{comp},\mathsf{int},\mathsf{minim}]$,333We consider that the operator $\mathsf{int}$ preserves analyticity (see [Campagnolo et al., 2000], [Campagnolo, 2002]). where $U$ is the set of projection functions $U_{i}(\vec{x})=x_{i}$, $\mathsf{comp}$ is composition, $\mathsf{int}$ is an operation that given $f$ and $g$ returns the solution of the initial value problem $h(\vec{x},0)=f(\vec{x})$ and $\partial_{y}h(\vec{x},y)=g(\vec{x},y,h)$ and $\mathsf{minim}$ returns the smallest zero $\mu_{y}f(\vec{x},y)$ of a given $f$. Moore also studied the weaker algebra $\mathcal{I}=[0,1,-1,U;\mathsf{comp},\mathsf{int}]$ and claimed its equivalence with the class of unary functions generated by the GPAC [Moore, 1996].

Many non recursively enumerable sets are $\mathbb{R}$-recursive. Since $\mathsf{minim}$ is the operation in $\mathcal{M}$ which gives rise to uncomputable functions, a natural question is to ask if $\mathsf{minim}$ can be replaced by some other operation of mathematical analysis. This was done in [Mycka and Costa, 2004] where $\mathsf{minim}$ is replaced by the operation $\mathsf{lim}$, which returns the infinite limits of the functions in the algebra. These authors stratify $[0,1,-1,U;\mathsf{comp},\mathsf{int},\mathsf{lim}]$ according to the allowed number ($\eta$) of nested limits and relate the resulting $\eta$-hierarchy with the arithmetical and analytical hierarchies. In [Loff et al., 2007a] it is shown that the $\eta$-hierarchy does not collapse (see also [Loff, 2007]), which implies that infinite limits and first order integration are not interchangeable operations [Loff et al., 2007b].

The algebra $\mathcal{I}$ only contains analytic functions and is not closed under iteration [Campagnolo et al., 2000]. However, if an arbitrarily smooth extension to the reals $\theta$ of the Heaviside function is included in the set of basic functions of $\mathcal{I}$, then $\mathcal{I}+\theta$ contains extensions to the reals of all primitive recursive functions.

The closure of fragments of $\mathcal{I}+\theta=[0,1,-1,\theta,U;\mathsf{comp},\mathsf{int}]$ under discrete operations like bounded products, bounded sums and bounded recursion, has been investigated in the thesis [Campagnolo, 2001] and also in the papers [Campagnolo et al., 2002], [Campagnolo, 2002], [Campagnolo, 2004].

In particular, several authors studied the function algebra $\mathcal{L}=[0,1,-1,\pi,\theta,U;$ $\mathsf{comp},\mathsf{LI}]$ where the $\mathsf{LI}$ is only able to solve linear differential equations (i.e., it restricts $\mathsf{int}$ to the case $\partial_{y}h(\vec{x},y)=g(\vec{x},y)\,h(\vec{x},y)$). The class $\mathcal{L}$ contains extensions to the reals of all the elementary functions, [Campagnolo et al., 2002].

Instead of asking which computable functions over $\mathbb{N}$ have extensions to $\mathbb{R}$ in a given function algebra, Bournez and Hainry consider classes of functions over $\mathbb{R}$ computable according to recursive analysis, and characterize them precisely with function algebras. This was done for the elementarily computable functions [Bournez and Hainry, 2005], characterized as $\mathcal{L}$ closed under a restricted limit schema. This was extended to yield a characterization of the whole class of computable functions over the reals [Bournez and Hainry, 2006], adding a restricted minimisation schema. Those results provide syntactical characterizations of real computable functions in a continuous setting, which is arguably more natural than the higher order Turing machines of recursive analysis.

A more general approach to the structural complexity of real recursive classes, developed in [Campagnolo and Ojakian, 2007], is based on the notion of approximation. This notion was used to lift complexity results from $\mathbb{N}$ to $\mathbb{R}$, and was applied in particular to characterize $\mathcal{L}$.

Somewhat surprisingly, the results above indicate that two distinct models of computation over the reals (computable analysis and real recursive functions) can be linked in an elegant way.

2.2 Models inspired by continuous time system theories

2.2.1 Hybrid Systems

An increasing number of systems exhibit some interplay between discrete and analog behaviors. The investigation of these systems has led to relevant new results about continuous time computation.

A variety of models have been considered: see for example the conference series Hybrid Systems Computation and Control or [Branicky, 1995a]. However, hybrid systems444“Hybrid” refers here to the fact that the systems have intermixed discrete and continuous evolutions. This differs from historical literature about analog computations, where “hybrid” often refers to machines with a mixture of analog and digital components. are essentially modeled either as differential equations with discontinuous right hand sides, as differential equations with continuous and discrete variables, or as hybrid automata. A hybrid automaton is a finite state automaton extended with variables. Its associated dynamics consists of guarded discrete transitions between states of the automaton that can reset some variables. Typical properties of hybrid systems that have been considered are reachability, stability and controllability.

With respect to the differential equation modeling approach, Branicky proved in [Branicky, 1995b] that any hybrid system model that can implement a clock and also implement general continuous ordinary differential equations is able to simulate Turing machines. Asarin, Maler and Pnueli proved in [Asarin et al., 1995] that piecewise constant differential equations can simulate Turing machines in $\mathbb{R}^{3}$, while the reachability problem for these systems in dimension $d\leq 2$ is decidable [Asarin et al., 1995]. Piecewise constant differential equations, as well as many hybrid systems models, exhibit the so-called Zeno’s phenomenon: an infinite number of discrete transitions may happen in a finite time. This has been used in [Asarin and Maler, 1998] to prove that arithmetical sets can be recognized in finite time by these systems. Their exact computational power has been characterized in terms of their dimension in [Bournez, 1999a] and [Bournez, 1999b]. The Jordan’s theorem based argument of [Asarin et al., 1995] to get decidability for planar piecewise constant differential equations has been generalized for planar polynomial systems [Ceraens and Viksna, 1996] and for planar differential inclusion systems [Asarin et al., 2001].

There is extensive literature on the hybrid automata modeling approach about determining the exact frontier between decidability and non-decidability for reachability properties, according to the type of allowed dynamics, guards, and resets. The reachability property has been proved decidable for timed automata [Alur and Dill, 1990]. By reduction to this result, or by a finite bisimulation argument in the same spirit, this has also been generalized to multirate automata [Alur et al., 1995], to specific classes of updatable timed automata in [Bouyer et al., 2000a], [Bouyer et al., 2000b], and to initialized rectangular automata in [Henzinger et al., 1998], [Puri and Varaiya, 1994]. There is a multitude of undecidability results, most of which rely on simulations of Minsky two-counter machines. For example, the reachability problem is semi-decidable but non-decidable for linear hybrid automata [Alur et al., 1995], [Nicollin et al., 1993]. The same problem is known to be undecidable for rectangular automata with at least $5$ clocks and one two-slope variable [Henzinger et al., 1998], or for timed automata with two skewed clocks [Alur et al., 1995]. For discussion of these results, see also [Asarin and Schneider, 2002]. Refer to [Blondel and Tsitsiklis, 1999], and [Collins and van Schuppen, 2004] or to the survey [Blondel and Tsitsiklis, 2000] for properties other than reachability (for example, stability and observability).

O-minimal hybrid systems are initialized hybrid systems whose relevant sets and flows are definable in an o-minimal theory. These systems always admit a finite bisimulation [Lafferriere and Pappas, 2000]. However, their definition can be extended to a more general class of “non-deterministic” o-minimal systems [Brihaye and Michaux, 2005], for which the reachability problem is undecidable in the Turing model, as well as in the Blum Shub Smale model of computation [Brihaye, 2006]. Upper bounds have been obtained on the size of the finite bisimulation for Pfaffian hybrid systems [Korovina and Vorobjov, 2004] [Korovina and Vorobjov, 2006] using the word encoding technique introduced in [Brihaye and Michaux, 2005].

2.2.2 Automata theory

There have been several attempts to adapt classical discrete automata theory to continuous time; this is sometimes referred to as the general program of Trakhtenbrot [Trakhtenbrot, 1995].

One attempt is related to timed automata, which can be seen as languages recognizers [Alur and Dill, 1994]. Many specific decision problems have been considered for timed automata: see survey [Alur and Madhusudan, 2004]. Timed regular languages are known to be closed under intersection, union, and renaming, but not under complementation. The membership and empty language problems are decidable, whereas inclusion and universal language problems are undecidable. The closure of timed regular languages under shuffling is investigated in [Finkel, 2006]. Several variants of Kleene’s theorem are established [Asarin et al., 1997], [Asarin, 1998], [Asarin et al., 2002], [Bouyer and Petit, 1999], [Bouyer and Petit, 2002], and in [Asarin and Dima, 2002]. There have been some attempts to establish pumping lemmas [Beauquier, 1998]. A review, with discussions and open problems related to this approach can be found in [Asarin, 2004].

An alternative and independent automata theory over continuous time has been developed in [Rabinovich and Trakhtenbrot, 1997], [Trakhtenbrot, 1999], [Rabinovich, 2003]. Here automata are not considered as language recognizers, but as computing operators on signals. A signal is a function from the non-negative real numbers to a finite alphabet (the set of the channel’s states). Automata theory is extended to continuous time, and it is argued that the behavior of finite state devices is ruled by so-called finite memory retrospective functions. These are proved to be speed-independant, i.e. independant under “stretchings” of the time axis. Closure properties of operators on signals are established, and the representation of finite memory retrospective functions by finite transition diagrams (transducers) is discussed. See also [Francisco, 2002] for a detailed presentation of Trakhtenbrot and Rabinovich’s theory, and for discussions about the representation of finite memory retrospective operators by circuits.

Finally, another independent approach is considered in [Ruohonen, 2004], where Chomsky-like hierarchies are established for families of sets of piecewise continuous functions. Differential equations, associated to specific memory structures, are used to recognize sets of functions. Ruohonen shows that the resulting hierarchies are not trivial and establishes closure properties and inclusions between classes.

2.3 Other computational models

In addition to the two previously described approaches, there are a number of other computational models that have lead to interesting developments in continuous time computation theory.

The question of whether Einstein’s general relativity equations admit space-time solutions that allow an observer to view an eternity in a finite time was investigated and proved possible in [Hogarth, 1992]. The question of whether this implies that super-tasks can in principle be solved has been investigated in [Earman and Norton, 1993], [Hogarth, 1996], [Hogarth, 1994], [Hogarth, 2006], [Etesi and N meti, 2002], [Németi and Andréka, 2006], [Németi and Dávid, 2006], and [Welch, 2006].

Some machine inspired models are neither clearly digital nor analog. For example, the power of planar mechanisms attracted great interest in England and France in the late 1800s, and in the 1940s in Russia. Specifically, these consisted of rigid bars constrained to a plane and joined at either end by rotable rivets. A theorem attributed555The theorem is very often attributed to Kempe [Artobolevskii, 1964], [Svoboda, 1948], even if he apparently never proved exactly that. to Kempe [Kempe, 1876] states that they are able to compute all algebraic functions: see for e.g. [Artobolevskii, 1964] or [Svoboda, 1948].

3 ODEs and properties

Most of the continuous time models described above have a continuous dynamics described by differential equations. In Shannon’s GPAC and Hopfield networks, the input corresponds to the initial condition while the output is, respectively, the time evolution or the equilibrium state of the system. Other models are language recognizers. The input again corresponds to the initial condition, or some initial control, and the output is determined by some accepting region in the state space of the system. All these systems therefore fall into the framework of dynamical systems.

In this section we will recall some fundamental results about dynamical systems and differential equations and discuss how different models can be compared in this general framework.

3.1 ODEs and dynamical systems

Let’s consider that we are working in $\mathbb{R}^{n}$ (in general, we could consider any vector space with a norm). Let us consider $f:E\to\mathbb{R}^{n}$, where $E\subset\mathbb{R}^{n}$ is open. An ODE is given by $y^{\prime}=f(y)$ and its solution is a differentiable function $y:I\subset\mathbb{R}\to E$ that satisfies the equation.

For any $x\in E$, the fundamental existence-uniqueness theorem (see e.g. [Hirsch et al., 2003]) for differential equations states that if $f$ is Lipschitz on $E$, i.e. if there exists $K$ such that $||f(y_{1})-f(y_{2})||<k||y_{1}-y_{2}||$ for all $y_{1},y_{2}\in E$, then the solution of

[TABLE]

exists and is unique on a certain maximal interval of existence $I\subset\mathbb{R}$. In the terminology of dynamical systems, $y(t)$ is referred to as the trajectory, $\mathbb{R}^{n}$ as the phase space, and the function $\phi(t,x)$, which gives the position $y(t)$ of the solution at time $t$ with initial condition $x$, as the flow. The graph of $y$ in $\mathbb{R}^{n}$ is called the orbit.

In particular, if $f$ is continuously differentiable on $E$ then the existence-uniqueness condition is fulfilled [Hirsch et al., 2003]. Most of the mathematical theory has been developed in this case, but can be extended to weaker conditions. In particular, if $f$ is assumed to be only continuous, then uniqueness is lost, but existence is guaranteed: see for example [Coddington and Levinson, 1972]. If $f$ is allowed to be discontinuous, then the definition of solution needs to be refined. This is explored by Filippov in [Filippov, 1988]. Some hybrid system models use distinct and ad hoc notions of solutions. For example, a solution of a piecewise constant differential equation in [Asarin et al., 1995] is a continuous function whose right derivative satisfies the equation.

In general, a dynamical system can be defined as the action of a subgroup $\mathcal{T}$ of $\mathbb{R}$ on a space $X$, i.e. by a function (a flow) $\phi:\mathcal{T}\times X\to X$ satisfying the following two equations

[TABLE]

[TABLE]

It is well known that subgroups $\mathcal{T}$ of $\mathbb{R}$ are either dense in $\mathbb{R}$ or isomorphic to the integers. In the first case, the time is called continuous, in the latter case, discrete.

Since flows obtained by initial value problems (IVP) of the form (1) satisfy equations (2) and (3), they correspond to specific continuous time and space dynamical systems. Although not all continuous time and space dynamical systems can be put in a form of a differential equation, IVPs of the form (1) are sufficiently general to cover a very wide class of such systems. In particular, if $\phi$ is continuously differentiable, then $y^{\prime}=f(y)$, with $f(y)=\left.\frac{d}{dt}\phi(t,y)\right|_{t=0}$, describes the dynamical system.

For discrete time systems, we can assume without loss of generality that $\mathcal{T}$ is the integers. The analog of of IVP (1) for discrete time systems is a recurrence equation of type

[TABLE]

A dynamical system whose space is discrete and that evolves discretely is termed digital, otherwise it is analog. A classification of some computational models according to the nature of their space and time can be found in Figure 3.

3.2 Dissipative and non-dissipative systems

A point $x^{*}$ of the state space is called an equilibrium point if $f(x^{*})=0$. If the system is at $x^{*}$ it will remain there. It is said to be stable if for every neighborhood $U$ of $x^{*}$ there is a neighborhood $W$ of $x^{*}$ in $U$ such that every solution starting from a point $x$ of $W$ is defined and is in $U$ for all time $t>0$. The point is asymptotically stable if, in addition to the properties above, we have $\lim y(t)=x^{*}$ [Hirsch et al., 2003].

Some local conditions on the differential $Df(x^{*})$ of $f$ in $x^{*}$ have been clearly established. If at an equilibrium point $x^{*}$ all eigenvalues of $Df(x^{*})$ have negative real parts, then $x^{*}$ is asymptotically stable, and furthermore nearby solutions approach $x^{*}$ exponentially. In that case, $x^{*}$ is called a sink. At a stable equilibrium point $x^{*}$, no eigenvalue of $Df(x^{*})$ can have a positive real part [Hirsch et al., 2003].

In practice, Lyapunov’s stability theorem applies more broadly (i.e., even if $x^{*}$ is not a sink). It states that if there exists a continuous function $V$ defined on a neighborhood of $x^{*}$, differentiable (except perhaps on $x^{*}$) with $V(x^{*})=0$, $V(x)>0$ for $x\neq x^{*}$, and $dV(x)/dt\leq 0$ for $x\neq x^{*}$ then $x^{*}$ is stable. If, in addition, $dV(x)/dt<0$ for $x\neq x^{*}$, then $x^{*}$ is asymptotically stable: see [Hirsch et al., 2003].

If the function $V$ satisfies the previous conditions everywhere, then the system is globally asymptotically stable. Whatever the initial point $x$ is, the trajectories will eventually converge to local minima of $V$. In this context, the Lyapunov function $V$ can be interpreted as an energy, and its minima correspond to attractors of the dynamical system. These are bounded subsets of the phase space to which regions of initial conditions of nonzero volume converge as time increases.

A dynamical system is called dissipative if the volume of a set decreases under the flow for some region of the phase space. Dissipative systems are characterized by the presence of attractors. By opposition, a dynamical system is said to be volume-preserving if the volume is conserved. For instance, all Hamiltonian systems are volume-preserving because of Liouville’s theorem [Arnold, 1989]. Volume preserving dynamical system cannot be globally asymptotically stable [Arnold, 1989].

3.3 Computability of solutions of ODEs

Here we review some results on the computability of solutions of IVPs in the framework of recursive analysis (see e.g. [Weihrauch, 2000] and the corresponding chapter in this volume).

In general, given a computable function $f$, one can investigate if the solution of the IVP (1) is also computable in the sense of recursive analysis. If we require that the IVP has a unique solution then that solution is computable. Formally, if $f$ is computable on $[0,1]\times[-1,1]$ and the IVP $y^{\prime}=f(t,y)$, $y(0)=0$ has a unique solution on $[0,b]$, $0<b\leq 1$, then the solution $y$ is computable on $[0,b]$.

This result also holds for a general $n$-dimensional IVP if its solution is unique [Ruohonen, 1996]. However, computability of solutions is lost as soon as uniqueness of solutions is relaxed, even in dimension $1$. Indeed, the famous result of [Pour-El and Richards, 1979] shows that there exists a polynomial-time computable function $f:[0,1]\times[-1,1]\to\mathbb{R}$, such that the equation $y^{\prime}=f(t,y)$, with $y(0)=0$, has non-unique solutions, but none of them is computable on any closed finite time interval.

Similar phenomena hold for other natural equations: the $3$-dimensional wave equation (which is a partial equation), with computable initial data, can have a unique solution which is nowhere computable666However, in all these cases, the problems under study are ill-posed: either the solution is not unique, or it is unstable  and the addition of some natural regularity conditions to prevent ill-posedness do yield computability [Weihrauch and Zhong, 2002]. [Pour-El and Richards, 1981], [Pour-El and Zhong, 1997]. Notice that, even if $f$ is assumed computable and analytic, and the solution unique, it may happen that the maximal interval $(\alpha,\beta)$ of existence of the solution is non-computable [Graça et al., 2006]. This same question is open if $f$ is polynomial. Those authors show, however, that if $f$ and $f^{\prime}$ are continuous and computable, then the solution of $y^{\prime}=f(y,t),y(0)=x$, for computable $x$, is also computable on its maximal interval of existence. Refer also to [Pour-El and Richards, 1989], [Ko, 1983] for more uncomputability results, and also to [Ko, 1983], [Ko, 1991] for related complexity issues.

3.4 Static undecidability

As observed in [Asarin, 1995] and in [Ruohonen, 1997b], it is relatively simple but not very informative to get undecidability results with continuous time dynamical systems, if $f$ encodes a undecidable problem. To illustrate this, we recall the following example in [Ruohonen, 1997b]. Ruohonen discusses the event detection problem: given a differential equation $y^{\prime}=f(t,y)$, with initial value $y(0)$, decide if a given condition $g_{j}(t,y(t),y^{\prime}(t))=0,j=1,\cdots,k$ happens at some time $t$ in a given interval $I$. Given the Turing machine $\mathcal{M}$, the sequence $f_{0},f_{1},\cdots$ of rationals defined by

[TABLE]

is not a computable sequence of rationals, but is a computable sequence of reals, following the nomenclature of [Pour-El and Richards, 1989]. Now, the detection of the event $y(t)=0$ for the ordinary differential equation $y^{\prime}=0$, given $n$, and the initial value $y(0)=f_{n}$, is undecidable over any interval containing [math], because $f_{n}=0$ is undecidable.

A further modification can be obtained as follows [Ruohonen, 1997b]. He defines the smooth function

[TABLE]

which is computable on $[0,\infty)$. The detection of the event $y_{1}(t)=0$ for the ODE

[TABLE]

given an initial value $y_{1}(0)=1$, $y_{2}(0)=n$, where $n$ is a nonnegative integer is then undecidable on $[0,1]$.

As put forth in [Asarin, 1995] undecidability results given by recursive analysis are somehow built similarly.

3.5 Dynamic undecidability

To be able to discuss in more detail computability of differential equations, we will focus on ODEs that encode the transitions of a Turing machine instead of the result of the whole computation simulation777This is called dynamic undecidability in [Ruohonen, 1993].. Typically, we start with some (simple) computable injective function which encodes any configuration of a Turing machine $M$ as a point in $\mathbb{R}^{n}$. Let $x$ be the encoding of the initial configuration of $\mathcal{M}$. Then, we look for a function $f:E\subset\mathbb{R}^{n+1}\to\mathbb{R}^{n}$ such that the solution of $y^{\prime}(t)=f(y,t)$, with $y(0)=x$, at time $T\in N$ is the encoding of the configuration of $\mathcal{M}$ after $T$ steps. We will see, in the remainder of this section, that $f$ can be restricted to have low dimension, to be smooth or even analytic, or to be defined on a compact domain.

Instead of stating that the property above is a Turing machine simulation, we can address it as a reachability result. Given the IVP defined by $f$ and $x$, and any region $A\subset\mathbb{R}^{n}$, we are interested in deciding if there is a $t\geq 0$ such $y(t)\in A$, i.e., if the flow starting in $x$ crosses $A$. It is clear that if $f$ simulates a Turing machine in the previous sense, then reachability for that system is undecidable (just consider $A$ as encoding the halting configurations of $\mathcal{M}$). So, reachability is another way to address the computability of ODEs and a negative result is often a byproduct of the simulation of Turing machines. Similarly, undecidability of event detection follows from Turing simulation results.

Computability of reachable and invariant sets have been investigated in [Collins, 2005] for continuous time systems and in [Collins and Lygeros, 2005] for hybrid systems.

In general, viewing Turing machines as dynamical systems provides them a physical interpretation which is not provided by the von Neumann picture [Campagnolo, 2001]. This also shows that many qualitative features of (analog or non-analog) dynamical systems, e.g. questions about basins of attraction, chaotic behavior or even periodicity, are non computable [Moore, 1990]. Conversely, this brings into the realm of Turing machines and computability in general questions traditionally related to dynamical systems. These include in particular the relations between universality and chaos [Asarin, 1995], necessary conditions for universality [Delvenne et al., 2004], the computability of entropy [Koiran, 2001], understanding of edge of chaos [Legenstein and Maass, 2007], and relations with the shadowing property [Hoyrup, 2006].

3.6 Embedding Turing machines in continuous time

The embedding of Turing machines in continuous dynamical systems is often realized in two steps. Turing machines are first embedded into analog space discrete time systems, and then the obtained systems are in turn embedded into analog space and time systems.

The first step can be realized with low dimensional systems with simple dynamics: [Moore, 1990], [Ruohonen, 1993], [Branicky, 1995b], [Ruohonen, 1997b] consider general dynamical systems, [Koiran et al., 1994] piecewise affine maps, [Siegelmann and Sontag, 1995] sigmoidal neural nets, [Koiran and Moore, 1999] closed form analytic maps, which can be extended to be robust [Graça et al., 2005], and [Kurganskyy and Potapov, 2005] one dimensional very restricted piecewise defined maps.