Pascal Koiran

Computability with low-dimensional dynamical systems

It has been known for a short time that a class of recurrent neural networks has universal computational abilities. These networks can be viewed as iterated piecewise-linear maps in a high-dimensional space. In this paper, we show that similar systems in dimension two are also capable of universal computations. On the contrary, it is necessary to resort to more complex systems (e.g., iterated piecewise-monotone maps) in order to retain this capability in dimension one.

Neural Networks with Quadratic VC Dimension

This paper shows that neural networks which use continuous activation functions have VC dimension at least as large as the square of the number of weightsw. This results settles a long-standing open question, namely whether the well-knownO(wlogw) bound, known for hard-threshold nets, also held for more general sigmoidal nets. Implications for the number of samples needed for valid generalization are discussed.

Computing over the reals with addition and order

We provide a fairly complete picture of the complexity theory of additive real machines. This model of computation is a restriction of the real Turing machine of Blum et al. (1989), since addition and subtraction are the only legal arithmetic operations. Removing the order relation < on R yields an even weaker class of machines, which is also studied. Our main results are: 1) characterizations of the classes of recognizable Boolean languages; 2) equivalence of real and digital nondeterminism; 3) a simpler proof of Meers P lin ¬= NP lin result

Closed-form analytic maps in one and two dimensions can simulate universal Turing machines

We show closed-form analytic functions consisting of a finite number of trigonometric terms can simulate Turing machines, with exponential slowdown in one dimension or in real time in two or more.

Deciding stability and mortality of piecewise affine dynamical systems

In this paper we study problems such as: given a discrete time dynamical system of the form x(t + 1)= f(x(t)) where f: R-n --> R-n is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this Attractivity Problem is undecidable as soon as n greater than or equal to2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (do all trajectories go through 0?). We then show that AM-activity and Stability become decidable in dimension 1 for continuous functions

A Weak Version of the Blum, Shub, and Smale Model

We propose a weak version of the Blum?Shub?Smale model of computation over the real numbers. In this weak model only a “moderate” usage of multiplications and divisions is allowed. The class of boolean languages recognizable in polynomial time is shown to be the complexity class P/poly. The main tool is a result on the existence of small rational points in semi-algebraic sets which is of independent interest. As an application, we generalize recent results of Siegelmann and Sontag on recurrent neural networks, and of Maass on feedforward nets. A preliminary version of this paper was presented at the1993 IEEE Symposium on Foundations of Computer Science. Additional results include: an efficient simulation of order-free real Turing machines by probabilistic Turing machines in the full Blum?Shub?Smale model; the strict inclusion of the real polynomial hierarchy in weak exponential time.

Decidable and Undecidable Problems about Quantum Automata

We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it is known that strict and nonstrict thresholds both lead to undecidable problems.

The Stability of Saturated Linear Dynamical Systems Is Undecidable

We prove that several global properties (global convergence, global asymptotic stability, mortality, and nilpotence) of particular classes of discrete time dynamical systems are undecidable. Such results had been known only for point-to-point properties. We prove these properties undecidable for saturated linear dynamical systems, and for continuous piecewise affine dynamical systems in dimension 3. We also describe some consequences of our results on the possible dynamics of such systems.

Computing over the Reals with Addition and Order

This paper deals with issues of structural complexity in a linear version of the Blum-Shub-Smale model of computation over the real numbers. Real versions of PSPACE and of the polynomial time hierarchy are defined, and their properties are investigated. Mainly two types of results are presented: ?Equivalence between quantification over the real numbers and over {0, 1};?Characterizations of recognizable subsets of {0, 1}* in terms of familiar discrete complexity classes. The complexity of the decision and quantifier elimination problems in the theory of the reals with addition and order is also studied.

Quantum automata and algebraic groups

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.