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19
A Survey of Computational Complexity Results in Systems and Control
, 2000
"... 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 fi ..."
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Cited by 118 (20 self)
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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, NPcompleteness, 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, timevarying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
Deciding Stability and Mortality of Piecewise Affine Dynamical Systems
, 2001
"... In this paper we studyproblJ: such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a piecewise a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem that this AttractivityProblc isundecidabl as soon as n2. The same is ..."
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Cited by 22 (1 self)
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In this paper we studyproblJ: such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a piecewise a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem that this AttractivityProblc isundecidabl as soon as n2. The same is true of tworelkMI problI+J Stabil+J (is thedynamical systemglJH #RI asymptotical# stablto andMortal#M (do al trajectories go through 0?). We then show that Attractivity andStabilI: becomedecidabl in dimension 1 for continuous functions. c # 2001El1/JkR Science B.V.Al rights reserved. Keywords: Discretedynamical systems; Piecewise a#ne systems; Piecewiselecew systems; Hybrid systems;Mortal/JM Stabil/JM Decidabilk: 1.IP141 In this paper we studyproblJ+ such as: given a discrete timedynamical system of the form x(t +1)=f(x(t)) where f : R n #R n is a(possibl discontinuous) piecewise # This research waspartl carried outwhil Bllkk was visitingTsitsiklJ at MIT (Cambridge) and Koiran at ENS (Lyon). This research was supported by the ARO under grant DAAL0392G0115, by the NATO under grant CRG961115 and by the European Commission under the TMR(AlMkI;/z network contract ERBFMRXCT960074. # Corresponding author. Email addresses: blmCppCpA/J#JM:/zRkJ; (V.D.BlD./kIH Ol./kIH:J/zRkJ;/lkJ;/l (O. Bournez), pascal),/;MJMI/zRkJ;/ll (P. Koiran), christos@cs.berkel/ll (C.H. Papadimitriou), jnt@mit.edu (J.N. TsitsiklM#/ 03043975/01/$  see front matter c # 2001El1/kRk Science B.V.Al rights reserved. PII: S03043975(00)003996 688 V.D. Blondel et al. / Theoretical Computer Science 255 (2001) 687696 a#ne function, decide whetheral trajectories converge to 0. We show in our main theorem (Theorem 2) that this AttractivityProblc isundecidabl as soon as n2. The same is true of t...
The Topological Entropy of Iterated Piecewise Affine Maps is Uncomputable
 Discrete Mathematics and Theoretical Computer Science
"... We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is ..."
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Cited by 8 (0 self)
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We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every nonnegative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.
On the decidability of termination of query evaluation in transitiveclosure logics for polynomial constraint databases
, 2005
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Observability of hybrid systems and Turing machines
 PROCEEDINGS OF THE 43RD IEEE CONFERENCE ON DECISION AND CONTROL
, 2004
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On the presence of periodic configurations in Turing machines and in counter machines
"... . A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kurka has conjectured that every Turing machine  when seen as a dynamical system on the space of its configurations  has at least one periodic orbit. In this paper, we provide a ..."
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Cited by 2 (0 self)
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. A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kurka has conjectured that every Turing machine  when seen as a dynamical system on the space of its configurations  has at least one periodic orbit. In this paper, we provide an explicit counterexample to this conjecture. We also consider counter machines and prove that, in this case, the problem of determining if a given machine has a periodic orbit in configuration space is undecidable. 1 Introduction A Turing machine is an abstract deterministic computer with a finite set Q of internal states. The machine operates on a doublyinfinite tape of cells ind exed by an integer i # Z. Symbols taken from a finite alphabet # are written on every cell; a tape content can thus be seen as an element of # Z . fait ceci. At every discrete time step, the Turing machine scans the cell indexed by 0 and, depending upon its internal state and the scanned symbol, the machine ...
On the difficulty of deciding asymptotic stability of cubic homogeneous vector fields
 In Proceedings of the 2012 American Control Conference
, 2012
"... Abstract — It is wellknown that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next inte ..."
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Cited by 1 (1 self)
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Abstract — It is wellknown that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NPhard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NPhardness result, we obtain a Lyapunovinspired technique for proving positivity of forms. We also show that for asymptotically stable homogeneous cubic vector fields in as few as two variables, the minimum degree of a polynomial Lyapunov function can be arbitrarily large. Finally, we show that there is no monotonicity in the degree of polynomial Lyapunov functions that prove AS; i.e., a homogeneous cubic vector field with no homogeneous polynomial Lyapunov function of some degree d can very well have a homogeneous polynomial Lyapunov function of degree less than d. A. Background I.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
Turing Machines Can Be Efficiently Simulated by the General Purpose Analog Computer
, 2013
"... The ChurchTuring thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reduc ..."
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The ChurchTuring thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the ChurchTuring thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as spaceefficient as computations done in the context of Computable Analysis.