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26
Reachability Analysis of Dynamical Systems having PiecewiseConstant Derivatives
 Theoretical Computer Science
, 1995
"... In this paper we consider a class of hybrid systems, namely dynamical systems with piecewiseconstant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector ..."
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Cited by 111 (18 self)
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In this paper we consider a class of hybrid systems, namely dynamical systems with piecewiseconstant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector field, hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction. With respect to such systems we investigate the reachability question: Given an effective description of the systems and of two polyhedral subsets P and Q of the statespace, is there a trajectory starting at some x 2 P and reaching some point in Q? Our main results are a decision procedure for twodimensional systems, and an undecidability result for three or more dimensions. 1 Introduction 1.1 Motivation Hybrid systems (HS) are systems that combine intercommunicating discrete and continuous components. Most embedded systems belong to this class since they operate...
Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems
, 1995
"... 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 ..."
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Cited by 68 (3 self)
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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...
A Survey of ContinuousTime Computation Theory
 Advances in Algorithms, Languages, and Complexity
, 1997
"... Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists o ..."
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Cited by 29 (6 self)
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Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuoustime computation. However, while specialcase algorithms and devices are being developed, relatively little work exists on the general theory of continuoustime models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...
Achilles and the Tortoise climbing up the hyperarithmetical hierarchy
, 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
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Cited by 26 (6 self)
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We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3 (respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyperarithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyperarithmetical and is \Sigma ! kcomplete (resp. \Sigma ! k +1 complete).
Reachability problems for sequential dynamical systems with threshold functions
 Theoretical Computer Science
, 2003
"... A sequential dynamical system (SDS) over a domain § is a triple ¨�©��������� � , where (i) ©�¨������� � is an undirected graph with � nodes with each node having a state value from §, (ii) ������ � ¤ �� � ¦ �������������� � is a set of local transition functions with �� � denoting the local transi ..."
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Cited by 22 (3 self)
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A sequential dynamical system (SDS) over a domain § is a triple ¨�©��������� � , where (i) ©�¨������� � is an undirected graph with � nodes with each node having a state value from §, (ii) ������ � ¤ �� � ¦ �������������� � is a set of local transition functions with �� � denoting the local transition function associated with node � � and (iii) � is a permutation of (i.e., a total order on) the nodes in �. A single SDS transition is obtained by updating the states of the nodes in � by evaluating the function associated with each of them in the order given by �. We consider reachability problems for SDSs with restricted local transition functions. Our main intractability results show that the reachability problems for SDSs are PSPACEcomplete when either of the following restrictions hold: (i) � consists of both simplethresholdfunctions and simpleinvertedthreshold functions, or (ii) � consists only of thresholdfunctions that use weights in an asymmetric manner. Moreover, the results hold even for SDSs whose underlying graphs have bounded node degree and bounded pathwidth. Our lower bound results also extend to reachability problems for Hopfield networks and communicating finite state machines. On the positive side, we show that when � consists only of threshold functions that use weights in a symmetric manner, reachability problems can be solved efficiently provided all the weights are strictly positive and the ratio of the largest to the smallest weight is bounded by a polynomial function of the number of nodes.
On the Computational Power of Dynamical Systems and Hybrid Systems
 Theoretical Computer Science
, 1996
"... We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as ..."
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Cited by 22 (5 self)
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We explore the simulation and computational capabilities of discrete and continuous dynamical systems. We introduce and compare several notions of simulation between discrete and continuous systems. We give a general framework that allows discrete and continuous dynamical systems to be considered as computational machines. We introduce a new discrete model of computation: the analog automaton model. We characterize the computational power of this model as P=poly in polynomial time and as unbounded in exponential time. We prove that many very simple dynamical systems from literature are able to simulate analog automata. From this results we deduce that many dynamical systems have intrinsically superTuring capabilities. 1 Introduction The computational power of abstract machines which compute over the reals in unbounded precision in constant time is still an open problem. We refer the reader to [18] for an upto date survey. Indeed, a basic model for their computations has been propose...
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 21 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
The Computational Power of Continuous Time Neural Networks
 In Proc. SOFSEM'97, the 24th Seminar on Current Trends in Theory and Practice of Informatics, Lecture Notes in Computer Science
, 1995
"... We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11 ..."
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Cited by 14 (8 self)
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We investigate the computational power of continuoustime neural networks with Hopfieldtype units. We prove that polynomialsize networks with saturatedlinear response functions are at least as powerful as polynomially spacebounded Turing machines. 1 Introduction In a paper published in 1984 [11], John Hopfield introduced a continuoustime version of the neural network model whose discretetime variant he had discussed in his seminal 1982 paper [10]. The 1984 paper also contains an electronic implementation scheme for the continuoustime networks, and an argument showing that for sufficiently largegain nonlinearities, these behave similarly to the discretetime ones, at least when used as associative memories. The power of Hopfield's discretetime networks as generalpurpose computational devices was analyzed in [17, 18]. In this paper we conduct a similar analysis for networks consisting of Hopfield's continuoustime units; however we are at this stage able to analyze only the gen...
Deciding reachability for planar multipolynomial systems
 In Hybrid Systems III, LNCS 1066
, 1996
"... Abstract. In this paper we investigate the decidability of the reachability problem for planar nonlinear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a nite number of (polyhedral) regions. To each ..."
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Cited by 13 (0 self)
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Abstract. In this paper we investigate the decidability of the reachability problem for planar nonlinear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a nite number of (polyhedral) regions. To each of these regions is assigned some vector eld which governs the dynamical behaviour of the system within this region. We prove the decidability of point to point and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree. 1