Transcript Monofolhas

There are several potencial applications of
computation and complexity theory for
continuous systems. Among these are
verification and control of continuous time
systems, analog computation and the
analysis of continuous models of large scale
discrete systems like distributed computing
in sensor and telecommunication networks.
Toward these goals, a variety of models of
continuous
time
computation
were
compared and explored. Their dynamics,
computational bounds and robustness to
noise have been assessed.
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Research team
> Manuel Campagnolo
> Paula Gouveia
> Daniel Graça
> Carlos Lourenço
> Kerry Ojakian
Contract number if applicable
ConTComp: Continuous time computation
and complexity
Context
Continuous time computation has been considered under two approaches. One is related to continuous
time analog machines and has his roots in models of natural or artificial analog machinery. The other,
which is broader in scope, arises from system theory, and encompasses for instance the computational
analysis of hybrid systems and timed automata. The problems addressed by the theory of continuous
time computation come from, among others, the fields of verification, control theory, VLSI design and
neural networks.
Unlike discrete computation which is unified under the Turing paradigm, continuous time computation
covers a wide variety of models which are apparently distinct. To build a fruitful theory of computation
for continuous systems one needs to unify the existent models under a common framework. Complexity
of continous time systems is still a ill-defined notion. In general, continuous systems can suffer space
and time contractions, which makes it difficult to compare them with respect to complexity. While the
notion of complexity is easier to define for dissipative systems that converge to attractors, this still relies
on the notion of “natural” time. Complexity of continuous time models can instead be studied with
respect to the computational complexity of the functions that those models allow to compute. In that
case, standard and continuous models match nicely for a wide range of complexity classes.
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Discrete computation by a continuous
dynamical system: the Poincaré map.
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Neural network inspired by Nature with excitatory
and inhibitory neurons for dynamical computation.
Results
Systems of polynomial differential equations have been thoroughly analized. These are equivalent to
Shannon’s GPAC, a model of the analog computer known as the Differential Analyzer. Turing machines
can be robustly simulated by such systems. Using a new and more natural notion of computability for
continuous dynamical systems, polynomial differential equations compute precisely the real functions
that type-2 Turing machines compute. This result establishes an equivalence between two major
paradigms of computation over the reals and set the foundations for a unified theory. On the negative
side, it was shown that the boundness of the domain of definition is undecidable which prevents in
principle the verification of processes modeled by such dynamical systems.
New techniques were developed to provide machine-independent proofs of the equivalence between
different notion of computability for functions over the reals. It was found that a correct notion of
approximation, more flexible than exact computation, is at the core of the equivalence between models.
One other area of application of the project has been neural networks described by ordinary or partial
differential equations, with rich dynamical features, and connectivity patterns that mimic the massively
parallel nets of Biology. The role of chaos in the computation achieved by those systems has been
elucidated. This has been applied to diverse tasks like Selenoprotein discovery and pattern processing.
Perspectives
Continuous time systems arise as soon as one attempts to model systems which evolve in continuous
space and time. They also emerge as natural descriptions of discrete time or space systems when a
huge population of agents (molecules, individuals, processors) is abstracted into real quantities such as
proportions or thermodynamic data. Therefore, continuous time models may have a prominent role in
analyzing massively parallel systems. The results about decidability, robustness to noise and
computational abilities of continuous time models resulting from the project may be used to better
understand such systems. In collaboration with INRIA (Nancy) these issues will be further explorer.
Contact information [email protected]