Affiliations: VERIMAG, Centre Équation 38610 Gières,
France. [email protected] | LIX, CNRS & École Polytechnique 91128
Palaiseau, France. [email protected]
Note: [] Address for correspondence: LIX, CNRS & École
Polytechnique, 91128 Palaiseau, France
Abstract: We develop a unified functional formalism for modelling complex
systems, that is to say systems that are composed of a number of heterogeneous
components, including typically software and physical devices. Our approach
relies on non-standard analysis that allows us to model continuous time in a
discrete way. S ystems are defined as generalized Turing machines with temporized
input, internal and output mechanisms. Behaviors of systems are represented by
transfer functions. A transfer function is said to be implementable if it is
associated with a system. This notion leads us to define a new class –
which is natural in our framework – of computable functions on (usual)
real numbers. We show that our definitions are robust: on one hand, the class of
implementable transfer functions is closed under composition; on the other
hand, the class of computable functions in our meaning includes analytical
functions whose coefficients are computable in the usual way, and is closed
under addition, multiplication, differentiation and integration. Our class of
computable functions also includes solutions of dynamical and Hamiltonian
systems defined by computable functions. Hence, our notion of system appears to
take suitably into account physical systems.
Keywords: Complex system, Hybrid system, Non-standard analysis, Physical system, Software system, System modelling, Temporized systems, Time scale, Turing machine
