Modeling of complex systems II: A minimalist and unified semantics for heterogeneous integrated systems

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Abstract

The purpose of this paper is to contribute to a unified formal framework for complex systems modeling. To this aim, we define a unified semantics for systems including integration operators. We consider complex systems as functional blackboxes (with internal states), whose structure and behaviors can be constructed through a recursive integration of heterogeneous components. We first introduce formal definitions of time (allowing to deal uniformly with both continuous and discrete times) and data (allowing to handle heterogeneous data), and introduce a generic synchronization mechanism for dataflows. We then define a system as a mathematical object characterized by coupled functional and states behaviors. This definition is expressive enough to capture the functional behavior of any real system with sequential transitions. We finally provide formal operators for integrating systems and show that they are consistent with the classical definitions of those operators on transfer functions which model real systems.

Introduction

The concept of complex systems has led to various definitions in numerous disciplines (biology, physics, engineering, mathematics, computer science, etc). One speaks for instance of dynamical, mechanical, Hamiltonian, hybrid, holonomic, embedded, concurrent or distributed systems (cf. [2], [4], [21], [24], [28], [32]). A minimalist fuzzy definition consistent with (almost) all those of the literature is that a ‘system’ is “a set of interconnected parts forming an integrated whole”, and the adjective ‘complex’ implies that a system has “properties that are not easily understandable from the properties of its parts”. In the mathematical formalization of “complex systems”, there are today two major approaches: the first one is centered on understanding how very simple, but numerous, elementary components can lead to complex overall behaviors (e.g. cellular automatas), the second one (that will also be ours) is centered on giving a precise semantics to the notion of system and to the integration of systems to build greater overall systems.

When mathematically apprehended, the concept of system (in the sense of this second approach) is classically defined with models coming from:

  • control theory and physics, that deal with systems as partial functions (dynamical systems may also be rewritten in this way), called transfer functions, of the form:tT,y(t)=F(x,q,t),where x,q and y are inputs, states and outputs dataflows, and where T stands for time (usually considered in these approaches as continuous (see [32], [1], [12]).

  • theoretical computer sciences and software engineering, with systems that can be depicted by models equivalent to timed Turing machines with input and output, evolving on discrete times generally considered as a universal predefined sequence of steps (see for instance [19], [5], [16]).

However all these models do not easily allow to handle layered systems with multiple time scales. The introduction of a more evolved notion of time within Turing-like models involves many difficulties, mainly the proper definition of sequential transitions or the synchronization of different systems exchanging dataflows without synchronization of their time scales. Dealing with evolved definitions of times will generally imply to introduce infinity and infinitesimal (for instance with non-standard real numbers). There is therefore a great challenge (which we propose to address in this paper) on being able to unify in a same formal framework mathematical methods dealing with the design of both continuous and discrete systems.

The theory of hybrid systems was developed jointly in control theory (see [32], [34]) and in computer science (see [2], [3], [22]) to address this challenge. A serious issue with this theory is however that the underlying formalism has some troubling properties such as the Zeno effect which corresponds to the fact that an hybrid system can change of state an infinite number of times within a finite time (because of the convergence of series of durations) that one usually prefers to avoid in a robust modeling approach. Moreover, it does not allow to consider various time scales of heterogeneous granularity (which will be the central point of our approach). Other interesting and slightly different attempts in the same direction can also be found in Rabinovitch and Trakhtenbrot (see [27], [33]) who tried to reconstruct a finite automata theory on the basis of a real time framework, or in [35].

In the literature about (complex) systems, the real object and its model are often confused and both called “system”. We will call a real system any object of the real world which transforms flows of data. We will call system the mathematical object introduced to model real systems. In this paper, we are interested in modeling the functional behavior of real systems, and their integration. Thus, we will model real systems as functional blackboxes (with an internal state), whose structure and behaviors can be described by the recursive integration of heterogeneous smaller subsystems (thus considering complex systems as heterogeneous integrated systems). We will thus focus on two aspects of the complexity of systems:

  • the heterogeneity of systems (modeled following continuous or discrete time, and exchanging data of different types, - informational, material or energetic).1

  • the integration of systems, i.e. the mechanism to construct a system resulting from the composition of smaller systems, whose behaviors may be described at a more concrete level (i.e. a finer grain).

We will assume that the observational behavior of any real system can be modeled by a functional machine processing dataflows (for related work on dataflow networks, see [19], [11], [10]) in a way that can be encoded by timed transitions for changing states and outputs in instantaneous reaction to the inputs (comparable with timed Mealy machines [25] with possibly infinite states). We show that our formalization makes it possible to model the basic kinds of real systems (physical, software and human/organizational), which is especially important in Systems Engineering [6], [23], [30].

This paper is the second of a series on “Modeling of Complex Systems”. Indeed, we generalize the approach of the first paper [7] (where a unified framework for continuous and discrete systems was defined by using non-standard infinitesimal and finite time steps) by dealing with time, data, and synchronization axiomatically, and by introducing integration operators. The purpose of this second paper is to give a unified and minimalist semantics for heterogeneous integrated systems and their integration. By “unified”, we mean that we propose a unified model of real systems that can describe the functional behavior of heterogeneous systems and that is closed under integration. By “minimalist” we mean that our formalization intends to provide a small number of concepts and operators to model the behaviors and the integration of (complex) real systems. We believe that our work allows to give a relevant formal semantics for concepts and models typically used in Systems Engineering, where semi-formal modeling is well-spread. The paper is organized as follows:

  • in Section 2 Time, 3 Data, we introduce unified definitions of time (both continuous and discrete) and data (with various behaviors) to handle heterogeneous components and encompass classical approaches. We also define a generic synchronization for dataflows,

  • in Section 4, we introduce a formal definition of systems as unified functional objects modeling heterogeneous real systems,

  • in Section 5, we introduce minimalist operators for integrating systems (with closure of the definition of system) and prove that they are consistent with classical concepts of integration formalized on transfer functions.

Section snippets

Time

Most of the challenges raised by the unified definition of heterogeneous integrated systems are coming from time. Indeed, real systems are naturally modeled according to various time scales (modeling discrete or continuous time), and we must therefore be able to define:

  • a unified model of time encompassing continuous and discrete times to later introduce a unified definition of heterogeneous systems,

  • the mixture of various time scales for integrating systems.

Unifying both discrete and continuous

Data

Another challenge to address to model complex systems is the heterogeneity of data (modeling any element that can be exchanged between real systems) and of their synchronization between different time scales. We introduce datasets that will be used for defining data transmitted by dataflows. The dataflows will be used to describe variables of systems (inputs, outputs and states), and we define the synchronization of dataflows between time scales.

Systems

We introduce the definition of a system as a timed Mealy machine, and show that it can be represented by a transfer function.

Integration operators

We propose three elementary operators allowing to model systems integration, i.e. to build greater systems from a set of elementary systems by recursive application of composition operators and abstraction operator.

Conclusion

We have introduced a minimalist and unified semantics for heterogeneous integrated systems. This semantics allows us to capture two very important properties of complex systems: heterogeneity (being able to deal with various types of systems through rich time & data) and recursive integration (taking into account the integrative dimension of complex systems that are build recursively with multiple levels of components).

This work is the theoretical part of a broader project aiming at building an

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