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## Towards a unified formalism

Various models & methods have been developed during the last decades to help modeling and designing specific types of systems. The idea introduced by Daniel Krob in [12] is that all those approaches share strong similarities at a certain level of abstraction, and could thus be formalized in a unified framework helping to deal with both the “big picture” and the more vertical models of systems. It also means synthesizing the “fundamental” characteristics of what is called systems architecting, i.e. the application of the systems approach to the design of complex industrial systems.

From a practical point of view, our aim is to give a unified formal semantics to the concepts manipulated on a daily basis by engineers from various fields working together on the design of complex industrial systems. Indeed, they need formal tools to reason on & model those systems in a unified & consistent way, with a clear understanding of the underlying concepts5. Of course, we do not intend to replace existing frameworks dedicated to specific systems, as such frameworks are much more accurate when dealing with the design of homogeneous systems. Our approach has strong benefits when dealing with heterogeneous systems at the right level of integration. Our aim is to give a unified formal semantics to the concepts manipulated on a daily basis by engineers from various fields working together on the design of complex industrial systems. Indeed, they need formal tools to reason on & model those systems in a unified & consistent way, with a clear understanding of the underlying concepts.

The purpose of the present work is thus to contribute to a unified formal framework for complex systems modeling & architecture. We will model the observational behavior of any real system through a functional machine pro- cessing dataflows (for related work on dataflow networks, see [37, 19, 18, 17]) 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 [46]). We show that our formalization makes it possible to model all kinds of real systems (physical, software and human/organizational), which is necessary in Systems Engineering.

An underlying assumption of our approach is that each system has its own rhythm, and that this rhythm cannot be changed by an interaction with another system, nor cause sampling problems when two systems of different time scales are integrated together. This means that each system somehow has a set of characteristic predefined moments of transitions that are generally based on its internal mechanisms seen at a certain level of abstraction. in a unified way.

• we separate the behavior of systems that can be observed (outputs and states) and their structure (how a system is built from elementary com- ponents).

• we view all behaviors as “algorithmic”. We define and model all objects so that a system behavior can be explained as a step-by-step transformation of dataflows.

• we model systems structure following a “Lego paradigm”. We explain the architecture of systems through the integration of smaller building blocks, themselves, modeled as systems.

• we consider that only three actions are possible during the design process: abstracting a system, composing together a set of systems, and verifying if a system behavior respects a set of requirements6.

### Structure of this manuscript

For a better understanding, this manuscript should be better read chapter af- ter chapter. Hence, we progressively build our framework. First, we define heterogeneous dataflows, and then systems as step-by-step machines transform- ing dataflows. Such systems can be integrated using operators, to build more complex systems, so that we can handle the two dimensions of the complexity (heterogeneity and integration). We then introduce a minimalist formalism for systems architecture to model requirements, underspecification & structure of systems through the design process. We finally open perspectives around systems optimization when fairness is required.

Chapter 2, Heterogeneous dataflows, introduces formal definitions of time, data and dataflows. Our unified definition of time allows to deal uniformly with both continuous and discrete times, while our definition of data allows to handle heterogeneous data having specific behaviors. This makes it possible to define heterogeneous dataflows with generic synchronization mechanisms allowing to mix dataflows together. The deliverable is a unified and well-formalized defi- nition of heterogeneous dataflows with properties that will be later needed to define & integrate systems.

Chapter 3, Systems, defines a system as a mathematical object characterized by coupled functional and states behaviors upon a time scale. This is a definition modeling a real system as a black box with observable functional behavior and an internal state (similarly to a timed Mealy machine). This definition is expressive enough to capture, at some level of abstraction, the functional behavior of any real industrial system with sequential transitions. We also express the functional behavior of systems via transfer functions transforming dataflows and show the equivalence. The deliverable is a unified definition of a system (viewed as a functional black box) and a proof of its equivalence with transfer functions.

Chapter 4, Integration operators, provides formal operators to integrate such systems. Those operators make it possible to compose systems together (i.e. interconnecting inputs and outputs of various systems) and to abstract a system (i.e. change the level of description of a system in term of granularity of all dataflows). We show that these operators are consistent with the natural

definitions of such operators on transfer functions. The deliverable is a set of integration operators that are proven to be consistent and whose expressivity allows to model systems integration. Chapters 2, 3 & 4 have been published as a journal article Complex Systems Modeling II: A minimalist and unified se- mantics for heterogeneous integrated systems [30] in Applied Mathematics and Computation (Elsevier), 2012.

Chapter 5, A logic for requirements, provides a minimalist logic to express re- quirements on systems. We first introduce an equivalent definition of systems using coalgebraic models. Based on these models, we define logical requirements to express properties on the observable behavior of systems. This chapter has been published as an article: An adequate logic for heterogeneous systems [4] at the 18th IEEE International Conference on Engineering of Complex Computer Systems (ICECCS 2013).

#### Time reference

A time reference is a universal time in which all systems will be defined. It captures the intuition we have of time: a linear quantity composed of ordered moments, pairs of which define durations. Such a modeling of “time” will be common to all the systems we want to integrate together.

**Table of contents :**

**1 Introduction **

1.1 Complex industrial systems

1.2 Systems Engineering

1.3 What is systems architecture?

1.3.1 A definition

1.3.2 Fundamental principles

1.4 Towards a unified formalism

1.5 Structure of this manuscript

**2 Heterogeneousdataflows **

2.1 Time

2.1.1 Time reference

2.1.2 Time scale

2.2 Data

2.2.1 Datasets

2.2.2 Implementation of standard data behaviors

2.3 Dataflows

2.3.1 Definition

2.3.2 Operators

2.3.3 Consistency of dataflows

**3 Systems **

3.1 Systems

3.1.1 Definition

3.1.2 Execution

3.1.3 Examples & expressivity

3.2 Transfer functions

3.2.1 Definition

3.2.2 Transfer function of a system

**4 Integration operators **

4.1 Composition

4.1.1 Timed extension

4.1.2 Product

4.1.3 Feedback

4.2 Abstraction

4.2.1 Nondeterminism

4.2.2 Abstraction

4.3 Integration of systems

4.3.1 Composition & abstraction

4.3.2 Example

**5 Alogic for requirements **

5.1 A coalgebraic definition of systems

5.1.1 Preliminaries

5.1.2 Transfer functions via coalgebras

5.1.3 Systems as coalgebras

5.2 A logic for system requirements

5.2.1 Definition

5.2.2 Examples of requirements

5.2.3 Adequacy of the logic

**6 Towards a framework for systemsarchitecture **

6.1 Handling underspecification

6.2 Modeling recursive structure

**7 Fair assignmentsbetweensystems **

7.1 Introduction

7.2 Inequality measurement with Lorenz dominance relations

7.2.1 Notations and definitions

7.2.2 Infinite order Lorenz dominance

7.3 Properties of infinite order Lorenz dominance

7.3.1 A representation theorem

7.3.2 Main properties of L∞-dominance

7.4 Solving multiagent assignment problems

7.4.1 Fair multiagent optimization

7.4.2 Linearization of the problem

7.4.3 Numerical tests

**8 Conclusion**