Formal Systems Concepts

CSER 2006 – Conference on Systems Engineering Research Formal System Concepts Joseph J. Simpson Mary J. Simpson April 7, 2006

Overview System and Meta-System Applications  Functional & Contruction Rules in Design & Discovery (Simpson & Simpson)  An Applied Science to Solve Complex Problems (Hall)  System Complexity Management & Control (Warfield)

Sequential Forms  Moore  Wymore

Abstraction Frames Abstraction Stacks Summary © 2006 Joseph J Simpson, Mary J Simpson

2

System Modes A Mapping Context for Complexity (Does Not Address System Boundary Directly )

Design Mode

Design Objects

Design Objects

Increasing Complexity

(Know Relationships)

Discovery Mode (Know Objects)

Discover Relationships

‘Real’ Objects © 2006 Joseph J Simpson, Mary J Simpson

Discover Relationships

‘Conceptual’ Objects 3

A.D. Hall, Cycle for Scientific Method, Model Building

alm e l R ed) a rm el Fo mod C (

Theory C C

C

Deduction

C

Predictions

C Confirm, Verify

Modify

Induction

P

P′

Compare

Data, Observations alm e l R e”) a u ct atur a F “n (

~ Error in Model, Theory

Where © 2006 Joseph J Simpson, Mary J Simpson

C

Data, Observations

Confidence Limits of Model or Theory

= Concepts 4

Warfield – Poly-Loop Model

C

Act on the Theory

Set of Processes

Act on the Process Set

2-Loop

B Theory 3-Loop

1-Loop

One Process Plan

Act on the Foundations E

Foundations 4-Loop

© 2006 Joseph J Simpson, Mary J Simpson

D

Act on the Problem 0-Loop A

5

Moore Type Sequential Machine Moore type sequential machine quintuple: M = (S, I, O, δ, λ)

δ

I I = Set of Inputs (finite, non-empty)

δ = Next state function (transition function) δ : S x I→S

O O = Set of Outputs (finite, non-empty)

λ

λ = Output Function of λ : S →O

S

S = Set of States (finite, non-empty)


6

Wymore – Model-Based Systems Engineering Discrete system quintuple: Z = (SZ, IZ, OZ, NZ, RZ)

NZ

IZ IZ = Set of Inputs of discrete system Z

RZ

SZ


N = Next state function of discrete system Z

OZ OZ = Set of Outputs of discrete system Z

RZ = Readout Function of discrete system Z

SZ = Set of States of discrete system Z

7

Abstraction Frame Sequencing Abstraction Frame (n-1)

Abstraction Frame (n)

Abstraction Frame (n+1)

Δaf n

Δaf n+1

Overall Environment Δaf n-1 © 2006 Joseph J Simpson, Mary J Simpson

8

CCFRAT Approach – Phases, Hierarchies, Content Meta Process Applies to:

Content Phases

• Technical • Management • Programmatic

• Over time • Over products • Over events

Hierarchies • Abstraction • Meta-levels • Levels of detail

Pick One Aspect from Each Axis © 2006 Joseph J Simpson, Mary J Simpson

9

Abstraction Stacks A House Consists of: Use Abstraction ‘Stacks’

Atoms

*“… house consists of several thousand pounds of carbon, some complex polymers, abount 5000 bricks, two bathrooms, and 13 ceilings…”

Molecules

Lumber, Bricks

Floors, Walls, Ceilings

Rooms * From Chapter 12, What do Classes Represent?, The C++ Programming Language, 2nd Edition, Bjarne Stroustrup, 1991.


10

Systems Engineering Conceptual Context T n-1

Tn

T n+1

Customer System Rule Based Interaction

System Design System Rule Based Interaction

System Under Design


11

Summary Sequential machines and systems provide a powerful conceptual pattern for system description and design. When vectors/groups of systems are used as inputs into a “sequential system,” each system abstraction level must be clearly defined. System abstraction frames and system abstraction stacks are used to help define and control system levels. The combination of meta-systems, system abstraction frames and system abstraction stacks provide the necessary context for the development of an executable systems engineering and design language.

 Provides context for objects and operations.  Provides framework for inter-relationship mapping © 2006 Joseph J Simpson, Mary J Simpson

12

Questions


13