A Denotational Mathematical Theory of System ...

Copyright © 2015 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Advanced Mathematics and Applications Vol. 4, 132–157, 2015

A Denotational Mathematical Theory of System Science: System Algebra for Formal System Modeling and Manipulations Yingxu Wang RESEARCH ARTICLE

International Institute of Cognitive Informatics and Cognitive Computing (ICIC), Laboratory for Denotational Mathematics, Cognitive Systems, and Software Science, Department of Electrical and Computer Engineering, Schulich School of Engineering and Hotchkiss Brain Institute, University of Calgary, 2500 University Drive, NW, Calgary, Alberta, T2N 1N4, Canada It is a great leap of humanity in abstraction and induction to represent the nature by a highly abstract and general concept known as systems. Systems are the most complicated entities and phenomena in abstract, physical, information, and social worlds across all science and engineering disciplines. A formal theory of system science is system algebra as an abstract mathematical structure for the formal treatment of general systems as well as their algebraic properties, relations, operations, and rules. This paper presents a denotational mathematical theory of system science and its applications in complex intelligent, knowledge, software, big data, and cognitive systems. A mathematical model of abstract systems is formally introduced. A set of algebraic operators on formal system relations, reproductions, and compositions is rigorously defined. System algebra provides a denotational mathematical means for modeling, specifying, and manipulating generic system structures and functions in system science, computer science, software science, knowledge science, and cognitive science.

Keywords: System Science, System Algebra, Denotational Mathematics, System Theory, Abstract Systems, System Engineering, Mathematical Models, Engineering Applications, Intelligent Systems.

1. INTRODUCTION System science is a discipline that studies the structures, mechanisms, behaviors, principles, properties, theories, and formal models of abstract systems and their applications in concrete systems in engineering and societies (von Bertalanffy, 1952; Boulding, 1956; Hall and Fagan, 1956; Ashby, 1958; Rapoport, 1962; Schedrovitzk, 1962; Ellis and Fred, 1962; Simon, 1965; Bunge, 1977; Gaines, 1984; Takahara and Takai, 1985; Klir, 1992; Wang, 2007a, 2008b, 2012c, 2015a; Wang and Zadeh et al., 2009a). Systems are the most complicated structures and mechanisms in abstract, physical, information, cognitive, brain, and social worlds across a wide range of science and engineering disciplines. Systems as a complex hyperstructure are characterized by a large set of heterogeneously configured and intricately interconnected components (Wang, 2015a). Changes at one point may affect the functioning of the entire whole due to propagations of interactions via highly coupled structures and dependent functions. The philosophy of systems is the most general scientific philosophy that intends to treat everything as a system. 132

J. Adv. Math. Appl. 2015, Vol. 4, No. 2

The universal recursive philosophy reveals that any system belongs to other supersystem and encompasses more subsystems. Although Rene Descartes (1596–1650) had presented the notion of systems by observing the interrelationships among scientific disciplines, the term of general systems was proposed by Ludwig von Bertalanffy in the 1920s (von Bertalanffy, 1952; Ellis and Fred, 1962). Max Planck described the essence of systems in Philosophy of Physics (1936) where he wrote: “Modern physics has taught us that the nature of any system cannot be discovered by dividing it into its component parts and studying each part by itself, since such a method often implies the loss of important properties of the system. We must keep out attention fixed on the whole and on the inter-connection between the parts.” He pointed out that “The whole is never equal simply to the sum of its various parts.” It has become the famous slogan of system science. It leads to Herman Haken’s assertion that: “The more science becomes divided into specialized disciplines, the more important it becomes to find unifying principles (Haken, 1977).” 2156-7565/2015/4/132/026

doi:10.1166/jama.2015.1082

Wang

A Denotational Mathematical Theory of System Science: System Algebra for Formal System Modeling and Manipulations

of general and abstract systems by denotational mathematic (Wang, 2003a, 2007b, 2008b, 2010a, b, 2012a–e, 2013, 2014b, c, 2015a–d; Wang et al., 2006, 2009b). Both the mathematical models of abstract system structures and their algebraic manipulations lead to the denotational mathematical structure known as system algebra (Wang, 2008a) and a set of general system properties and principles (Wang, 2007a, 2015a). This paper presents a revisited denotational mathematical structure of system algebra, which provides a fundamental theory for rigorously manipulating the structures and behaviors of formal systems in system science. In the remainder of this paper, the universe of discourse and abstract models of systems are formally defined in Section 2. The framework of system algebra is introduced in Section 3 on the basis of the mathematical model of formal and general systems. The relational, reproductive, and compositional operators of system algebra on formal systems are elaborated in Sections 4 though 6. A set of formal properties and rules of system algebra is summarized in Section 7. Applications of system algebra are demonstrated throughout the paper.

2. THE ABSTRACT SYSTEM THEORY OF SYSTEM SCIENCE Systems are characterized by a large set of heterogeneously configured and intricately interconnected components, relations, and functions. This section introduces the abstract system theory (Wang, 2015a) for the formal

Yingxu Wang is professor of cognitive informatics, brain science, software science, and denotational mathematics, President of International Institute of Cognitive Informatics and Cognitive Computing (ICIC, http://www.ucalgary.ca/icic/). He is a Fellow of ICIC, a Fellow of WIF (UK), a P.Eng of Canada, and a Senior Member of IEEE and ACM. He was visiting professor (on sabbatical leave) at Oxford University (1995), Stanford University (2008), UC Berkeley (2008), and MIT (2012), respectively. He received a Ph.D. in Computer Science from the Nottingham Trent University in 1998 and has been a full professor since 1994. He is the founder and steering committee chair of the annual IEEE International Conference on Cognitive Informatics and Cognitive Computing (ICCI ∗ CC) since 2002. He is founding Editor-in-Chief of Int. Journal of Cognitive Informatics and Natural Intelligence, founding Editor-in-Chief of Int. Journal of Software Science and Computational Intelligence, Associate Editor of IEEE Trans. on SMC—Systems, and Editor-in-Chief of Journal of Advanced Mathematics and Applications. Dr. Wang is the initiator of a few cutting-edge research fields such as cognitive informatics, denotational mathematics (concept algebra, process algebra, system algebra, semantic algebra, inference algebra, big data algebra, fuzzy truth algebra, and fuzzy probability algebra, visual semantic algebra, granular algebra), abstract intelligence (I), mathematical models of the brain, cognitive computing, cognitive learning engines, cognitive knowledge base theory, and basic studies across contemporary disciplines of intelligence science, robotics, knowledge science, computer science, information science, brain science, system science, software science, data science, neuroinformatics, cognitive linguistics, and computational intelligence. He has published 420+ peer reviewed papers and 29 books in aforementioned transdisciplinary fields. He has presented 30 invited keynote speeches in international conferences. He has served as general chairs or program chairs for more than 20 international conferences. He is the recipient of dozens international awards on academic leadership, outstanding contributions, best papers, and teaching in the last three decades. He is a top 2.5% scholar worldwide and top 10 at University of Calgary according to the Research Gate big data statistics. J. Adv. Math. Appl. 4, 132–157, 2015

133

RESEARCH ARTICLE

Theories of system science have evolved from classic theories (Boulding, 1956; Ashby, 1958; Rapoport, 1962; Klir, 1992) to complex systems theories (Simon, 1965; Zadeh, 1973; Klir, 1992; Hassanien et al., 2015; Wang, 2008b, 2015a; Wang and Zadeh et al., 2009a), fuzzy system theories (Zadeh, 1965, 1973, 1982; Pedrycz, 1981; Negoita, 1989; Wang, 2014b, 2014c, 2015b), chaotic system theories (Ford, 1986), and intelligent system theories (Bender, 2000; Castillo et al., 2013; Wang, 2003a, 2003b, 2009b, 2009c, 2010a, 2010b, 2012a–e, 2015c–h; Wang and Wang, 2006; Wang et al., 2006, 2009b). With regards to the metrics of system complexity, a novel type of long life-span systems is identified in (Wang, 2012c), which reveals a new form of system complexity in the time domain supplement to the conventional focus on the size-oriented complexities and magnitudes of systems. System science and philosophy are a great leap of humanity in abstraction and induction in order to represent the natural, abstract, and mental worlds. It is recognized that there is not only a general theory of diverse systems, but also a fundamental theory of the general system (von Bertalanffy, 1952; Boulding, 1956; Gaines, 1984; Wang, 2015a). The former are denoted by the framework of general theories for the formal structures, properties, behaviors, and operations across all forms of systems; while the latter are represented by the mathematical model of abstract systems (Rapoport, 1962; Zadeh, 1965, 1973, 1982; Pedrycz, 1981; Klir, 1992; Wang, 2008b, 2015a). Therefore, the major problem centric in system science is the formal modeling, representation, and manipulation


RESEARCH ARTICLE

structures and properties of abstract systems, which provides a general mathematical structure known as the hyperstructure for rigorously modeling real-world systems. 2.1. The Universe of Discourse of Formal Systems as the System of Systems The conceptual model of a system is a collection of a set of interactive entities that has a stable structure, a set of coherent functions, and a clear boundary with the external environment. The natural and abstract worlds as typical systems can be perceived as an enclosure of entities and relations (Rapoport, 1962; Zadeh, 1965; Klir, 1992; Wang, 2007a, 2008b, 2015a, g, h). Therefore, the discourse of universal systems can be defined as follows. Definition 1. Let be a finite nonempty set of components, a finite nonempty set of behaviors, a finite nonempty set of systems, and # a finite nonempty set of relations where # = × × × × . The universe of discourse of general systems, , is denoted as a triple, i.e.: ∧

= # where # = #f #i #o = #f × →

(1)

#i × → #o × → where #f , #i , and #o is called the functional, input, and output relations of a system, respectively, is a system distinct from the target system usually known as the external system, and | demotes alternative relations. The universe of discourse of general systems, , provides the overarching context for individual systems or system of systems. Any abstract or concrete system can be formally modeled in . 2.2. Mathematical Model of Abstract Systems Although there are a wide variety of concrete systems in both the natural and symbolic worlds, there is a unified model of abstract systems that constitutes the common properties of real-world systems in . Definition 2. The abstract system S in the universe of discourse of general systems is a 6-tuple, i.e.: ∧

S = C B R R R f

i

o

(2)

where • C is a finite set of components of system S, C ⊂ þ where þ denotes a power set, and represents a hyperstructural inclusion between an elemental set or dimension and a multidimensional hyperstructure. • B is a finite set of behaviors (or functions), B ⊂ þ . 134

Wang

• Rf = B × C is a finite set of functional relations, Rf ⊂ þRf . • = S1 S2 Sn is the environment of the system S encompassing a set of external systems Si ∧ Si S. • Ri = × S is a finite set of input relations, Ri ⊂ þi . • Ro = S × is a finite set of output relations, Ro ⊂ þo . The formal model of abstract systems as described in Definition 2 does not only elicit the generic model of various real-world systems, but also represent the most common attributes and properties of abstract systems. The structure of the formal system model S = C B Rf Ri Ro is illustrated in Figure 1 where C, B, , and , = Rf Ri Ro , denote the components, behaviors, environment, as well as its functional/input/output relations, respectively. The internal relations in C and B are omitted in the figure for clarity, which may be recursively reduced to a lower-level system. Example 1. An abstract system, S0 , as given in Figure 1 is formally modeled according to Definition 2 as follows: S0 = S0 C B Rf Ri Ro ⎧ ⎪ C = C1 C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = B1 B2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rf = B × C ⎪ ⎪ ⎪ ⎨ = S0 = B1 C1 B1 C2 B2 C1 B2 C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Ri = Ri1 Ri2 Ri2 = S0 ⊆ × S0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Ro = Ro Ro Ro = S ⊆ S × ⎪ 0 0 ⎪ 1 2 2 ⎪ ⎪ ⎪ ⎪ ⎩ = S

S C

Ri i

R1

Ro

C1

C2

Ro1

Θ

Ri3

Ri2

Ro3

Rf

B2

B1

Ro2

B

Fig. 1. The hyperstructural model of abstract open systems.

J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Example 2. A clock system, S1 Clock, can be formally modeled according to Definition 2 as follows: ∧

f

In S1 Clock, the static components of the clock system specified in C1 may be further refined by structural relations, which are represented by a set of structure models (SMs) and their interactions in Real-Time Process Algebra (RTPA) (Wang, 2002, 2008c, 2014a). The dynamic behaviors of the clock system specified in B1 may be further refined by behavioral relations and functional relations. The former is embodied by a set of interactive relations among the process models (PMs) in RTPA. The latter (Rf is embodied by a set of cross relations between PMs and SMs. The RTPA models for system behaviors can be implemented in any programming language in order to realize the expected functions and behaviors of the system on the basis of its structural models. Example 3. Similarly, an alarm system, S2 Alarm, can be formally modeled according to Definition 2 as follows: f

S2 Alarm C2 B2 R2 Ri2 Ro2 2 ⎧ C = Processor Keypad LED Bell ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ B2 = SetAlarm ShowAlarm CheckAlarm ⎪ ⎪ ⎪ ⎪ ⎪ Ring ReleaseAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ R2 = SetAlarmPMKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ ShowAlarmPMProcessor LED ⎪ ⎪ ⎨ CheckAlarmPMKeypad Processor = S2 ⎪ ⎪ RingPMProcessor Bell ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarmPMKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ ⎪R2 = User Keypad ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro2 = LED User Bell User ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 = User Example 4. An integrated alarm-clock system, S3 AlarmClock, can be formally modeled based on the J. Adv. Math. Appl. 4, 132–157, 2015

S3 AlarmClock ∧

= S3 C3 B3 R3 Ri3 Ro3 3 ⎧ ⎪ C3 = Processor Keypad Pulse LED Bell ⎪ ⎪ ⎪ ⎪ ⎪ B3 = SetTime Tick ShowTime SetAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ShowAlarm CheckAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarm Ring SelectFunc ⎪ ⎪ ⎪ ⎪ f ⎪ R3 = SetTimePMKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ TickPMPulse Processor ⎪ ⎪ ⎪ ⎪ ⎪ ShowTimePMProcessor LED ⎪ ⎪ ⎪ ⎨ SetAlarmPMKeypad Processor ∧ = S3 ⎪ ShowAlarmPMProcessor LED ⎪ ⎪ ⎪ ⎪ CheckAlarmPMKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ RingPMProcessor Bell ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarmPMKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ SelectFuncClock Alarm ⎪ ⎪ ⎪ ⎪ i ⎪ = User Keypad R ⎪ 3 ⎪ ⎪ ⎪ o ⎪ ⎪ R3 = LED User Bell User ⎪ ⎪ ⎪ ⎩ = User 3 f

RESEARCH ARTICLE

S1 Clock = S1 C1 B1 R1 Ri1 Ro1 1 ⎧ ⎪ C = ProcessorKeypadPulseLED ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪B1 = SetTimeTickShowTime ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ ⎪ ⎪ ⎪R1 = SetTimePMKeypadProcessor ⎪ ⎪ ⎨ TickPMPulseProcessor = S1 ShowTimePMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ri1 = UserKeypad ⊆ 1 ×S1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro1 = LEDUser ⊆ S1 ×1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 = User

results of Examples 2 and 3 according to Definition 2 as follows:

It is noteworthy in the mathematical models of formal systems where any formal system possesses the ability to associate it to other systems in via its input/output relations to the environment. This structural and functional property of formal systems allows a hierarchical architecture of systems to be coherently established among individual systems. Definition 3. A primitive system, S 0 , is a terminal system at the bottom layer of a system hierarchy that only possesses a single element in each of its configuration according to the formal model of general systems, i.e.: ∧

S 0 = C0 B0 R0 Ri0 Ro0 0 f

C0 = B0 = R0 = Ri0 = Ro0 = 0 = 1 f

(3)

Primitive systems play an important role in system deductive analysis and inductive synthesize, as well as system design and comprehension. A primitive system services as the terminal system whose structure and function have already been known in order to express higher-layer systems based on it in a general system hierarchy. Further details will be explained by Theorem 1 after the hierarchy of general systems is formally modeled. Definition 4. The environment S k of a system S k at the kth layer of a system hierarchy encompasses its parent system at a higher layer S k+1 and a set of n peer systems at the same layer (S1k S2k Snk , i.e.: S k = S k+1

n

Sik Sik

S k ∧ Sik S k

(4)

i=1

135

Wang


RESEARCH ARTICLE

where denotes the union of systems, and interactions between the system and its environment are via the sets of input relations Rik and output relations Rok . As indicated in Examples 2 through 4, users are a typical parent system or a default peer system for any arbitrary system in the formal model of system environment. Abstract systems may be classified into two categories known as the open and closed systems. Most practical and useful systems in nature are open systems that interact with external world known as the environment . However, a closed system has no interaction to its environment in terms of inputs and outputs, which will be formally described as a special case of the general abstract system in the following section. 2.3. Mathematical Model of Closed Systems A closed system can be derived as a special case of the formal model of abstract systems as given in Definition 2. Definition 5. An abstract closed system S is a 3-tuple in the universe of discourse of general systems , i.e.: ∧ S = C B Rf Ri Ro Ri0 = Ro0 = = $

= C B Rf

S

C ⊂ þ B ⊂ þ Rf ⊂ þ#

(5)

where there is no input relation (Ri nor output relation (Ro involved in the system, and the environment is treated as empty. Example 5. According to Definition 5, the primitive 0 closed system, S , is the terminal system with only a single component, behavior, and functional relation, i.e.: ∧ S 0 = C0 B0 Rf0

C0 = B0 = Rf0 = 1

Lemma 1. The pair of unique singularities of the empty system and the universal system are both a closed system in , i.e.: ⎧ ⎪ = S C B Rf C = B = Rf = $ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = $ $ $ (6) f f ⎪ ⎪ ⎪ = S C B R C = B = R = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = The relationship between a closed and an open system can be analyzed by contrasting Definitions 2 and 5. Corollary 1. The relationship between an open and a closed system in states that the open system is an extension of the closed system, and the closed system is a special case of the open system, i.e.: ⎧ ⎨S = S Ri Ro (7) ⎩ S = S\Ri Ro 136

Proof. Corollary 1 can be directly proved according to Definitions 2 and 5 as follows: S = S Ri Ro = C B Rf Ri Ro = C B Rf Ri Ro = S S = S\Ri Ro = C B Rf Ri Ro \Ri Ro = C B Rf = S

(8)

It is noteworthy that most practical systems in the real world are open rather than closed, because a general system needs to interact with external world in order to exchange energy, matter, and/or information via the environment. According to Corollary 1, a closed system is merely a special case of open systems. Therefore, all abstract systems will be denoted by the unified mathematical model of open systems as given in Definition 2 unless there is a specific need to distinguish a closed system from a general open system. 2.4. The Hierarchical and Recursive Principle of Formal Systems The key principle of the formal system theory is the generality of systems as formally described as follows. Theorem 1. The generality principle of systems states that a system S is a recursively embedded topological structure in where each kth layer of it, S k , in the system hierarchy can be represented or refined by its next layer, S k−1 , with the same structure as generally specified in Definition 2, i.e.: ∧

S=

n

R S k S k−1 S 0 = C0 B0 Rf0 Ri0 Ro0 0 k=1

= S n S n−1 S 1 S 0

(9) n

where S 0 is a known primitive system, and Ri=1 Si is an iterative calculus known as the big-R notation (Wang, 2007a), which denotes a set of recursive systems, repetitive functions, or recurrent structures (Wang, 2007a, 2008c). Proof. The recursive structure of a general system S in Theorem 1 can be inductively proved based on Definition 2 as follows: f Given S 0 = C0 B0 R0 Ri0 Ro0 0 as a primitive system and suppose all its attributes are known and terminal, f i.e., C0 = B0 = R0 = Ri0 = Ro0 = 1, then each of the higher layer systems can be inductively realized by its immediate low-layer subsystems until the top layer of the system is formally represented, i.e.: k1

S1 =

k1

R Si0 = iR=1Ci0 Bi0 0i i0 i =1 1

1

1

1

1

1

1

i1 = Ri1 Rii1 Roi1 f

J. Adv. Math. Appl. 4, 132–157, 2015

Wang


S2 =

k2

k2

2

2

=

k1

R Si1 = iR=1 iR=1 Si0 i i =1 k2

2

2 1

1

k1

R R Ci0 i Bi0 i 0i i i0 i

i2 =1 i1 =1

2 1

2 1

2 1

(10)

2 1

Sn =

kn

n

=

k2

k1

R iR=1 iR=1 Si0 i i i =1 2

kn

n

1

k2

2 1

k1

R iR=1 iR=1Ci0 i i Bi0 i i 0i i i i0 i i i =1 n

2

= S S n

n−1

n

1

2 1

n

n

2 1

S S = S 1

0

2 1

n

2 1

specify, and manipulate system designs, analyses, syntheses, refinements, and validations in a wide range of applications in system science, system engineering, cognitive informatics, cognitive computing, software engineering, cognitive systems, and intelligent systems. The formal operators of system algebra are summarized in Table I. Among the n-ary operators of system algebra, the abbreviation i represents Rii ∩ Roj∧j=i −1 and/or Roi ∩ Rij∧j=i −1 in Part 1 of the table; and = Rf Ri Ro represents the sets of functional, input, and output relations in Part 2 of the table. Each of the algebraic operators of system algebra will be formally described and elaborated in the following sections.

4. RELATIONAL OPERATIONS ON FORMAL SYSTEMS IN SYSTEM ALGEBRA

On the basis of the mathematical models of abstract systems and the theory of general systems as developed in preceding section, formal manipulations of abstract systems can be described by a mathematical structure known as system algebra. System algebra is a denotational mathematics for the formal treatment of abstract and general systems as well as their algebraic relations, operations, and rules in rigorous system modeling, analysis, and synthesis. System algebra deals with any concrete system in the real-world based on the general abstract system model and rigorous rules of formal system manipulations. Definition 6. System algebra, SA, in the universe of discourse of general systems is a triple, i.e.:

Relationships between systems in may be related, independent, equivalent, nonequivalent, subsystem, and supersystem. The relational operations of system algebra are comparative that do not change the structure, property, or status of the systems involved in relational operations by system algebra. Definition 7. The relational operators •r of system algebra encompass six associative and comparative operators for manipulating the algebraic relations between formal systems, i.e.:

∧

SA = S • = C B Rf Ri Ro •r •p •c

(11)

where • = •r •p •c denotes the sets of relational, reproductive and compositional operators, respectively, on formal systems. The framework of system algebra is illustrated as shown in Figure 2 where the type suffixes H and L denote a hyperstructure and a logical variable, respectively. A set of 16 system operators in the categories of relational, reproductive, and compositional operations on formal systems is summarized in Figure 2. In the architecture of system algebra, the relational operators encompass those of related/independent, equivalent/nonequivalent, and sub/super systems. The reproductive operators encompass those of system inheritance, tailoring, extension, substitution, and instantiation. The compositional operators encompass those of synergy, dissipation, composition, decomposition, recursive synthesis, and recursive analysis for manipulating multiple and complex systems. Each operator of system algebra represents an algebraic rule between a pair or a set of formal systems. System algebra provides a denotational mathematical means for algebraic manipulations of abstract, general, and formal systems. System algebra can be used to model, J. Adv. Math. Appl. 4, 132–157, 2015

∧

•r = ↔ = =

(12)

where each of the relational operators represent related, independent, equivalent, nonequivalent, subsystem, and supersystem, respectively. 4.1. The Related/Independent Operators on Formal Systems Definition 8. Two arbitrary systems S1 and S2 are independent, denoted by S1 S2 , iff they do not share common components and there is no mutual intersection between the sets of input and output relations in , i.e.: S1 C1 B1 R1 Ri1 Ro1 1 S2 C2 B2 R2 Ri2 Ro2 2 f

f

∧

= C1 ∩ C2 = $ ∧ Ri1 ∩ Ro2 −1 = $ ∧ Ro1 ∩ Ri2 −1 = $

(13)

where R−1 denotes an inverse relation, i.e., ∀ a ∈ S1 ∧ b ∈ S2 , ra b ∈ Ro1 ⇒ rb a ∈ Ri2 = Ro1 −1 . Otherwise, they are related, denoted by S1 S2 in , i.e.: S1 C1 B1 Rf1 Ri1 Ro1 1 ↔ S2 C2 B2 Rf2 Ri2 Ro2 2 ∧

= C1 ∩ C2 = $ ∨ Ri1 ∩ Ro2 −1 = $ ∨ Ro1 ∩ Ri2 −1 = $

(14)

Example 6. The relationship between the formal systems, S1 Clock and S2 Alarm, as defined in 137

RESEARCH ARTICLE

3. THE FRAMEWORK OF SYSTEM ALGEBRA


Wang

RESEARCH ARTICLE

Table I. Summary of formal operators of system algebra.

138

J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Relational Operations (••r)

Reproductive Operations (•p)

→

Operation Related

→

Related|L

→

Independent

→

Independent|L

→

Equivalent

=

→

Equivalent|L

→

Nonequivalent

≠

→

Inequivalent|L

→

Subsystem

→

Subsystem|L

→

Supersystem

→

Supersystem|L

s1|H s1|H s1|H s1|H

→

Inheritance

→

Tailoring

→

→

Extension

→

→

Substitute

→

s2|H s2|H s2|H s2|H

Synergy

→

R|H

Dissipation

→

R1|H, R2|H,…, R n |H

→

Composition

→

→

Decomposition

→

→

Recursive synthesis

→

→

Recursive analysis

→

s |H s1|H, s2|H, …, sn|H s |H s0|H , s1|H, …, sn|H

R1|H, R 2|H, …, R n |H → RH → Compositional Operations (•c)

Fig. 2.

S1|H,

s2|H, …, sn|H s|H S0|H, S1|H, …, Sn |H s|H

Operator ↔

⇒

→

The architecture of system algebra.

Examples 2 and 3 can be rigorously determined according to Definition 8 as follows: S1 ClockC1 ∩ S2 AlarmC2 = Processor Keypad Pulse LED ∩

Example 7. According to Definition 9, the equivalency between the systems of clock and alarm, S1 Clock and S2 Alarm, as given in Examples 2 and 3 can be formally determined as follows: C1 = Processor Keypad Pulse LED =

Processor Keypad LED Bell

C2 = Processor Keypad LED Bell

= Processor Keypad LED = $

⇒ S1 C1 B1 Rf1 Ri1 Ro1 1

⇒ S1 Clock ↔ S2 Alarm

= S2 C2 B2 R2 Ri2 Ro2 2 f

That is, S1 Clock and S2 Alarm are related but not independent. It is obvious that related and independent systems are mutually exclusive. That is, if S1 ↔ S2 , then ¬S1 S2 ; and vice versa. It is noteworthy that there is no closed system that is related to any external system by definition. 4.2. The Equivalent/Nonequivalent Operators on Formal Systems Definition 9. Two related systems S1 and S2 are equivalent, denoted by S1 = S2 , iff all sets of components, behaviors, and functional relations are identical in , respectively, i.e.:

That is, S1 Clock and S2 Alarm are nonequivalent. 4.3. The Subsystem/Supersystem Operators on Formal Systems Definition 10. A system S is a subsystem of a system S, denoted by S S, if both sets of components and behaviors are subsets of the counterparts of S in , respectively, i.e.:

S C B Rf Ri Ro SC B Rf Ri Ro ∧

= C ⊂ C ∧ B ⊂ B

(17)

Otherwise, they are nonequivalent, denoted by S1 = S2 , if any pair of the sets of components or behaviors is different.

where the dominative condition for determining a subsystem is that C ⊂ C. Definition 11. A system S is a supersystem of system S , denoted by S S , if both sets of components and behaviors are supersets of the counterparts of S in , respectively, i.e.:

S1 C1 B1 Rf1 Ri1 Ro1 1 = S2 C2 B2 Rf2 Ri2 Ro2 2

SC B Rf Ri Ro S C B Rf Ri Ro

S1 C1 B1 Rf1 Ri1 Ro1 1 = S2 C2 B2 Rf2 Ri2 Ro2 2 ∧

= C1 = C2 ∧ B1 = B2

∧

= C1 = C2 ∨ B1 = B2 J. Adv. Math. Appl. 4, 132–157, 2015

(15)

(16)

∧

= C ⊃ C ∧ B ⊃ B

(18) 139

RESEARCH ARTICLE

s1|H, s2|H s1|H, s2|H s1|H, s2|H s1|H, s2|H s1|H, s2|H s1|H, s2|H


Example 8. According to Definitions 10 and 11, the inclusive relationship between the systems of Clock and Alarm, S1 Clock and S2 Alarm, as given in Examples 2 and 3 can be formally determined as follows: C1 ⊂ C2 ∧ C2 ⊂ C1 ∧ B1 ⊂ B2 ∧ B2 ⊂ B1 ⇒ S1 C1 B1 Rf1 Ri1 Ro1 1 S2 C2 B2 R2 Ri2 Ro2 2 ∧ f

S1 C1 B1 Rf1 Ri1 Ro1 1

RESEARCH ARTICLE

S2 C2 B2 Rf2 Ri2 Ro2 2 That is, S1 Clock and S2 Alarm are neither mutually as a subsystem nor a supersystem. Example 9. The systems of Clock and Alarm Clock, S1 Clock and S3 AlarmClcok, as given in Examples 2 and 4 possess a mutual subsystem or supersystem relation that can be verified according to Definitions 10 and 11 as follows: C1 ⊂ C3 ∧B1 ⊂ B3 ⇒ S1 C1 B1 R1 Ri1 Ro1 1 S3 C3 B3 R3 Ri3 Ro3 3 or f

f

⇒ S3 C3 B3 R3 Ri3 Ro3 3 S1 C1 B1 R1 Ri1 Ro1 1 f

f

That is, S1 Clock is a subsystem of S3 AlarmClock, or inversely, S3 AlarmClock is a supersystem of S1 Clock. So do S2 Alarm and S3 AlarmClock.

5. REPRODUCTIVE OPERATIONS ON FORMAL SYSTEMS IN SYSTEM ALGEBRA Reproductive operations on formal systems in are a category of clone operations in system algebra that derives a similar or variant system on the basis of an existing parent system. Once a system is reproduced based on the given system, both the parent and derived systems become related in general, and equivalent in particular when the operation is a direct inheritance. Definition 12. The reproductive operators •p of system algebra encompass four clone operators for deriving similar systems based on existing ones in , i.e.: ∧

−

+

∼

•p = ⇒ ⇒ ⇒ ⇒

(19)

where the reproduction operators represent system inheritance, tailoring, extension, and substitute, respectively. The reproductive operators of systems algebra are formally described in the following subsections. 5.1. The Inherence Operator on Formal Systems Definition 13. The inheritance of a formal system Sh from a given parent system S, denoted by 140

Wang

S ⇒ Sh , is a direct clone of Sh based on S in , i.e.: SCBRf Ri Ro ⇒ Sh Ch Bh Rfh Rih Roh h ⎧ Ch = C ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bh = B ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ ⎨Rh = Rf ∧ (20) = Sh ⎪ ⎪ ⎪Rih = Ri ∪SSh ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Roh = Ro ∪Sh S ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h = ' SCBRf Ri = Ri ∪Sh SRo = Ro ∪SSh where ' reads ‘in parallel’ which denotes that the inheritance operation mutually affects both the parent and inherited systems by newly established input/output relations between S and Sh in both directions. Example 10. According to Definition 13, an instance of the clock system S11 Clock 11 that inherits the existing system S1 Clock as given in Example 2, denoted by S1 Clock ⇒ S11 Clock 11 , reproduces a clone of S1 as follows: S1 C1 B1 R1 Ri1 Ro1 1 ⇒ f

S11 Clock 11 C11 B11 Rf11 Ri11 Ro11 11 ⎧ C11 = C1 = ProcessorKeypadPulseLED ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B11 = B1 = SetTimeTickShowTime ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f f ⎪ R11 = R1 = SetTimePMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ TickPMPulseProcessor ⎪ ⎪ ⎪ ⎨ ShowTimePMProcessorLED = S11 i i ⎪ R11 = R1 ∪S1 S11 ⎪ ⎪ ⎪ ⎪ ⎪ = UserKeypad∪S1 S11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro11 = Ro1 ∪S11 S1 ⎪ ⎪ ⎪ ⎪ ⎪ = LEDUser∪S11 S1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 11 = User ' S1 C1 B1 Rf1 Ri1 = Ri1 ∪S11 S1 o Ro 1 = R1 ∪S1 S11 1

The pairwise system inherence as modeled in Definition 13 can be generally extended to n-ary inherences. Definition 14. The multiple inheritance of a system S0 from n parent systems S1 S2 Sn , denoted by J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Rni=1 Si ⇒ni=1 S0 , is a set of n pairwise inherence operations between S0 and each Si in , i.e.: n

n

f R Si Ci Bi Rfi Rii Roi i ⇒ S0 C0 B0 R0 Ri0 Ro0 0 i=1 i=1

∧

= S0

⎧ n ⎪ ⎪ C = Ci ⎪ 0 ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ B0 = Bi ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ f f ⎪ ⎪ ⎨R0 = Ri i=1

(21)

i=1

'R

n f i i i=1 Si Ci Bi Ri Ri = Ri ∪S0 Si

' S1 C1 B1 Rf1 Ri1 = Ri1 ∪ S12 S1

o Ro i = Ri ∪Si S0 i

5.2. The Tailoring Operator on Formal Systems Definition 15. The tailoring of a system St from a − parent system S, denoted by S ⇒ St , is a special inheritance from S according to a given tailoring strategy that removes a specific subset of components Ct Ct ⊂ SC, and/or of behaviors Bt Bt ⊂ SB in , i.e.: −

SCBRf Ri Ro ⇒ St Ct Bt Rft Rit Rot t

∧

= St

Ct ⊂ SC ∧Bt ⊂ SB ⎧ Ct = C\Ct ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Bt = B\Bt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Rft = Rf \Bt ×Ct ∪Bt ×C ∪B ×Ct

(22)

' SCBRf Ri = Ri ∪St SRo = Ro ∪SSt where new associations between both systems are established in parallel via their I/O relations. Example 11. Given S1 Clock and a tailoring strat = Keypad ⊂ S1 C1 and B12 = SetTime ⊂ egy as C12 S1 B1 . According to Definition 15, the tailoring operation − S1 Clock ⇒ S12 Clock 12 reproduces a modified clone S12 Clock 12 as follows: −

⇒ S12 Clock 12 C12 B12 Rf12 Ri12 Ro12 12 J. Adv. Math. Appl. 4, 132–157, 2015

where the tailored component(s), behavior(s), and functional relation(s) are indicated by xxx, and the newly established input/output relations between the parent and tailored systems are shown by S1 S12 and S12 S1 , respectively. The pairwise system tailoring as modeled in Definition 15 can be generally extended to n-ary tailoring operations. Definition 16. The multiple tailoring of a system S0 from n parent systems S1 S2 Sn , denoted by ¯ n S , is a set of n pairwise tailoring operations Rni=1 Si ⇒ i=1 0 between S0 and each Si in , i.e.: n

n

⎪ ⎪ Rit = Ri ∪SSt \Ct ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rot = Ro ∪St S\Ct ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t =


o Ro 1 = R1 ∪ S1 S12 1

¯ R Si Ci Bi Rfi Rii Roi i ⇒ i=1

i=1 n n f i o S0 C0 B0 R0 R0 R0 0 Ci ⊂ Ci ∧ Bi ⊂ Bi i=1 i=1 ⎧ n ⎪ ⎪ = C \C C ⎪ 0 i i ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪B0 = Bi \Bi ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ f f ⎪ ⎪ ⎨R0 = Ri \Bi ×Ci ∪Bi ×Ci ∪Bi ×Ci ∧ i=1 = S0 n ⎪ ⎪ ⎪ Ri0 = Rii ∪Si S0 \i Ci ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ o ⎪ R = Roi ∪S0 Si \Ci i ⎪ ⎪ 0 ⎪ ⎪ i=1 ⎪ ⎪ n ⎪

R

⎪ ⎪ ⎩0 =

n

R

(23)

i

i=1

' R Si Ci Bi Rfi Rii = Rii ∪S0 Si i=1

o Ro i = Ri ∪Si S0 i

141

RESEARCH ARTICLE

n ⎪ ⎪ ⎪ Ri0 = Rii ∪Si S0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ o ⎪ = Roi ∪S0 Si R ⎪ 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ ⎩0 = i

C12 = Keypad ⊂ C1 B12 = SetTime ⊂ B1 ⎧ ⎪ C = Processor Keypad Pulse LED ⎪ ⎪ 12 ⎪ ⎪ B12 = SetTime Tick ShowTime ⎪ ⎪ ⎪ ⎪ ⎪Rf12 = SetTimePM (Keypad, Processor) ⎪ ⎪ ⎪ ⎨ TickPMPulse Processor = S12 ⎪ ShowTimePMProcessor LED ⎪ ⎪ ⎪ i ⎪ ⎪ = (User, Keypad) ∪ S1 S12 R ⎪ 12 ⎪ ⎪ o ⎪ = LED User ∪ S12 S1 R ⎪ 12 ⎪ ⎪ ⎩ 12 = 1 = User ⎧ C12 = Processor Pulse LED ⎪ ⎪ ⎪ ⎪ ⎪ = Tick ShowTime ⎪ ⎪B12 ⎪ f ⎪ ⎪ R ⎨ 12 = TickPMPulse Processor = S12 ShowTimePMProcessor LED ⎪ ⎪ i ⎪ R12 = S1 S12 ⎪ ⎪ ⎪ ⎪ ⎪Ro12 = LED User S12 S1 ⎪ ⎪ ⎩ 12 = User


5.3. The Extension Operator on Formal Systems Definition 17. The extension of a system Se from + the parent system S, denoted by S ⇒ Se , is a special inheritance that creates Se based on S with an additional subset of components Ce Ce ⊂ SC and/or of behaviors Be Be ⊂ SB in , i.e.: +

RESEARCH ARTICLE

SCBRf Ri Ro ⇒ Se Ce Be Rfe Rie Roe e Ce ⊂ SC ∧Be ⊂ SB ⎧ Ce = C ∪Ce ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Be = B ∪Be ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨Rfe = Rf ∪Be ×Ce ∪Be ×Ce ∪Be ×Ce ∧ (24) = Se ⎪ i i ⎪ ⎪Re = R ∪SSe ∪e Ce ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Roe = Ro ∪Se S∪Ce e ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ e = ' SCBRf Ri = Ri ∪Se S Ro = Ro ∪SSe

Wang

⎧ ⎪ C13 = ProcessorKeypadPulseLEDButtons ⎪ ⎪ ⎪ ⎪ ⎪B13 = SetTimeTickShowTimeSelectFunc ⎪ ⎪ ⎪ ⎪ Rf13 = SetTimePMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ TickPMPulseProcessor ⎪ ⎪ ⎪ ⎨ ShowTimePMProcessorLED = S13 ⎪ SelectFuncButtonsProcessor ⎪ ⎪ ⎪ ⎪ ⎪ Ri13 = UserKeypad UserButtons ⎪ ⎪ ⎪ ⎪ S1 S13 ⎪ ⎪ ⎪ ⎪ ⎪Ro13 = LEDUserS13 S1 ⎪ ⎪ ⎪ ⎩ = User 13 ' S1 C1 B1 Rf1 Ri1 = Ri1 ∪S13 S1 o Ro 1 = R1 ∪S1 S13 1

where the extended component(s), behavior(s), and functional relation(s) are indicated by xxx, and the newly established relations between the parent and extended systems are shown by User Buttons S1 S13 and S13 S1 , respectively. The pairwise system extension as modeled in Definition 17 can be generally extended to n-ary extensions. Definition 18. The multiple extension of a formal system S0 from n parent systems S1 S2 , and Sn , denoted + n

where new associations between the two systems are established in parallel via their I/O relations. Example 12. Given S1 Clock and the subsets of com = Buttons ponents and behaviors to be extended as C13 and B13 = SelectFunc. According to Definition 17, the

by Ri=1 Si ⇒i=1 S0 , is a set of n pairwise extensions operations between S0 and each Si in , i.e.:

extension operation S1 Clock ⇒ S13 Clock 13 as follows:

i=1 S0 C0 B0 Rf0 Ri0 Ro0 0 n n Ci ⊂ Si Ci ∧ Bi ⊂ Si Bi i=1 i=1 ⎧ n ⎪ ⎪ ⎪ C0 = Ci ∪ Ci ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ B = Bi ∪ Bi ⎪ 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ f ⎪ = Rfi ∪ Bi × Ci ∪ Bi × Ci R ⎪ ⎪ 0 ⎪ ⎨ i=1 ∧ = S0 ∪Bi × Ci ⎪ n ⎪ ⎪ ⎪ ⎪ Ri0 = Rii ∪ Si S0 ∪ i Ci ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ o ⎪ = Roi ∪ S0 Si ∪ Ci i R ⎪ 0 ⎪ ⎪ i=1 ⎪ ⎪ n ⎪

+

f

S1 C1 B1 R1 Ri1 Ro1 1 +

f ⇒ S13 C13 B13 R13 Ri13 Ro13 13 C13 = Buttons ⊂ S1 C1 = SelectFunc ⊂ S1 B1 B13 ⎧ ⎪ ⎪C13 = C1 ∪C13 ⎪ ⎪ ⎪ ⎪ = ProcessorKeypadPulseLEDButtons ⎪ ⎪ ⎪ ⎪ = B1 ∪B13 B ⎪ 13 ⎪ ⎪ ⎪ ⎪ = SetTimeTickShowTimeSelectFunc ⎪ ⎪ ⎪ f ⎪ ⎪ R13 = Rf1 ∪B13 ×C13 ∪B13 ×C1 ∪B1 ×C13 ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimePMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ TickPMPulseProcessor ⎨ ∧ = S13 ShowTimePMProcessorLED ⎪ ⎪ ⎪ SelectF uncButtonsProcessor ⎪ ⎪ ⎪ ⎪ ⎪Ri13 = Ri1 ∪S1 S13 ∪13 C13 ⎪ ⎪ ⎪ ⎪ ⎪ = UserKeypad∪ ⎪ ⎪ ⎪ ⎪ ⎪ UserButtonsS1 S13 ⎪ ⎪ ⎪ o o ⎪ = R R ⎪ 1 ∪S13 S1 ∪C13 13 ⎪ 13 ⎪ ⎪ ⎪ = LEDUserS13 S1 ⎪ ⎪ ⎪ ⎩ = = User 13 1

142

n

n

R

i=1

Si Ci Bi Rfi Rii Roi i

R

⇒

R

⎪ ⎪ ⎪ ⎩0 =

n

n +

(25)

i

i=1

' R Si Ci Bi Rfi Rii = Rii ∪ S0 Si i=1

o Ro i = Ri ∪ Si S0 i

5.4. The Substitution Operator on Formal Systems Definition 19. The substitution of a system Sb from ∼ the parent system S, denoted by S ⇒ Sb , is a flexible inheritance that creates Sb based on S according to a given J. Adv. Math. Appl. 4, 132–157, 2015

Wang


substitution strategy where the specific inherited subset of components C C ⊂ SC and/or of behaviors B B ⊂ SB, are replaced by the given counterparts, C ⊂ SC and/or B ⊂ SB in , respectively, i.e.: ∼

SC B Rf Ri Ro ⇒ Sb Cb Bb Rfb Rib Rob b

b

' SC B Rf Ri = Ri ∪ Sb S Ro = Ro ∪ S Sb where new associations between the two systems are established in parallel via their I/O relations. Example 13. Given S1 Clock and a substitute strategy for replacing components C = Keypad by C = Buttons, and behaviors B = SetTime by B = SelectFunc, respectively. According to Definition 19, ˜ 14 Clock 14 reprothe substitution operation S1 Clock⇒S duces a modified clone S14 Clock 14 as follows: f

S1 C1 B1 R1 Ri1 Ro1 1 ∼

⇒S14 C14 B14 Rf14 Ri14 Ro14 14 = Keypad ⊂ C1 B14 = SetTime ⊂ B1 C14 C14 = Buttons ⊂ CB14 = SelectFunc ⊂ B ⎧ C14 = C1 ∪C14 \C14 ⎪ ⎪ ⎪ ⎪ ⎪ = ProcessorKeypadPulseLEDButtons ⎪ ⎪ ⎪ ⎪ ⎪ \B14 B14 = B1 ∪B14 ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimeTickShowTime SelectFunc ⎪ ⎪ ⎪ f f ⎪ ⎪ = R ⎪ 14 R1 ∪B14 ×C14 ∪B14 ×C1 ∪B1 ×C14 \ ⎪ ⎪ ⎪ B14 ×C1 ∪B1 ×C14 ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimePM(Keypad, Processor) ⎪ ⎪ ⎪ ⎨ TickPMPulseProcessor ∧ = S14 ⎪ ShowTimePMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ SelectFuncButtonsProcessor ⎪ ⎪ ⎪ i i ⎪ ⎪ = R R 14 1 ∪S1 S14 ∪14 C14 \14 C14 ⎪ ⎪ ⎪ ⎪ = (User, Keypad))UserButtons ⎪ ⎪ ⎪ ⎪ ⎪ S1 S14 ⎪ ⎪ ⎪ ⎪ o o ⎪ = R R ⎪ 14 1 ∪S14 S1 ∪C14 14 \C14 14 ⎪ ⎪ ⎪ ⎪ = LEDUserS14 S1 ⎪ ⎪ ⎩ 14 = 1 = User

J. Adv. Math. Appl. 4, 132–157, 2015

' S1 C1 B1 Rf1 Ri1 = Ri1 ∪S14 S1 o Ro 1 = R1 ∪S1 S14 1

where the tailored or extended component(s), behavior(s), and functional relation(s) are indicated by xxx or xxx, respectively; and the newly established relations between the parent and tailored systems are shown by User Buttons S1 S14 and S14 S1 . The pairwise system substitution as modeled in Definition 19 can be generally extended to n-ary substitutions. Definition 20. The multiple substitution of a system S0 from n parent systems S1 S2 , and Sn , denoted by ∼ n Rni=1 Si ⇒i=1 S0 , is a set of n pairwise substitution operations between S0 and each Si in , i.e.: n

R Si Ci Bi Rfi Rii Roi i i=1 n ∼

⇒ S0 C0 B0 Rf0 Ri0 Ro0 0 i=1

n

n

R Ci ⊂ Si Ci ∧ R Bi ⊂ Si Bi i=1 i=1 n n R Ci ⊂ Si Ci ∧ R Bi ⊂ Si Bi i=1 i=1

⎧ n ⎪ ⎪C0 = Ci ∪Ci \Ci ⎪ ⎪ ⎪ ⎪ i=1 ⎪ n ⎪ ⎪ ⎪ ⎪B0 = Bi ∪Bi \Bi ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ f ⎪ ⎪R0 = Rfi ∪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎨ Bi ×Ci ∪Bi ×Ci ∪Bi ×Ci ∧ = S0 ⎪ \Bi ×Ci ∪Bi ×Ci ∪Bi ×Ci ⎪ ⎪ ⎪ n ⎪ ⎪ i ⎪ R0 = Rii ∪Si S0 ∪i Ci \i Ci ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ o o ⎪ = Ri ∪S0 Si ∪Ci i \Ci i R ⎪ 0 ⎪ ⎪ ⎪ i=1 ⎪ n ⎪ ⎪ ⎪ ⎪ ⎩0 = i

n

(27)

i=1

' R Si Ci Bi Rfi Rii = Rii ∪S0 Si i=1 o Ro i = Ri ∪Si S0 i

6. COMPOSITIONAL OPERATIONS ON FORMAL SYSTEMS IN SYSTEM ALGEBRA Compositional operations on formal systems in are a category of constructive operations in system algebra 143

RESEARCH ARTICLE

Cb ⊂ SC ∧ Bb ⊂ SB Cb ⊂ SC ∧ Bb ⊂ SB ⎧ ⎪ ⎪ ⎪Cb = C ∪ Cb \Cb ⎪ ⎪ ⎪ ⎪ Bb = B ∪ Bb \Bb ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Rf = Rf ∪ B × C ∪ B × C ⎪ ⎪ b b b ⎪ b ⎪ ⎪ ⎪ ⎨ (26) ∪B × Cb ∧ = Sb ⎪ ⎪ \Bb × Cb ∪ Bb × C ∪ B × Cb ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rib = Ri ∪ S Sb ∪ b Cb \b Cb ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rob = Ro ∪ Sb S ∪ Cb b \Cb b ⎪ ⎪ ⎪ ⎪ ⎩ =

⎧ ⎪ C14 = ProcessorPulseLEDButtons ⎪ ⎪ ⎪ ⎪ B14 = TickShowTimeSelectFunc ⎪ ⎪ ⎪ ⎪ ⎪Rf14 = TickPMPulseProcessor ⎪ ⎪ ⎪ ⎨ ShowTimePMProcessorLED = S14 ⎪ SelectFuncButtonsProcessor ⎪ ⎪ ⎪ ⎪ ⎪Ri14 = UserButtonsS1 S14 ⎪ ⎪ ⎪ o ⎪ ⎪ ⎪R14 = LEDUserS14 S1 ⎪ ⎩ 14 = User


RESEARCH ARTICLE

for building complex supersystems based on simple ones. Formal system synergy and composition are algebraic operations at the same layer on a certain system hierarchy. However, the recursive system operation integrates multiple subsystems from the bottom up via system syntheses. Inversely, a supersystem may be hierarchically decomposed into multiple subsystems from the top down via system analyses. Definition 21. The compositional operators •c of system algebra encompass a set of six synthetic and analytic operations for creating or decomposing complex systems based on existing ones according to certain algebraic rules in , i.e.: ∧ (28) •c = * ⇑ ⇓ where the compositional operators represent system synergy/dissipation, composition/decomposition, and recursive synthesis/analysis, respectively. The three fundamental forms of system compositions/decompositions are formally described in the following subsections. A hierarchical synthesis of systems is recursively defined based on lower layer system compositions and relational synergies. The compositional operations of system algebra are interactive, which affect the input/output relations of the original component systems as well as the newly generated system. 6.1. The Synergy of Formal Systems Systems are widely needed due to the advantage of system gains via system synergy in physical, abstract, and social worlds. The synergic effect of systems is a unique property of systems that is not possessed by any of its components before it is composed into the system. This effect can be formally explained by the incremental union of two sets of relations into a coherent system (Wang, 2007a, 2015e), which results in a set of newly generated relations as shown in Figure 3 as highlighted by dashed lines beyond the traditional simple conjunction of relations. Therefore, a key operation identified in relational and system synergy and composition is the incremental union of relations determined by a bi-Cartesian product between two sets of components in given systems (Wang, 2015a, 2015e). Definition 22. A bidirectional Cartesian product, shortly bi-Cartesian product, between two sets of elements X and Y , denoted by X ⊗ Y , is a set of ordered

S32

s21

S33 S2

S3

s22 R23= R2

s33

R3

Fig. 3. System synergy via of the incremental union of relations.

144

pairs of joint elements from each set in both directions, Y Y X RX i=1 Rj=1 xi yj and Rj=1 Ri=1 yj xi , i.e.: ∧

X ⊗Y = X ×Y ∪Y ×X X Y = R R xi yj xi ∈ X ∧ yj ∈ Y ∪ i=1 j=1

Y

X

R Ryj xi xi ∈ X ∧ yj ∈ Y j=1 i=1

(29)

where the size of the bi-Cartesian product is X ⊗ Y = 2X × Y = 2X • Y . Definition 23. The synergy of two systems S1 and S2 , denoted by S = S1 S2 , yields the sets of newly generated behaviors SB = B1 B2 and functional relations SRf = Rf1 Rf2 via the mechanism of incremental unions of systems in , i.e.: SC B Rf Ri Ro = S1 C1 B1 R1 Ri1 Ro1 1 f


⎧ ⎪ SB = S1 B1 S2 B2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = S1 B1 ∪ S2 B2 ∪ S B12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = C1 × C1 ∪ C2 × C2 ∪ C1 ⊗ C2 ⎪ ⎪ ⎪ ⎨ ∧ = SRf = S1 Rf1 S2 Rf2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = S1 Rf1 ∪ S2 Rf2 ∪ S Rf12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = B1 × C1 ∪ B2 × C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∪B1 × C2 ∪ B2 × C1

(30)

where SR represents system behaviors SB or functional relations SRf in which the R represents the dimension R of the hyperstructure S. The set of newly generated global relations and its size by the incremental union of system synergy are determined by: ⎧ ⎨ R12 = C1 ⊗ C2 (31) ⎩ R12 = C1 ⊗ C2 = 2C1 • C2 It is noteworthy where R12 ⊂ R1 R2 but R12 ⊂ R1 ∧ R12 ⊂ R2 . Example 14. Given two relations R2 S2 = s21 s22 and R3 S3 = s31 s32 s33 as shown in Figure 3, the expected set of relations for the incremental union of S23 , R23 = R2 R3 , in system synergy is determined according to Definition 23 as follows: R23 = R2 + S3 + R23

S31

⇒

S31 s22

s32

s21

Wang

= S2 2 + S3 2 + 2S2 • S3 = S23 2 S23

= 22 + 32 + 22 • 3 = 52 = 25 where R23 = 2S2 • S3 = 22 • 3 = 12 J. Adv. Math. Appl. 4, 132–157, 2015

Wang


R23 = R2 R3 = R2 ∪ R3 ∪ R23

generated between the subsystem and other counterparts in the system in , i.e.:

= S2 × S2 ∪ S3 × S3 ∪ S2 ⊗ S3

S C B Rf Ri Ro = SC B Rf Ri Ro

= s21 s21 s21 s22 s22 s21 s22 s22 ∪ s31 s31 s31 s32 s31 s33 s32 s31 s32 s32 s32 s33 s33 s31 s33 s32 s33 s33 ∪ s21 s31 s21 s32 s21 s33 s22 s31 s22 s32 s22 s33 s31 s21 s32 s21 s33 s21 s31 s22 s32 s22 s33 s22

∧

SR =

⎪ ⎪ S Rf = SRf S0 Rf0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SRf \S0 Rf0 ∪ S Rf0S ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = SRf B × C\B0 × C0 ∪ B0 × C

where the relations refer to the behaviors S B = B B1 f and functional relations S Rf = Rf R1 of the system. The mechanism of system dissipation is determined by the decremental disunion of relations in a system (Wang, 2015a, 2015e), which is an elimination of both R2 S2 and the previously generated incremental relations R12 S1 S2 from RS. As illustrated in Figure 3 in a inversed direction, a decremental disunion of relational decomposition eliminates the relations originally gained in an incremental composition, R12 = 2S1 • S2 , once another set of relations is removed from the system. Example 15. Reusing the layout as shown in Figure 3, the expected result of a decremental disunion R2 = R23

R3 in system dissipation is derived according to Definition 25 as follows:

n

Si Ri i=1

= S1 R1 S2 R2 S3 R3 Sn Rn

R2 = R23 S2 S3 R3 S3 = R23 − R3 − R23 = 25 − 9 − 12 = 4

(32)

∧

where SR represents system behaviors SB or functional relations SRf .

R2 = R23 R3 = R23 \R3 ∪ R23 S2 S2 = S2 × S2 = R R xi xj xi xj ∈ S2 i=1 j=1

6.2. The Dissipation of Formal Systems The dissipation of formal systems is an inverse operation of system synergy by decremental disunion of formal relations of the system. A decremental disunion of two sets of relations in a system results in the dissolution of related set of relations and the disappearance of previously obtained incremental relations as shown in Figure 3 by dash lines. The decremental disunion of relations indicates the difference between the dynamic disunion and the traditional static difference operation of relations. Definition 25. The dissipation of a formal system S by eliminating a subsystem S2 , denoted by S = S S0 , is a decremental disjunction of formal relations for both local relations in the subsystem and incremental relations J. Adv. Math. Appl. 4, 132–157, 2015

= s21 s21 s21 s22 s22 s21 s22 s22 The pairwise dissipation of system relations and behaviors as modeled in Definition 25 can be generally extended to n-ary system dissipations as follows. Definition 26. The general dissipation of n sets of n ∧ system relations, denoted by S R = SR i=1 Si Ri , is a series of pairwise decremental disunion of given relations in as follows: n

SR = Si Ri Si Ri SR ∧

i=1

(34)

= SR S1 R1 S2 R2 Sn Rn where SR represents system behaviors SB or functional relations SRf . 145

RESEARCH ARTICLE

The result of Example 14 indicates a fast increase in both function and complexity gains during the incremental union of system relations, which is a fundamental property of general systems. It also demonstrates that the physical meaning of a bi-Cartesian product in system synergy is the generation of a set of undirected and fully connected relations between the components in both systems, which are symmetric but irreflexive. The pairwise synergy of system relations and behaviors as modeled in Definition 23 can be generally extended to n-ary system synergies. Definition 24. The general synergy of n sets of sysn ∧ tems relations, denoted by SR = i=1 Si Ri , is a series of pairwise incremental union of given relations in as follows:

∧

=

S0 C0 B0 Rf0 Ri0 Ro0 0 S0 S ⎧ S B = SB S0 B0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SB\S0 B0 ∪ S B0S ⎪ ⎪ ⎪ ⎪ ⎪ (33) ⎪ ⎪ ⎨ = SBC\C0 × C0 ∪ C0 ⊗ C\C0


RESEARCH ARTICLE

6.3. The Composition of Formal Systems On the basis of the algebraic operation of system synergy expressed by the incremental union of system relations and functions, system composition can be rigorously defined by a set of hybrid operations of normal and incremental unions of the elements among target systems. Definition 27. The composition of two formal systems S1 and S2 , denoted by S = S1 * S2 , yields a supersystem S in by incremental unions of the sets of behaviors and relations, B = B1 B2 and Rf = Rf1 Rf2 , as well as conjunctions of the sets of components, I/O relations, and environments, respectively, i.e.: ∧

SCBRf Ri Ro = S1 C1 B1 Rf1 Ri1 Ro1 1 *S2 C2 B2 R2 Ri2 Ro2 2 ⎧ ⎪ C = C1 ∪C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = B1 B2 = B1 ∪B2 ∪ B12 ⊆ C1 ⊗C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Rf = Rf1 Rf2 ⎪ ⎪ ⎪ ⎨ =S = Rf1 ∪Rf2 ∪ Rf12 ⊆ B1 ×C2 ∪B2 ×C1 (35) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ri = Ri1 ∪Ri2 ∪S1 SS2 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Ro = Ro ∪Ro ∪SS SS ⎪ 1 2 ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎩ = 1 ∪2 f

2

' R Si Ci Bi Rfi Rii = Rii ∪SSi i=1

o Ro i = Ri ∪Si Si

where the operator of relational and functional synergy ( ) is given in Definition 23, and the newly created associations between the supersystem and original subsystems are established in parallel via their I/O relations. It is noteworthy that a system compositions do not only create incremental usage such as new behaviors and relations, but also results in additional complexity when multiple subsystems are composed into a supersystem. Example 16. A supersystem SAlarmClock can be composed using the subsystems S1 Clock and S2 Alarm as given in Examples 2 and 3 according to Definition 27 as follows: SAlarmClock CBRf Ri Ro = S1 Clock C1 B1 Rf1 Ri1 Ro1 1 * 146

Wang

f

S2 Alarm C2 B2 R2 Ri2 Ro2 2 ⎧ C = C1 ∪C2 ⎪ ⎪ ⎪ ⎪ ⎪ = ProcessorKeypadPulseLED∪ ⎪ ⎪ ⎪ ⎪ ⎪ ProcessorKeypadLEDBell ⎪ ⎪ ⎪ ⎪ ⎪ = ProcessorKeypadPulseLEDBell ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = B1 B2 = B1 ∪B2 ∪ B12 ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimeTickShowTime∪ ⎪ ⎪ ⎪ ⎪ ⎪ SetAlarmShowAlarmCheckAlarmRing ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarm∪ SelectFunc ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimeTickShowTimeSetAlarm ⎪ ⎪ ⎪ ⎪ ShowAlarmCheckAlarmRing ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarm SelectFunc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ f f f f f ⎪ f ⎪ ⎪R = R1 R2 = R1 ∪R2 ∪ R12 ⎪ ⎪ ⎪ ⎨ = SetTimePMKeypadProcessor =S TickPMPulseProcessor ⎪ ⎪ ⎪ ⎪ ShowTimePMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ SetAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ ShowAlarmPMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ CheckAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ RingPMProcessorBell ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ SelectFuncPMClockAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ri = Ri1 ∪Ri2 ∪Ri12 ⎪ ⎪ ⎪ ⎪ = UserKeypadS1 SS2 S ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro = Ro1 ∪Ro2 ∪Ro12 ⎪ ⎪ ⎪ ⎪ ⎪ = LEDUserBellUserSS1 SS2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = 1 ∪2 = User ' S1 C1 B1 Rf1 Ri1 = Ri1 ∪SS1 o Ro 1 = R1 ∪S1 S1

' S2 C2 B2 Rf2 Ri2 = Ri2 ∪SS2 o Ro 2 = R2 ∪S2 S2

where the newly created behaviors B12 and incremental relations Rf12 yielded in the composition are identified as SelectFunc and SelecFunctPMClock Alarm, respectively. The updated input relations are S1 S and S2 S, while the updated output relations are S S1 and S S2 . The incremental counterparts updated in S1 Clock and S2 Alarm are shown in parallel. The pairwise composition of systems as modeled in Definition 27 can be generally extended to n-ary system compositions. Definition 28. The multiple composition of subsys tems, denoted by S = ni=1 Si , is an n-ary composition that J. Adv. Math. Appl. 4, 132–157, 2015

Wang


creates a supersystem S based on the set of given subsystems in , i.e.:

(36)

i=1

n

'R

Si Ci Bi Rfi Rii = Rii ∪ S Si i=1 o Ro i = Ri ∪ Si S i

6.4. The Decomposition of Formal Systems The decomposition of a formal system is an inverse operation of system composition, which yields two or more subsystems based on a given decomposition strategy in the form of a set of component, environ behavior, and ment partitions such as C = ni=1 Ci B = ni=1 Bi , and = ni=1 i . System decomposition can be described according to the mechanism of system decremental disunion. Definition 29. The decomposition of a system S, ∧ denoted by S = S1 S2 , is a partition of S into two subsystems in based on a specified partition strategy for the sets of components C = C1 ∪ C2 , behaviors B = B1 ∪ B2 , and environments = 1 ∪ 2 , i.e.:

' SCBRf Ri = Ri ∪S1 S∪S2 S Ro = Ro ∪SS1 ∪SS2

(37)

where newly created associations between the derived subsystems and the supersystem are established in parallel via their I/O relations. It is noteworthy that a system decomposition may yield different solutions dependent on the given partition strategy. As specified in Definition 29, the decomposition operation results in the removal of all the incremental funcf tional relations Rij = Bi × Cj 1 ≤ i j ≤ n that are no longer belong to any of the subsystems as the results of decomposition. Example 17. Given the following partition strategies: C = C1 ∪ C2 = Processor Keypad Pulse LED ∪ Processor Keypad LED Bell B = B1 ∪ B2 = SetTime ShowTime Tick ∪

2

SCBRf Ri Ro Si Ci Bi Rfi Rii Roi i

SetAlarm ShowAlarm CheckAlarm

i=1

C=

2

Ci

B=

i=1 ∧

2

Bi

=

i=1

2

Ring ReleaseAlarm i

= S1 C1 B1 R1 Ri1 Ro1 1 f

C1 ⊂ C ∧B1 ⊂ B ∧1 ⊂

⎧ ⎪ C1 = C1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B1 = B1 ⎪ ⎪ ⎪ ⎪ f f f f ⎨ R1 = Rf R2 = Rf \R2 ∪ R12 = B1 ×C2 = S1 ⎪Ri = ×S1 ⊂ Ri ∪SS1 S2 S1 ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪Ro1 = S1 ×1 ⊂ Ro ∪S1 SS1 S2 ⎪ ⎪ ⎪ ⎪ ⎩ = 1 1

S2 C2 B2 Rf2 Ri2 Ro2 2 J. Adv. Math. Appl. 4, 132–157, 2015

= 1 ∪ 2 = User

i=1

the decomposition of the supersystem SAlarmClock as obtained in Example 16 yields two subsystems S1 Clock the S2 Alarm according to Definition 29 as follows: SAlarmClock C B Rf Ri Ro 2

Si Ci Bi Rfi Rii Roi i i=1 C=

2 i=1

∧

= S1 Clock

Ci

B=

2

Bi

=

i=1

C1 B1 Rf1 Ri1 Ro1 1

2

i

i=1

S1 S 147

RESEARCH ARTICLE

SC B Rf Ri Ro n ∧ = Si Ci Bi Rfi Rii Roi i i=1 ⎧ n ⎪ ⎪ C = Ci ⎪ ⎪ ⎪ ⎪ i=1 ⎪ n ⎪ ⎪ ⎪ ⎪ B = Bi ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ f ⎪ f ⎪ Ri ⎨R = i=1 n =S ⎪ Rii = Rii ∪ Si S ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ o ⎪ Ri = Roi ∪ S Si ⎪ ⎪ ⎪ ⎪ i=1 ⎪ n ⎪ ⎪ ⎪ ⎪ = i ⎩

C2 ⊂ C ∧B2 ⊂ B ∧2 ⊂ ⎧ C2 = C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = B2 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ f f f f ⎪ ⎨R2 = Rf R1 = Rf \R1 ∪ R12 = B2 ×C1 = S2 ⎪ ⎪ Ri2 = 2 ×S2 ⊂ Ri ∪SS2 S1 S2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro2 = S2 ×2 ⊂ Ro ∪S2 SS2 S1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 = 2


RESEARCH ARTICLE

⎧ ⎪ C1 = C1 = ProcessorKeypadPulseLED ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B1 = B1 = SetTimeTickShowTime ⎪ ⎪ ⎪ ⎪ f f f f ⎪ ⎪ ⎪R1 = Rf R2 = Rf \R2 ∪ R12 ⊆ B1 ×C1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimePMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎨ TickPMPulseProcessor = S1 ShowTimePMProcessorLED ⎪ ⎪ ⎪ ⎪ i ⎪ ⎪ R1 = 1 ×S1 ⊂ Ri ∪SS1 S2 S1 ⎪ ⎪ ⎪ ⎪ ⎪ = UserKeypadSS1 S2 S1 ⎪ ⎪ ⎪ ⎪ ⎪ o o ⎪ ⎪ ⎪R1 = S1 ×1 ⊂ R ∪S1 SS1 S2 ⎪ ⎪ ⎪ = LEDUserS1 SS1 S2 ⎪ ⎪ ⎪ ⎩ = = User 1 1 f C2 B2 R2 Ri2 Ro2 2 S2 S

S2 Alarm ⎧ ⎪ ⎪C2 = ProcessorKeypadLEDBell ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = SetAlarmShowAlarmCheckAlarm ⎪ ⎪ ⎪ 2 ⎪ ⎪ RingReleaseAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Rf = Rf Rf = Rf \Rf ∪ Rf ⊆ B ×C ⎪ ⎪ 2 2 2 1 1 12 ⎪ ⎪ ⎪ ⎪ = SetAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ShowAlarmPMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ CheckAlarmPMKeypadProcessor ⎪ ⎪ ⎨ RingPMProcessorBell = S2 ⎪ ⎪ ReleaseAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Ri = ×S ⊂ Ri ∪SS S S ⎪ 2 2 1 2 2 ⎪ ⎪ 2 ⎪ ⎪ = UserKeypadSS2 S1 S2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro2 = S2 ×2 ⊂ Ro ∪S2 SS2 S1 ⎪ ⎪ ⎪ ⎪ ⎪ = LEDUserBellUserS2 S ⎪ ⎪ ⎪ ⎪ ⎪ S2 S1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 2 = 1 = User ' SCBRf Ri = Ri ∪S1 SS2 S Ro = Ro ∪SS1 SS2

where the interactive impacts on the input/output relations of the supersystem and subsystems are updated in parallel. After the decomposition of the supersystem S, the incremental behaviors and relations gained during system composition as described in Example 16, such as SelectFunc and SelectFuncPMClock Alarm, are no longer existent because they do not belong to any of the individual subsystems. The pairwise decomposition of systems as modeled in Definition 29 can be generally extended to n-ary system decompositions. Definition 30. The multiple decomposition of a sysn ∧ tem S, denoted by S = i=1 Si , is a partition of S into n subsystems according to a given decomposition 148

strategy represented by C = = ni=1 i , i.e.:

n

i=1 Ci B

=

Wang

n

i=1 Bi

and

n

SC B Rf Ri Ro Si Ci Bi Rfi Rii Roi i i=1

C=

n i=1

∧

n

= R Si i=1

Ci B =

n

Bi =

i=1

n

i

i=1

⎧ ⎪ ⎪Ci = Ci ⎪ ⎪ ⎪ ⎪ ⎪ ⎪Bi = Bi ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ Rfi = Rf Rfi ⊆ Bi × Ci ⎪ ⎪ j=1∧j=i ⎨ n

⎪ ⎪ ⎪ Ri = i × Si ⊂ Ri ∪ R Sj Si ⎪ ⎪ i j=1∧j=i ⎪ ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ o o ⎪ R = S × ⊂ R ∪ R Si Sj ⎪ i i i ⎪ j=1∧j=i ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ i = i

(38)

n

' SC B Rf Ri = Ri ∪ R Si S i=1

n

Ro = Ro ∪ RS Si i=1

6.5. The Hierarchical Recursive Synthesis of Formal Systems The recursive system synthesis and analysis, or system integration and specification, are hierarchical operations of formal systems in system design, modeling, and pattern recognition. Recursive and hierarchical synthesis and analysis are formal methodologies for building or decomposing complex systems. Both analysis and synthesis embody a system’s hierarchical structure and complex relations/functions in a series of recursive deductions or inductions, respectively. The recursive process is a coherent integration of the cognitive processes of analysis and synthesis, which indicates that system analysis must be complement by synthesis in order to form a closed loop in system inference (Wang, 2015a). System synthesis is a typical operation on formal systems for building complex systems by an inductive process. The synthesis or integration of formal systems can be generally expressed as a hierarchical and recursive process of induction from the bottom up where each level of the system is composed according to the methodologies as formally described in Sections 6.1 and 6.3. Synthesis is usually embodied by a series of system compositions that incrementally yields a more complex system from that of lower layer systems. The recursive methodology applies to both structural and functional synthesis on formal systems. Definition 31. The recursive synthesis of a supersystem S from two hierarchical subsystems S 0 and S 1 , J. Adv. Math. Appl. 4, 132–157, 2015

Wang


denoted by S ⇑k=0 S k , is an inductive integration of S = S 2 from a set of low-layer subsystems from the bottom up in , i.e.: 2

2

i2 =1

where the newly created associations between the supersystem and the original subsystem are established in parallel via their I/O relations. The expression of system syntheses in Definition 31 is based on the generality principle of systems as states in n1 Theorem 1. Once all subsystems at layer k = 0, Ri=1 Si0 , are given, the entire hierarchy of the supersystems at all n higher layers, Rk=1 Sik , is recursively determinable. Example 18. Given a hierarchical abstract system as shown in Figure 4, the three-layer supersystem, S, can be hierarchically synthesized according to Definition 31 as follows: 2 3 2 Given S = S 2 R Si12 R Si02 i1 ' R Si02 i1 i2 =1

i1 =1

i1 =1

S10 = s11 s12 s13 where S20 = s21 s22 S

S1

S11

Fig. 4.

S12

S2

S13

S21

S22

An abstract hierarchical system with a three-layer structure.

J. Adv. Math. Appl. 4, 132–157, 2015

2

⇑ S k = k=1 R S k S k−1 = S 2

k=0

n2 i2 =1

Si12

n1 i1 =1

Si01

⎧ ⎧ 3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S10 = Si011 ⎪ ⎪ 1 2 ⎨ ⎪ ⎪ i11 =1 0 ⎪ 1 ⎪ S = ⇑ R Si1 = ⎪ 2 ⎪ ⎪ ⎪ k=0 i1 =1 ⎪ 0 ⎪ ⎪ ⎪ Si012 ⎪ ⎨ ⎩S 2 = i12 =1 = 2 2 ⎪ ⎪ 1 ⎪ ⎪ S 2 = ⇑ R Si2 ⎪ ⎪ ⎪ k=1 i2 =1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ Si12 Ci2 Bi2 Rfi2 Rii2 Roi2 i2 ⎩ = i2 =1

Example 19. Suppose the system as given in Example 18 is realized by a set of concrete subsystems where S = S3 (AlarmClock), S1 = S1 (Clock), and S2 = S2 (Alarm), as given in Examples 2 through 4. Then, the integration of S(AlarmClock) can be rigorously described by a hierarchical system synthesis from the bottom up according to Definition 31 as follows: Given S10 ClockC1 = Processor Keypad Pulse LED and S20 AlarmC2 = Processor Keypad LED Bell 2

SAlarmClock =

2

⇑ S k = k=1 R S k S k−1

k=0

= S2 1

= S1 =

i2 =1

2

⇑ iR=1 Si0

k=0

n2

1

Si12

n1 i1 =1

Si01

1

⎧ 3 ⎪ ⎪ S11 Clock = Si011 Rf1 C1 ⎪ ⎪ ⎪ ⎪ i11 =1 ⎪ ⎪ ⎪ ⎪ ⎪ f ⎪ = S 0 R Processor Keypad Pulse LED ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⊆ B 1 × C1 ⎪ ⎪ ⎪ ⎪ = SetTimeKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ TickPulse Processor ⎪ ⎪ ⎪ ⎪ ⎪ ShowTimeProcessor LED ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5 ⎪ ⎨ 1 f Alarm = Si012 R2 C2 S 2 = i12 =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = S20 Rf2 Processor Keypad LED Bell ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⊆ B2 × C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetAlarmKeypad Processor ⎪ ⎪ ⎪ ⎪ ShowAlarmProcessor LED ⎪ ⎪ ⎪ ⎪ ⎪ CheckAlarmKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ RingProcessor Bell ⎪ ⎪ ⎩ ReleaseAlarmKeypad Processor 149

RESEARCH ARTICLE

SCBRf Ri Ro ⇑ S k Ck Bk Rfk Rik Rik k k=0 n1 n2 2 0 ∧ k k−1 2 = R S S = S Si12 Si1 k=1 i =1 i =1 ⎧ 2 n1 1 ni10 1 n1 ⎪ ⎪ 0 1 0 1 ⎪ = = ⇑ R S R S R S S ⎪ i1 i i i ⎪ i1 =1 1 i0 =1 1 0 ⎪ k=0 i1 =1 ⎪ ⎪ ⎪ n1 ni10 ⎪ ⎪ ⎪ = R S 0 C B Rf Ri Ro (39) ⎪ i1 i0 i1 i0 i1 i0 i1 i0 i1 i0 i1 i0 ⎪ ⎪ i1 =1 ⎪ i0 =1 ⎪ ⎨ = i1 i0 k = 1 ⎪ ⎪ ⎪ ⎪ n 2 2 ⎪ ⎪ 1 2 ⎪ ⎪ = ⇑ R S S i2 ⎪ ⎪ ⎪ k=1 i2 =1 ⎪ n2 ⎪ ⎪ ⎪ ⎪ = Si12 Ci2 Bi2 Rfi2 Rii2 Roi2 i2 k = 2 ⎪ ⎩

2

S=


{Editorial note: expression.} 2

⇑ S2 = ⇑

k=1

=

2

RESEARCH ARTICLE

i2 =1

cont’d

to

the

2nd

part

of

the

2

R Si1

i2 =1

2

Si12 Ci2 Bi2 Rfi2 Rii2 Roi2 i2

⎧ C = S11 C1 ∪S21 C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = ProcessorKeypadPulseLEDBell ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = S11 B1 S21 B2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimeTickShowTimeSetAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ShowAlarmCheckAlarmRing ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarm SelectFunc ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Rf = S11 Rf1 S21 Rf2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimePMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ TickPMPulseProcessor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ShowTimePMProcessorLED ⎪ ⎪ ⎪ ⎨ SetAlarmPMKeypadProcessor = ⎪ ⎪ ⎪ ShowAlarmPMProcessorLED ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ CheckAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ RingPMProcessorBell ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ReleaseAlarmPMKeypadProcessor ⎪ ⎪ ⎪ ⎪ ⎪ SelectFuncClockAlarm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ri = S11 Ri1 S21 Ri2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = UserKeypadS11 S 2 S21 S 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Ro = S11 Ro1 S21 Ro2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = LEDUserBellUserS 2 S11 S 2 S21 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = S11 1 ∪S21 2 = User o 1 2 ' S11 C1 B1 Rf1 Ri1 = Ri1 ∪S 2 S11 Ro 1 = R1 ∪S1 S 1 o 1 2 ' S21 C2 B2 Rf2 Ri2 = Ri2 ∪S 2 S21 Ro 2 = R2 ∪S2 S 2

The pairwise recursive synthesizes of formal systems as modeled in Definition 31 can be generally extended to n-ary recursive syntheses. Definition 32. The multiple recursive synthesis of a formal system S from n-layer subsystems in , denoted n ∧ by S = ⇑k=0 S k , n > 2, is a series of hierarchical inductive 150

Wang

syntheses from the bottom up where all subsystems at the 0th layer are given, i.e.: n

SC B Rf Ri Ro ⇑ S k Ck Bk Rfk Rik Rik k ∧

=

n

k=0

R S k S k−1 k=1

nn n2 n1 n−1 1 = Sn Sin Si2 Si01 i2 =1 i1 =1 ⎧ in =1 n1 1 ⎪ ⎪ 0 1 ⎪ S = ⇑ R S ⎪ i ⎪ 1 ⎪ k=0 i1 =1 ⎪ ⎪ ⎪ ⎪ ⎪ n1 n10 ⎪ 0 ⎪ ⎪ ⎪ = R Si1 i0 Ci1 i0 Bi1 i0 Rfi1 i0 Rii1 i0 ⎪ ⎪ i =1 1 ⎪ i0 =1 ⎪ ⎪ ⎪ ⎪ ⎪ o ⎪ R k=1 ⎪ i1 i0 i1 i0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 n2 ⎪ ⎪ 1 ⎪ 2 ⎪ S = ⇑ R S ⎪ i2 ⎪ ⎪ k=1 i2 =1 ⎪ ⎪ ⎪ ⎪ n2 n21 ⎪ ⎪ 1 ⎪ f ⎨ = R Si2 i1 Ci2 i1 Bi2 i1 Ri2 i1 Rii2 i1 i2 =1 = i1 =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ o ⎪ k=2 R ⎪ i2 i1 i2 i1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n nn ⎪ ⎪ ⎪ n−1 n ⎪ S = ⇑ R Sin ⎪ ⎪ ⎪ k=n−1 in =1 ⎪ ⎪ ⎪ ⎪ ⎪ nn ⎪ ⎪ f ⎪ ⎪ Sin−1 Cin Bin Rin Riin Roin in = ⎪ n ⎪ ⎪ in =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k=n

(40)

6.6. The Hierarchical Recursive Analysis of Formal Systems The analysis or specification of formal systems can be generally expressed as a hierarchical and recursive deduction process from the top down where each layer of the system is decomposed according to the methodologies as formally described in Sections 6.2 and 6.4. Analysis is typically embodied by a series of system refinements that decrementally reduces a complex system onto the terminal layer where all individual elements and properties are known or determinable. System analysis is an inverse operation of system synthesis for specifying a complex system onto simple ones by a deductive process from the top down. System analysis results in a hierarchical specification that deductively specifies a formal system into a set of constituent subsystems and their relations. This is also known as system refinement. The recursive methodology applies to both structural and functional analyses on formal systems. J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Definition 33. The recursive analysis of a supersystem S into two hierarchical subsystems S 0 and S 1 in , 0 denoted by S ⇓k=2 S k , is a deductive refinement of S into a set of subsystems at two layers from the top down, i.e.:

0

k=1

=

n1

0

k=2

∧

1

= R S k S k−1 = S 2 k=2

n2

n1

Si1 i=1 Si0 i =1 2

2

1

1

(41)

where newly created associations between the subsystems and the original supersystem are established in parallel via their I/O relations. Example 20. A recursive analysis of the formal system, S3 (AlarmClock), as obtained in Example 19 can be rigorously derived by a hierarchical system refinement from the top down according to Definition 33, which results in two formal subsystems S1 = S1 (Clock) and S2 = S2 Alarm as follows:

n10

n1

=

R Si1 iR=1 Si0 i i =1 1

1

1 0

0

ni0

R Si0 i Ci i Bi i Rfi i Rii i Roi i i i i =1 i =1 1 0

0

1 0

1 0

1 0

1 0

1 0

1 0

k=0 ⎧ 3 ⎪ ⎪ f 0 ⎪ ⎪ S Clock = Si0 R1 C1 1 ⎪ ⎪ i11 =1 11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = S 0 Rf Processor Keypad Pulse LED ⎪ ⎪ 1 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⊆ B1 × C1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetTimeKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ TickPulse Processor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ShowTimeProcessor LED ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 5 = S 0 Alarm = S 0 Rf C2 2 2 i ⎪ i12 =1 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = S 0 Rf Processor Keypad LED Bell ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⊆ B ⎪ 2 × C2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = SetAlarmKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ShowAlarmProcessor LED ⎪ ⎪ ⎪ ⎪ ⎪ CheckAlarmKeypad Processor ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ RingProcessor Bell ⎪ ⎪ ⎪ ⎪ ⎩ ReleaseAlarmKeypad Processor

RESEARCH ARTICLE

⎧ 1 n2 ⎪ ⎪ 1 1 ⎪ = ⇓ R S S ⎪ i2 ⎪ ⎪ k=2 i2 =1 ⎪ ⎪ ⎪ ⎪ ⎪ n2 ⎪ ⎪ ⎪ ⎪ = Si1 Ci2 Bi2 Rfi2 Rii2 Roi2 i2 k = 1 ⎪ ⎪ i2 =1 2 ⎪ ⎪ ⎪ ⎪ ⎪ n1 ni10 ⎨ 0 n1 = S 0 = ⇓ R Si0 = R Si1 R Si0 i 1 ⎪ i1 =1 1 i0 =1 1 0 ⎪ k=1 i1 =1 ⎪ ⎪ ⎪ ⎪ ⎪ n1 ni10 ⎪ ⎪ ⎪ ⎪ = R =1 Si01i0 Ci1 i0 Bi1i0 Rfi1 i0 Rii1 i0 ⎪ ⎪ ⎪ i =1 i 1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Roi1 i0 i1 i0 k = 0 ⎩

1

1

1

SCBRf Ri Ro ⇓ S k Ck Bk Rfk Rik Rik k

2

⇓ iR=1 Si0

⇓ S0 =

' S11 C1 B1 R1 Ri1 = Ri1 ∪ S 2 S11 f

o 1 2 Ro 1 = R1 ∪ S1 S 1

' S21 C2 B2 R2 Ri2 = Ri2 ∪ S 2 S21 f

Given

S10 ClockC1

= Processor Keypad Pulse LED

and

S20 AlarmC2

= Processor Keypad LED Bell

0

SAlarmClock =

1

⇓ S k = k=2 R S k S k−1

k=2

= S2

n2

i2 =1

Si12

1

= S1 =

n1 i1 =1

Si01

2

⇓ iR=1 Si1

SCBRf Ri Ro ⇓ S k Ck Bk Rfk Rik Rik k

2

=

Si1 Ci Bi Rfi Rii Roi i i =1 2

2

2

2

The pairwise recursive analysis of formal systems as modeled in Definition 33 can be generally extended to n-ary recursive system analyses. Definition 34. The multiple recursive analysis of a supersystem S into n-layer subsystems in , denoted by 0 S ⇓k=n S k n > 2, is a series of hierarchical deductive analyses from the top down where all subsystems at the 0th layer are given, i.e.: 0

2

2

k=2

o 1 2 Ro 2 = R2 ∪ S2 S 2

2

2

2

2

= S11 Clock C1 B1 Rf1 Ri1 Ro1 1 S21 Alarm C2 B2 Rf2 Ri2 Ro2 2 J. Adv. Math. Appl. 4, 132–157, 2015

k=n

k=1

∧

1

= R S k S k−1 k=n

=S

n

nn

in =1

Sin−1 n

n2 n1 1 0 Si2 Si1 i2 =1

i1 =1

151


RESEARCH ARTICLE

⎧ n−1 nn ⎪ ⎪ n−1 n ⎪ = ⇓ R S S ⎪ in ⎪ ⎪ k=n in =1 ⎪ ⎪ ⎪ ⎪ ⎪ nn ⎪ ⎪ f ⎪ n−1 i o ⎪ ⎪ = Sin Cin Bin Rin Rin Rin in ⎪ in =1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k=n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 n2 ⎪ ⎪ ⎪ ⎪S 2 = ⇓ R Si2 ⎪ 3 ⎪ ⎪ k=2 i2 =1 ⎨ = n2 n21 ⎪ ⎪ ⎪ R Si1 i Ci2 i1 Bi2 i1 Rfi2 i1 Rii2 i1 = ⎪ ⎪ i2 =1 i2 =1 2 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Roi2 i1 i2 i1 k = 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 n1 ⎪ ⎪ ⎪ 1 1 ⎪ S = ⇓ R S ⎪ i ⎪ 1 ⎪ k=1 i1 =1 ⎪ ⎪ ⎪ ⎪ ⎪ n1 n10 ⎪ ⎪ ⎪ ⎪ R Si0 i Ci1 i0 Bi1 i0 Rfi1 i0 Rii1 i0 = ⎪ ⎪ ⎪ i1 =1 i1 =1 1 0 ⎪ ⎪ ⎪ ⎪ o ⎪ Ri1 i0 i1 i0 k = 1 ⎩

and analyzed with increasing details at different layers, 0 ≤ k ≤ n, from the top down. k Corollary 4. Any subsystem S k−1 of a closed system S is an open system, i.e.:

∀ S k−1 S ⇒ Rik−1 = $ ∧ Rok−1 = $ k

⇒ S k−1 = S

(42)

7. PROPERTIES OF FORMAL SYSTEMS AND ALGEBRAIC RULES OF SYSTEM ALGEBRA System algebra provides a denotational mathematical means for rigorous representation and manipulation of both abstract and concrete systems. The mathematical model of formal systems, the framework of system algebra, and the algebraic operators on formal systems as developed in Sections 2 through 6 enable rigorous analyses of the nature, properties, and behaviors of formal systems as well as their algebraic operations. This section describes the general properties of formal systems as well as the algebraic properties and rules of system algebra. 7.1. Formal Properties of Recursive Systems Hierarchy The key principle of formal system theories is the generality of systems as formally described in Theorem 1. According to Theorem 1, the following properties of general and abstract systems can be derived. Corollary 2. The abstraction principle of systems states that any system S in can be inductively integrated and synthesized with decreasing details at different layers, 0 ≤ k ≤ n, from the bottom up. Corollary 3. The refinement principle of systems states that any system S in can be deductively specified 152

Wang

k−1

Rik−1 Rok−1

(43)

However, an inverse assertion is not true. That is, a subsystem of an open system is not necessarily a closed system. Theorem 1 and Corollaries 2 through 4 explain the fundamental principles of general systems and the mathematical model of formal systems. Other formal properties of formal system theory may refer to (Wang, 2015a). Corollary 5. The following relations hold for any given pairs of system compositions and decompositions according to Definitions 27 and 29: ⎧ n −1 n ⎪ ⎪ S = Si S = ⎪ i ⎪ i=1 ⎨ i=1 (44) n −1 n ⎪ ⎪ ⎪ ⎪ Si ⎩S = S i = i=1

i=1

which indicates that system composition and decomposition is a pair of inverse operations. 7.2. Properties of Formal System Synergy and Fusion C1 Theorem 2. Given two sets of components C1 = Ri=1 xi C2 and C2 = Rj=1 yj , the incremental union of relations between the sets of relations on C1 and C2 , R1 C1 and R2 C2 , denoted by R = R1 R2 , is a conjunction of the local relations R1 C1 and R2 C2 plus a set of newly generated incremental relations R12 C1 C2 interacting between the sets of components in , i.e.: ∧

R = R1 R2 = R1 C1 ∪ R2 C2 ∪ R12 C12 = C1 × C1 ∪ C2 × C2 ∪ C1 ⊗ C2

(45)

Proof. Theorem 2 can be proved by analyzing the local and interactive relations in a system as follows: C1 C2 ∀ C1 = R xi C2 = R yj i=1

j=1

(a) The local relations R1 and R2 are symmetric and reflexive: C1 C1 R1 = R R xi xj xi ∈ C1 ∧ xj ∈ C1 i=1 j=1

= C1 × C1 C2 C2 R2 = R R yi yj yi ∈ C2 ∧ yj ∈ C2 i=1 j=1

= C2 × C2 J. Adv. Math. Appl. 4, 132–157, 2015

Wang


(b) The interactive incremental relations R12 symmetric: C1 C2 R12 = R R xi yj xi ∈ C1 ∧ yj ∈ C2 ∪

are

R = R1 + R2 + R12 = C1 2 + C2 2 + 2C1 • C2

i=1 j=1

= C1 + C2 2 = C2

C2 C1

R R yj xi xi ∈ C1 ∧ yj ∈ C2 j=1 i=1

= C1 ⊗ C2 Therefore, the total system relations R are symmetric, reflexive, and transitive, i.e.: R = R1 ∪ R2 ∪ R12 = C1 × C1 ∪ C2 × C2 ∪ C1 ⊗ C2

(46)

The discovery in Theorem 2 reveals that the formal explanation for system utilities is the newly gained relations, behaviors, and functions during the synergy and composition of systems. The empirical awareness of this key system property has been intuitively or qualitatively described in the literature of system science. However, Theorem 2 provides a formal mathematical model for explaining system gains during system synergies and compositions. Theorem 2 can be used to predict the maximum number of newly established relations and behaviors via system synergies. Corollary 6. The size of an incremental union of relations, RC1 C2 = R1 R2 , is the sum of the sizes of R1 C1 and R2 C2 plus the size of the newly generated incremental relations R12 C1 C2 in as follows: R = R1 R2 = R1 + R2 + R12 = C1 2 + C2 2 + 2C1 • C2

(47)

= C1 + C22 = C2 where the incremental R12 = 2C1 • C2.

size

of

relations

is

Proof. Corollary 6 is proved according to Theorem 2 on the basis of the antisymmetric, symmetric, reflexive, and transitive properties of relations (Wang, 2015e), as follows: C1 ∀ C1 = R xi i=1

C2 C2 = R yj

R2 = C2 × C2 = C2 2 (b) The size of the incremental relations R12 are: R12 = 2C1 × C2 = 2C1 • C2 J. Adv. Math. Appl. 4, 132–157, 2015

Corollary 7. The principle of system synergy states that the composition of two systems, S = S1 * S2 , results in the creation of new relations or functions R12 , which solely belong to the supersystem S but not belong to any of the original individual subsystems S1 or S2 . The principle can be generally extended to n-ary dimensions of system fusions where S = ni=1 Si . According to Theorem 2, the structural complexity of systems represents a fully bidirectionally connected system that is constrained by the asymmetric relations of the system. Therefore, the structural complexity of systems is the theoretical upper-bound of system complexity where all components in a system are potentially fully interconnected with each other. Certain real-world systems may only possess partial connections where their particular structural complexity is constrained by the upper bound. 7.3. Properties of Structures and Configurations of Formal Systems The structural properties of formal systems are modeled by the degrees of cohesion and coupling of systems on the basis of internal relations Rf S and external relations Ri S and Ro S) according to Definition 2, where the total number of system relations is denoted as RS = Rf S + Ri S + Ro S. Definition 35. The cohesion of a formal system S, S, is determined by a ratio between the sizes of its internal relations Rf S and of the total relations RS, i.e.: ∧

S =

Rf S RS Rf S f R S + Ri S + Ro S

(49)

Corollary 8. Properties of system cohesion are as follows:

(a) The sizes of the local relations R1 and R2 are: R1 = C1 × C1 = C1 2

It is noteworthy that the new relations gained in an incremental union of system synergy and composition, R12 = 2C1 • C2 , represent an important property of general systems. However, it could not be formal explained in traditional system theory when system relational conjunctions and synergies were treated as static operations.

=

j=1

(48)

(a) Nonnegative: ∀S S ≥ 0 (b) Normalized: (c) Null cohesion:

∀S0 ≤ S ≤ 1 ∃SRf S = $ ⇒ S = 0

(50)

(d) Full cohesion: ∃SRi S∪Ro S = $ ⇒ S = 1 153

RESEARCH ARTICLE

= R1 R2

Therefore, the size of total system relations R is:

A Denotational Mathematical Theory of System Science: System Algebra for Formal System Modeling and Manipulations Formal properties of system algebra.

RESEARCH ARTICLE

Table II.

Wang

154

J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Definition 36. The coupling of a formal system S, S, is determined by a ratio between the sizes of its external relations Ri S and Ro S as well as the total relations RS, i.e.: ∧

S =

Ri S + Ro S RS

(51)

Ri S + Ro S = f R S + Ri S + Ro S

Corollary 9. Properties of system coupling are as follows:

8. CONCLUSIONS

(a) Nonnegative: ∀SS ≥ 0 ∀S0 ≤ S ≤ 1

(c) Full coupling: ∃SRf S = $ ⇒ S = 1

(52)

(d) Null coupling: ∃SRi S∪Ro S = $ ⇒ S = 0 It is obvious that the cohesion of a closed system are always 1; and the coupling of a closed system is always 0 according to Corollaries 8 and 9. Corollary 10. The cohesion and coupling of any system S are complement, i.e.: S + S ≡ 1

(53)

Proof. Corollary 10 can be directly proved based on Definitions 35 and 36 as follows: ∀ S S + S =

Rf S Ri S + Ro S + RS RS

=

Rf S + Ri S + Ro S Rf S + Ri S + Ro S

(54)

=1 7.4. Algebraic Properties and Rules of System Algebra It has been demonstrated that each operator of system algebra denotes an algebraic rule of formal system manipulations. On the basis of the definitions of system algebra as elaborated in Sections 2 through 6, a set of 92 formal properties and rules of system algebra is elicited and summarized in Table II. The basic rules of system algebra in the universe of discourse of general systems can be expressed in categories of the commutative, associative, transitive, reflexive, identity, inclusive, and partial order rules. The algebraic rules of system algebra can be applied to rigorously derive and simply complex system operations in system modeling, analysis, and engineering. The algebraic rules in Table II are mapped to each of the relational, reproductive, and compositional operators of system algebra. J. Adv. Math. Appl. 4, 132–157, 2015

Systems have been recognized as the most complicated entities and phenomena in abstract, physical, information, cognitive, brain, and social worlds across almost all science and engineering disciplines. This paper has presented a fundamental theory of system science for system representation, modeling, analysis, synthesis, and inference. A rigorous treatment of complex systems as a hyperstructure has been presented by a unified mathematical model of formal and abstract systems. It has been recognized that a general form of systems is a denotational mathematical structure that represents commonly shared properties of real-world systems. In system algebra, any abstract or concrete system has been modeled as a formal system, and system manipulations are reduced to formal algebraic operations. A set of 92 algebraic rules has been formally elicited for elaborating the mathematical properties and rules of system algebra. Based on system algebra, applied systems can be rigorously designed, modeled, and manipulated. Real-world case studies have been demonstrated that system algebra is not only a general and rigorous modeling methodology for systems, but also an efficient and formal manipulation framework for the composition and decomposition of complex systems in system engineering, software engineering, knowledge engineering, big data engineering, brain systems, cognitive robotics, computational linguistics, and cognitive systems. Acknowledgments: The author would like to acknowledge the support in part of a discovery fund granted by the Natural Sciences and Engineering Research Council of Canada (NSERC). The author would like to thank the anonymous reviewers for their valuable suggestions and comments on the previous version of this paper.

References Ashby, W. R. (1958). Requisite variety and implications for control of complex systems. Cybernetica 1: 83–99. Bender, E. A. (2000). Mathematical Methods in Artificial Intelligence (IEEE CS Press, Los Alamitos, CA). Boulding, K. (1956). General systems theory—The skeleton of science. General Systems Yearbook 1: 11–17. Bunge, M. (1977). General systems theory challenge to classical philosophy of science. Int. J. Gen. Sys. 4(1): 29–37.

155

RESEARCH ARTICLE

(b) Normalized:

Each algebraic rule and property of system algebra on formal systems as summarized in Table II can be rigorously proved by directly applying related definitions as given in Sections 4 through 6. The theories of formal systems, system algebra, and their algebraic properties establish a rigorous theoretical framework, which can be applied not only for mathematically abstracting and denoting formal systems, but also for rigorously manipulating formal systems in system analysis, synthesis, and computation in system engineering, cognitive systems, software science, and computational intelligence.

A Denotational Mathematical Theory of System Science: System Algebra for Formal System Modeling and Manipulations Castillo, O., Melin, P., and Kacprzyk, J. (2013). Recent Advances on Hybrid Intelligent Systems (Springer, Berlin).

Wang, Y. (2009c). On cognitive computing. International Journal of Software Science and Computational Intelligence 1(3): 1–15.

Descartes, R. (1979). Meditations on First Philosophy, D. Cress trans., Hackett Publishing Co. Inc., Indianapolis.

Wang, Y. (2010a). Cognitive robots: A reference model towards intelligent authentication. IEEE Robotics and Automation 17(4) 54–62.

Ellis, D. O. and Fred, J. L. (1962). Systems Philosophy (Prentice-Hall).

Wang, Y. (2010b). On formal and cognitive semantics for semantic computing. International Journal of Semantic Computing 4(2) 203–237.

Ford, J. (1986). Chaos: Solving the unsolvable, predicting the unpredictable, Chaotic Dynamics and Fractals (Academic Press, London). Gaines, B. R. (1984). Methodology in the large: Modeling all there is. Systems Research 1(2): 91–103. Haken, H. (1977). Synergetics (Springer-Verlag, NY). Hall, A. S. and Fagan, R. E. (1956). Definition of system. General Systems Yearbook 1: 18–28. Klir, G. J. (1992). Facets of Systems Science (Plenum, NY).

RESEARCH ARTICLE

Wang

Pedrycz, W. (1981). On approach to the analysis of fuzzy systems. Int. J. of Control 34: 403–421. Negoita, C. V. (1989). Review: Fuzzy sets, uncertainty, and information. Cybernetes 18(1): 73–74. Planck, M. (1936). The Philosophy of Physics (W. W. Norton & Company, Inc., NY). Rapoport, A., (1962). Mathematical aspects of general systems theory. General Systems Yearbook 11: 3–11. Schedrovitzk, G. P. (1962). Methodological problems of systems research. General Systems Yearbook 11: 27–53. Simon, H. (1965). Architecture of complexity. General Systems Yearbook 10: 63–76. Takahara, Y. and Takai, T. (1985). Category theoretical framework of general systems. Int. J. Gen. Sys. 11(1): 1–33. von Bertalanffy, L. (1952). Problems of Life: An Evolution of Modern Biological and Scientific Thought (C. A. Watts, London). Wang, Y. (2002). The Real-Time Process Algebra (RTPA). Annals of Software Engineering 14: 235–274. Wang, Y. (2003a). On cognitive informatics. Brain and Mind: A Transdisciplinary Journal of Neuroscience and Neurophilosophy 4(2) 151–167.

Wang, Y. (2012a). On denotational mathematics foundations for the next generation of computers: Cognitive computers for knowledge processing. J. Adv. Math. Appl. 1(1): 118–129. Wang, Y. (2012b). Keynote: Towards the next generation of cognitive computers: Knowledge versus data computers, Proc. 12th International Conference on Computational Science and Applications (ICCSA’12) (Salvador, Brazil, Springer), June, pp. 18–21. Wang, Y. (2012c). On long lifespan systems and applications. J. Comput. Theor. Nanosci. 9(2): 208–216. Wang, Y. (2012d). In search of denotational mathematics: Novel mathematical means for contemporary intelligence, brain, and knowledge sciences. J. Adv. Math. Appl. 1(1) 4–25. Wang, Y. (2012e). On abstract intelligence and brain informatics: Mapping cognitive functions of the brain onto its neural structures. International Journal of Cognitive Informatics and Natural Intelligence 6(4): 54–80. Wang, Y. (2013). On semantic algebra: A denotational mathematics for cognitive linguistics, machine learning, and cognitive computing. J. Adv. Math. Appl. 2(2): 145–161. Wang, Y. (2014a). Software science: On general mathematical models and formal properties of software. J. Adv. Math. Appl. 3(2): 130–147. Wang, Y. (2014b). Towards a theory of fuzzy probability for cognitive computing, Proc. 13th IEEE International Conference on Cognitive Informatics and Cognitive Computing (ICCI ∗ CC 2014) (IEEE CS Press, London, UK), August, pp. 19–28. Wang, Y. (2014c). Fuzzy causal inferences based on fuzzy semantics of fuzzy concepts in cognitive computing. WSEAS Transactions on Computers 13: 430–441.

Wang, Y. (2003b). Using process algebra to describe human and software system behaviors. Brain and Mind 4(2): 199–213.

Wang, Y. (2015a). Towards the abstract system theory of system science for cognitive and intelligent systems. Springer Journal of Complex and Intelligent Systems 1(3): 11–32.

Wang, Y. (2007a). Software Engineering Foundations: A Software Science Perspective, CRC Series in Software Engineering (Auerbach Publications, NY), Vol. II.

Wang, Y. (2015b). Fuzzy Probability Algebra (FPA): A theory of fuzzy probability for fuzzy inference and computational intelligence. J. Adv. Math. Appl. 4(1): 38–55.

Wang, Y. (2007b). The theoretical framework of cognitive informatics. The International Journal of Cognitive Informatics and Natural Intelligence 1(1): 1–27.

Wang, Y. (2015c). Keynote: On cognitive robotics and theories of abstract intelligence, 3rd International Conference on Automatic Control, Soft Computing and Human-Machine Interaction (ASME ’15) (Salerno, Italy), June, p. 3.

Wang, Y. (2008a). On contemporary denotational mathematics for computational intelligence. Transactions on Computational Science 2: 6–29. Wang, Y. (2008b). On system algebra: A denotational mathematical structure for abstract system modeling. International Journal of Cognitive Informatics and Natural Intelligence 2(2): 20–42.

Wang, Y. (2015d). Keynote: Big data algebra: A rigorous approach to big data analytics and engineering, 17th International Conference on Mathematical and Computational Methods in Science and Engineering (MACMESE ’15) (Kuala Lumpur), April, p. 14.

Wang, Y. (2008c). RTPA: A denotational mathematics for manipulating intelligent and computing behaviors. International Journal of Cognitive Informatics and Natural Intelligence 2(2): 44–62.

Wang, Y. (2015e). On the incremental union of relations: A key property of general systems explained. International Journal of Software Science and Computational Intelligence 7(1): 39–62.

Wang, Y. (2009a). Paradigms of denotational mathematics for cognitive informatics and cognitive computing. Fundamenta Informaticae 90(3) 282–303.

Wang, Y. (2015f). On cognitive learning methodologies for cognitive robotics, 15th International Conference on Robotics, Control and Manufacturing Technology (ROCOM’15) (Kuala Lumpur), April, pp. 11–19.

Wang, Y. (2009b). On abstract intelligence: Toward a unified theory of natural, artificial, machinable, and computational intelligence. International Journal of Software Science and Computational Intelligence 1(1): 1–17.

Wang, Y. (2015g). Keynote: Cognitive robotics and mathematical engineering, 14th IEEE International Conference on Cognitive Informatics and Cognitive Computing (ICCI ∗ CC 2015) (Tsinghua University, IEEE CS Press, Beijing, China), July, pp. 4–6.

156

J. Adv. Math. Appl. 4, 132–157, 2015

Wang


Wang, Y. (2015h). On mathematical theories and cognitive foundations of information. International Journal of Cognitive Informatics and Natural Intelligence 9(3): 41–63.

Wang, Y., Kinsner, W., and Zhang, D. (2009b). Contemporary cybernetics and its faces of cognitive informatics and computational intelligence. IEEE Trans. on System, Man, and Cybernetics (Part B) 39(4): 1–11.

Wang, Y. and Wang, Y. (2006). Cognitive informatics models of the brain. IEEE Transactions on Systems, Man, and Cybernetics (Part C) 36(2) 203–207.

Zadeh, L. A. (1965). Fuzzy sets and systems, edited by Fox, J., Systems Theory (Polytechnic Press, Brooklyn NY), pp. 29–37.

Wang, Y., Wang, Y., Patel, S., and Patel, D. (2006). A Layered Reference Model of the Brain (LRMB). IEEE Trans. on Systems, Man, and Cybernetics (Part C) 36(2): 124–133. Wang, Y., Zadeh, L. A., and Yao, Y. (2009a). On the system algebra foundations for granular computing. International Journal of Software Science and Computational Intelligence 1(1): 64–86.

Zadeh, L. A. (1973). Outline of a new approach to analysis of complex systems. IEEE Transactions on Systems, Man, and Cybernetics 1(1): 28–44. Zadeh, L. A. (1982). Fuzzy systems theory: Framework for analysis of buerocratic systems, edited by Cavallo, R. E., System Methods in Social Science Research (Kluwer-Nijhoff, Boston), pp. 25–41.

Received: 6 November 2014. Accepted: 16 August 2015.

RESEARCH ARTICLE

J. Adv. Math. Appl. 4, 132–157, 2015

157