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The function structure analysis theory based on the factor space and space fault tree

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Abstract

In order to analyze system function structure, the system function structure analysis theory is put forward based on factor space theory. Factors space is more suitable for describing cognition process of intelligence science than the qualitative cartesian space. Based on factor logic, the justice system of the function structure analysis was built. It is proved that the system function logic structure is a minimal disjunctive normal form from the system function analysis. The relationship is discussed between the classification reasoning method of inward analysis of system structure in space fault tree (SFT) and the function structure analysis. The process of inward analysis of system function structure in SFT is realized by the function structure analysis theory. The original classification reasoning method is enhanced to the level of logic mathematics. The system function structures of both incomplete information and complete information are analyzed with the method respectively, and the minimal disjunctive normal forms obtained from the analyses are \(T = x_{1}x_{4} + x_{3}x_{5} + x _{1}x_{2}\) and \(T = x_{1}x_{4} + x_{3}x_{5} +x_{1}x_{2} x_{3}\). The findings indicate that there are some implicit relationships between \(A_{3}\) and \(A_{2}\) , \(A_{1}\). They added the phase set to the background sets and converted uncertain problems to certain ones.

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Abbreviations

SFT:

Space fault tree

\(f_{1}\),...\(,f_{n}\) :

1th\(\sim i \)th condition factors

U :

Universe

F :

A set of condition factors

g :

Result(function) factors

\(f_{i}\) :

\(U\rightarrow X(f_{i})\): \(f_{i}(u)\) express phase of system components \(A_{i}\) in a moment

\(A_{i}\) :

ith component

X(f):

Phase space

\(\nabla \) :

Comprehensive operation

\(\Delta \) :

Decomposition operation

\(B=F(U)\) :

Background relationship between the factors

[a]=\(f_{j}^{-1}(a)\) :

Structure factor phase

\(d=h\)/m :

Decision ratio

\(L_{f}\) :

Factor logic system

S= X \(=X(f_{1})\cup \)...\(\cup X(f_{n})\) :

Symbol set of factor logic system

\(x_{ij}\) :

Phase

ij :

The word \(x_{ij}\) is the ith phase of factor \(f_{j}\).

F(S):

Formula set

(F(S), \(\vee \), \(\wedge \), \(\lnot \), \(\rightarrow )\) :

Boolean algebra

\(x_{i(1)j(1)}\wedge \)...\(\wedge x_{i(k)j(k)}\) :

Conjunction, simply \(x_{i(1)j(1)}\)...\(x_{i(k)j(k)}\)

\(r_{1}\vee \)...\(\vee r_{t}\) :

Disjunction, general \(r_{1}\)+...+\(r_{t}\)

p :

A tautology form

b :

A disjunctive normal form

\(W_{2}\) :

Value domain

MP :

Reasoning rules

\(\varSigma \) :

Axiom sets

W :

Assignment set

\(\varGamma \)-theorem:

The subsystem of the theorem

\(\Gamma \)1:

Word name axiom

\(\Gamma \)2:

Background axiom

\(F_{b}(S)=\{{p\wedge b}{\vert }{p\in F}(S)\}\) :

Formula set of factor logic

\(F^{+}(S)\) :

Proposition set or predicate set

\(U'=U_{/F}\) :

Quotient space

\(P^{\mathrm{c}}\) :

A subset of B

T :

Positive class

F :

Negative class

CSFT:

Continuous space fault tree

DSFT:

Discrete space fault tree

IASS:

Inward analysis of system structure

IASFS:

Inward analysis of system factor structure

IASCS:

Inward analysis of system component structure

01SFT:

01 Space fault tree

TM:

Table method

IIA:

Item by item analyses

CRM:

Classification reasoning method

O :

A system

\(A_{1}\sim A_{N}\) :

N basic events or components compose system O

\(x_{j}\) :

Component normal state

x \(_{j}\) :

Component failure state

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Acknowledgements

The author wishes to thank all his friends for their valuable critics, comments and assistances on this paper. This study was partially supported by the grants (Grant Nos. 51474121, 61350003, 51674127) from the Natural Science Foundation of China.

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Authors and Affiliations

  1. College of Safety Science and Engineering, Liaoning Technical University, Fuxin, 123000, China

    Tie-Jun Cui & Sha-Sha Li

  2. Key Laboratory of Mine Thermodynamic Disasters and Control of Ministry of Education, Fuxin, 123000, China

    Tie-Jun Cui & Sha-Sha Li

  3. Tunnel & Underground Structure Engineering Center of Liaoning, Dalian Jiaotong University, Dalian, 116028, China

    Tie-Jun Cui

  4. College of Intelligence Engineering and Mathematics, Liaoning Technical University, Fuxin, 123000, China

    Pei-Zhuang Wang

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  1. Tie-Jun Cui
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Correspondence to Tie-Jun Cui.

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Cui, TJ., Wang, PZ. & Li, SS. The function structure analysis theory based on the factor space and space fault tree. Cluster Comput 20, 1387–1399 (2017). https://doi.org/10.1007/s10586-017-0835-2

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