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Function packet sets (P-sets) is a novel model with law and dynamic characteristics, which is composed of function internal P-set $$S^{{\bar{F}}}$$ and function outer P-set $$S^{F}$$ . Using the structure, law and dynamic characteristic of function P-sets, we present the concepts of P-information law and band information law and their generation. Subsequently, the attribute theorem of P-information, the hiding and hiding theorem of P-information in band information law and the recovery theorem of information law are studied. Based on the above theory results, an application example of information law hiding is given and validated.

Firstly by improving inverse P-sets, inverse P-data model is proposed, it consists of internal and external inverse P-data models, and demonstrates dynamic characteristics; then the expansion-contraction generations of attribute disjunctive normal form for inverse P-data model are shown, and inverse P-data reasoning is proposed; finally intelligent data separation-discovery is given, and verified by an application.

By improving function inverse P-sets, inverse P-information law model and its structures are proposed. Then attribute disjunctive expansion-contraction generations of inverse P-information law models are presented. Furthermore, inverse P-information reasoning and reasoning structures are proposed Finally intelligent generating theorems of information law are proposed, the reality-camouflage intelligent transformations of information image and their intelligent generations and applications are also given.

P-sets (packet sets) is improved by introducing function into it, and the concept and structure of function P-sets (function packet sets) is proposed, which is a set pair composed of the function internal P-set \( S^{{\bar{F}}} \)(function internal packet set \( S^{{\bar{F}}} \)) and the function outer P-set SF (function outer packet set SF). Function P-sets has dynamic characteristics and law characteristics. Under certain conditions, function P-sets can return to the “origin” of function ground sets S. In the paper, it is proved that function P-sets is extension of P-sets, conversely, P-sets is special case of function P-sets. Applications of function P-sets in discovery of unknown information laws hidden information systems, are provided. Function P-sets is a new theory and a new method to research dynamic information laws for information systems.

Function S-rough sets are defined by R-function equivalence class, which have dynamic characteristic. Function S-rough sets have dynamic characteristic, law characteristic and law-hiding characteristic. Function S-rough sets can generate f-hiding law and $$ \bar f $$ -hiding law. By engrafting, crossing, and penetrating between the information security theory and function S-rough sets, the security hiding and the authentication of f-hiding law and $$ \bar f $$ -hiding law are given respectively in this paper. The fusion and share between function S-rough sets and information security theory is a new research direction of the application of information law in information system.

By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets (singular rough sets). S-rough sets are defined by element equivalence class that proposes dynamic characteristic. S-rough sets have dynamic characteristic. By introducing the function equivalence class (law equivalence class) that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets (function singular rough sets). Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.

Using S-rough sets, this paper gives the concepts of f -heredity knowledge and its heredity coefficient, and f -variation coefficient of knowledge; presents the theorem of f -attribute dependence of variation coefficient and the relation theorem of heredity-variation. The attribute dependence of f -variation coefficient and the relation of heredity-variation are important characteristics of S-rough sets. From such discussion, this paper puts forward the heredity mining of f -knowledge and the algorithm of heredity mining, also gives its relative application.

In order to recognize and control effectively dynamic information, the paper proposes the concept of F-recursive genetic information by introducing the concept of heredity in biology and using P-sets, including the concepts of F-recursive dominant inheritance information and -F-recursive recessive inheritance information. Meanwhile, the structure and characteristics about the F-recursive genetic information are provided. The paper gives the identification theorem and identification criterion as well as the reduction method for the F-recursive genetic information. Finally, the application of computer vision recognition of F-recursive genetic information is provided. The research indicates that the generated F-recursive genetic information is identifiable and the proposed recursive reduction method is more reliable than the non-recursive reduction method. The-recursive genetic information is a new tool to discover information, to forecast information and to control information.

In this paper, we define grey set relation and grey group relation. Their general problems are studied. Some basic results are obtained.

Based on P-augmented matrices theory, this paper put forwards the concepts of the dynamic internal segmentation of internal P-augmented matrix, the dynamic outer segmentation of outer P-augmented matrix, and the dynamic internal-outer segmentation of P-augmented matrix, and gives the segmentation structures and attribute relationships; then based on the concepts above, the concepts of internal P-augmented matrix reasoning, outer P-augmented matrix reasoning and P-augmented matrix reasoning, as well as their structures are proposed; finally, the intelligent acquisition theorems and acquisition criterion of P-information are presented, moreover, based on dynamic internal segmentation and dynamic internal P-augmented matrix reasoning, an application on the intelligent acquisition of internal P-information is shown.

Function S-rough sets, which was defined by R-function equivalence class, has dynamic characteristic and law characteristic., By combining dual of function one direction singular rough sets with intuitionistic fuzzy sets in this paper, the concepts such as adjoint intuitionistic fuzzy sets of law, degree of law variation and f-transfer coordination law were proposed; Based on adjoint intuitionistic fuzzy sets of law, the recognition theorem of law variation and the application of f transfer coordination law were given.

Two direction S-rough sets has two direction dynamic characteristics. By employing two direction S-rough sets, a generation method of two direction S-rough decision law is presented. The concepts of upper-decision law, lower-decision law, two direction S-rough decision law, two direction S-rough decision law kernel, two direction S-rough decision law band and two direction S-rough decision law hull are proposed. The theorems about decision law transfer and decision law mining are presented. The criterion of rough decision law mining is given.

Function S-rough sets is defined by function equivalence class, function is a kind of feature. By using of function S-rough sets, this paper gives a new feature selection method, gives concepts of lower approximation features and upper approximation features, furthermore, it presents the structure of F-feature pair, gives its characteristic discussion. F-feature selection method given in this paper has gotten applied in image processing, object recognition and etc, it is becoming a new research direction in recognition theory.

Function S-rough sets is defined by function equivalence class, function is a kind of law, law has heredity-variation characteristic, by using of this heredity- variation characteristic of function S-rough sets, this paper presents concept of f - heredity law, \(\overline{f}\) - variation law of image feature, gives image feature law generation theorem, and applications of image feature law heredity-variation in image hiding. All results of this paper have important application value in economic, military region.

Rough graph is the graph that can be used to research uncertainty problems, and it is a new research tool to analyze the rough characteristic of the system according to Tong He and Kaiquan Shi. This paper makes further research on rough graph, and gives the definitions of class connection of rough graph and some subgraphs of rough graph. Furthermore, this paper presents the definition of weighted rough graph by enduing the edges of rough graph with weight attribute, and gives the algorithm of exploring the class optimal tree in weighted rough graph, which generalizes the classical Kruskal algorithm of exploring the optimal tree. Finally, the application in relationship analysis shows the effectiveness of the algorithm

Recognition of financial risk (investment risk and profit risk) has attracted more and more attention of investors, because each investor is threaten by the financial risk. Function S-rough set (function singular rough set) has law characteristic and the law has heredity characteristic. Using function S-rough set, this paper advances the recognition of financial risk law and gives its recognition model and an application example. Function S-rough set is defined by R-function equivalence class [u], ui∈[u] is a function (or a law). Function S-rough set is the general form of S-rough set (singular rough set) and S-rough set is the special case of function S-rough set. The results of this paper have lots of important applications

This paper is on fuzzy decision theory and information security theory which are mutually independent and engrafted. This paper presents both-branch fuzzy decision and problems on decision encryption-authentication, puts forward two kinds of both-branch fuzzy decision onX: both-branch fuzzy decision onX having bounded domainsXo =X+ ∩X− = {x0}, both-branch fuzzy decision onX having overlapping domainsX* =X+ ∩X− = {x*1,x*2,…,x*t }. It presents single branch separate-separate out theorem of both-branch fuzzy decision onX, single branch superposition theorem of both-branch fuzzy decision, exclusive-decompose principle of both-branch fuzzy decision. This paper presents encryption-authentication theorem of decision by engrafting information security theory and technique with both-branch fuzzy decision, and provides encryption-decryption and signature-authentication of both-branch fuzzy decision.