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Function packet set is a dynamic model of information law, which introduces function concept into packet set and improves it; function packet set is a function set pair composed of function internal packet set $$ S^{{\bar{F}}} $$ and function outer packet set $$ S^{F} $$ ; or, $$ (S^{{\bar{F}}} ,S^{F} ) $$ is a function packet set. Packet reasoning is a dynamic reasoning generated from packet set, and it’s composed of internal packet reasoning and outer packet reasoning. This paper introduces function into packet reasoning, improves packet reasoning and gives packet information law reasoning; by cross-penetration of function outer packet set with outer packet information law reasoning, the studies on intelligent fusion of outer packet information law and its outer separating are given. The generating of outer packet information law reasoning of intelligent fusion of outer packet information law, intelligent fusion of outer packet information law with attribute conjunctive reduction theorem, outer separating and reducing of intelligent fusion of outer packet information law and outer separating of intelligent fusion of outer packet information law with discovery-application of unknown information law are given in this paper.

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.

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.

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.

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.

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