Production-Tabular Knowledge Base Correctness Checking Tools

Production-tabular knowledge bases are widely used in commercial expert systems. One of the main problems arising from the operation of such knowledge bases is their correctness. The reliability of the inference mechanism and the robustness of the expert system at “paradigm shift” depend largely on successful resolution of this problem. The paper gives a formal definition of “correctness” of extended entry production-tabular knowledge bases and proposes an algorithm to control their correctness. The obtained results create theoretical preconditions to ensure the reliability and robustness of the production-tabular technologies widely used in expert systems of diagnostics, monitoring, management, forecasting, decision-making


INTRODUCTION
Production-tabular knowledge bases under investigation are a class of hybrid structures of knowledge representation, in which production systems are described in terms of extended entry decision tables [1][2][3]. The formalism of decision tables can significantly expand the application of popular expert systems based on production rules.
In particular, they can be used to check the correctness (completeness and consistency) of knowledge bases, which is a critical issue for these systems.
The paper considers production-tabular knowledge bases, which are a class of hybrid structures of knowledge representation, in which production systems are described in terms of extended entry decision tables [1][2][3].
However, said tools of control are generally focused on too closed and static problem environment models, i.e. on the situations, in which inference mechanisms are determined at the stage of their formation and are not corrected during operation. This paper discusses tools for testing the correctness both at the stage of formation of inference mechanisms and at their possible modification during operation.
Tools of this kind are relevant for expert systems operating in open and dynamic problem environments typical for the vast majority of reallife problems.
The paper attempts to solve this issue in the framework of the mathematical model of decision table, regarded as the isomorphism of the production-tabular knowledge base.
For the analysis of correctness, we use the modified technique developed in [14,15] for limited entry decision tables (tables with doubledigit "Yes / No" terms).

BASIC CONCEPTS AND DEFINITIONS
Formally, the decision table is given [14] by the is the set of conditions or identifiers of conditions regarded as the coordinates of a set of data vectors representing elementary states of the problem environment; are the matrices establishing the relationship between the data vectors (or states) and solutions.
The general structure of a decision table is shown in Table 1.  Table name Rule The pair are the vector-columns of matrices C and D is called the solutions rule (rule R).
The pair , , where the symbol " * " means that the first element of the pair is undetermined, is called the "otherwise" rule (rule E).
Rule E is used for fixing the situations which are anomalous in terms of semantics of the problem environment, and entered into a decision table to rectify possible incompleteness of the knowledge base.
A set of states of the problem environment is the set consisting of the data vectors λ establish the relationship between the data vectors (or states) and solutions.
The values of the matrix elements C and D have the following meaning: In this case, we say that the data vector causes inconsistency of decision tables for rules j R and p R .

Definition 3. A decision table is called correct
concerning S, if it is complete and consistent relative to S. Otherwise, a decision table is called incorrect relative to S.
The correctness of a decision table relative to S is also called semantic correctness or correctness relative to a given problem interpretation. The correctness of a decision table relative to the set N is the syntactic correctness or correctness relative to any problem interpretation.
Before proceeding to describe the correctness checking algorithms, we will make some remarks.

Remark 1.
Since S is determined by the specifics of a current problem and is usually given implicitly (through a system of constraints), let us take the set N as S for universality. Accordingly, will check the correctness of decision tables relative to the set N.

Remark 2.
In the event of inconsistency or incompleteness of a decision table against N, we assume that there is a processor (e.g., the compiler of decision tables), capable of establishing the correctness or incorrectness of decision tables relative to S based on the output of the algorithm.
Thus, the issue of semantic correctness in this case depends on the processor. ,

CONSISTENCY CHECK
Accordingly, the necessary and sufficient condition of inconsistency of a decision table relative to S for the rules j R and p R is the The scheme for the proof of the lemma is borrowed from [15].
Consistency check algorithm is determined.

COMPLETENESS CHECK
According to Definition 1 (with the substitution of N for S), a decision is the number of data vectors contained in various intersections of the elements from 1 S to Z, or the number of data vectors satisfying simultaneously Z solution rules.

CONCLUSION
The isomorphism between decision tables and production structures gives grounds for regarding the proposed scheme of correctness control as basic for production-tabular systems in general, both for limited-entry and extended-entry systems.
It should also be noted that the scheme can be used both at the stage of development of production-tabular systems, and their possible modifications during operation. This is important in open and dynamic problem environment characterized by high requirements to reliability and promptness of decisions.
The proposed verification algorithm was used in the "System of reactive diagnostics of Ethernet LAN" [16], "System of on-line diagnostics of power plants" [17], and in the "System of predicting the preservation of sinus rhythm after the elimination of a cardiac fibrillation" [