JPH02284230A - Deductive computation system based on derivation principle - Google Patents

Abstract

PURPOSE:To obtain the conclusion of inference through less number of times of compu tation by converting the form of an inputted logical expression into a standard form, and obtaining a solution by executing the deductive cumulative computation of the logical expression of the standard form. CONSTITUTION:The logical expression is converted into the standard form, and the conclusion is obtained by executing the deductive cumulative computation of the logical expression of the standard form. In this case, in respect of a first logical expres sion, for instance, 'a' among N-pieces of the logical expressions, all of the logical conversion of '-a' are found out from among (N-1) pieces of the remaining logical equations, and are eliminated. As the result of this operation, it is assumed that the solution of 'b', for instance, is obtained. Next, all the variables of '-b' are found out from among all the remaining logical expressions, and are eliminated. When such the operation is repeated, the number of times required for finding out all the variables is N(N-1)/2, that is, the order of N<2>. Thus, the number of times of the computation for obtaining the solution by executing the inference of N pieces of the logical expressions can be made the order of N<2>, and the number of times of the computation is reduced.

Description

[Detailed Description of the Invention] (Industrial Application Field) The present invention relates to inference methods in expert systems, etc., which are a branch of artificial intelligence, and in particular, by performing deductive operations based on derivation principles, the number of operations is limited to a finite number of times. This paper concerns an arithmetic method based on derivation principles that allows correct conclusions to be obtained.

(Prior Art) As a field of artificial intelligence, expert systems are being used in various fields that express the knowledge of a specific expert on a computer and allow the computer to solve problems in that field.

FIG. 4 is a block diagram showing an example of the configuration of such an expert system. In addition, in the following explanation, "~
” is negation, “v” is disjunction, “→” is implication, rP, Q, a
, b, . . .'' indicate logical variables. The knowledge base storage means 50 stores the knowledge of experts in the field as rules (logical formulas).

This rule is generally used for affirmative expressions in syllogisms (Mo
For example, in a syllogism, we know that ``A is true,'' and if there is a rule J such as ``If A, then B, then ``B is true.'' Adopt a form of reasoning that concludes that "is true." The user interface means 52 inputs a question from the user and passes it to the inference means 51. The inference means 51 performs inference on all the logical rings stored in the knowledge base storage means 50 and obtains an answer. . In general, when any two logical expressions contain the same logical variable, one with a negation symbol and one without, a new logical expression can be derived (
Resolution). This derivation is the following calculation. The purpose is to remove the logical formula 'QJ and its negation "~Q" from two logical formulas, such as "~PVQJ and "~QVRJ," and derive a new logical formula "~PVRJ.

Such processing is performed for all N rules stored in the knowledge base storage means 50.

Regarding the logical formula obtained by the derivation, the above-mentioned derivation is also performed between any two logical formulas between it and other logical formulas to obtain a solution.

(Issues to be solved by the invention 11J) As mentioned above, the conventional inference method performs derivation when two arbitrary logical expressions include the same logical variable, one with a negation symbol and one without. The number of such combinations for N logical formulas will not be less than 29 orders, therefore, as the number N of logical formulas increases, the number of combinations will increase explosively (NP-Comp I et e)
L. There are cases where a solution cannot be found within a realistic calculation time. As described above, conventional inference methods have problems that need to be solved.

The present invention has been made in view of these circumstances, and its purpose is to provide a deductive calculation method based on a derivation principle that can obtain an inference conclusion with fewer calculations than conventional methods.

(Means for Solving the Problem) In order to achieve the above object, the deductive calculation method based on the derivation principle of the present invention has an input means for inputting a logical formula, and a standard format for the logical formula input by the input means. The present invention has a converting means for converting the standard form into a form, and a deductive calculating means for performing a deductive calculation on a logical formula in a standard form output from the converting means to obtain a solution.

(Operation) In the deductive calculation method based on the derivation principle of the present invention, the input means inputs a logical formula, and the conversion means converts the logical formula input by the input means into a standard form. Then, the deductive calculation means performs a deductive calculation on the logical formula converted into the standard form by the conversion means to obtain a solution.

(Principle of the Invention) In the deductive calculation method based on the derivation principle of the present invention, a logical formula is converted into a standard form, and a deductive calculation is performed on the logical formula 12 in the standard form to obtain a conclusion. Multiple logical expressions of any form by performing deductive operations on logical expressions of standard form? i
It is possible to obtain a logically correct conclusion with a finite number of calculations.

That is, the following processing is performed. Here, the first logical expression is, for example, "aJ." The derivation for "a" is the remaining N-1
This corresponds to finding all the logical variables "~a" from among the logical equations and deleting them. The number of times this search is required is on the order of N-1. As a result of this operation, it is assumed that, for example, a solution "b" is obtained in the 0th order, and all variables "~b" are searched out from all remaining logical expressions and deleted. The number of searches required for this is on the order of N-2. Assume that a solution r CJ is obtained as a result.Next, ``~C'' is searched out from the remaining logical formulas N-3g and deleted. The number of searches required to repeat this operation and search for all variables is (N1) + (N-2) +...
12+1=N (N-1)/2, that is, on the order of N2. Therefore, in a method that performs inference using such a calculation method, the number of searches is not NP-Complete.

The principle of the present invention will be specifically explained using the following seven logical expressions as examples.

p formula (1 p″g formula 2~r→~g
Formula 3~S
Formula 4S→ (~rVj) Formula 5
S→ (tVu) Formula 6t→ (pV
s) Equation 7 First, Equation (1) to Equation (7)
If we take the standard form, that is, the form that does not include the "implication (→)", we obtain equations (8) to (14).

p Formula (8) ~PVg Formula (9) rV~g Formula 10~S
Formula 11~gV(~r vj
) Formula 12~5V (tVu)
Equations 13 - tV (pVs) Equation 14 Next, perform the following deductive operation on equations (8) - (14). First, since equation (8) is a single logical variable 'PJ,
Find the logical variable "~ρ" in the other six equations and delete them all. As a result, the solution 'gJ is obtained from equation (9).
The negation of this logical variable "g", that is, "~g" contained in equations (10) to (14) are all deleted. As a result,
The solution ``rJ'' is obtained from equation (10). Next, all variables ``~r'' are deleted from the remaining logical expressions. This operation is repeated until no new solution can be obtained; 6×5/2=15 searches are sufficient to complete all operations. In the conventional search method, it is necessary to perform 2' = 64 searches for a deductive operation including the six logical equations as described above. If the number N of logical expressions is large, 2N and N
The difference between 2 and 2 is even greater.

(Example) Next, an example of the present invention will be described in detail with reference to the drawings.

FIG. 1 is a block diagram of an embodiment of the present invention. In the figure, 1 is an input means for inputting rules (logical formulas) and questions, 2 is a knowledge base storage means for storing the logical formulas input by the input means 1, and 3 is stored in the knowledge base storage means 2. Converting means 5 converts a logical formula into a standard form and stores it in the work memory 4; input means 1 performs inference based on the standard form logical formula stored in the work memory 4;
6 is an output means for outputting the solution obtained by the deductive calculation means 5.

FIG. 2 shows an example of a logical formula input to the embodiment of FIG. 1. FIG. 3 shows a logical formula obtained by converting the logical formula of FIG. 2 into a standard form by the conversion means 3.

Hereinafter, an explanation will be given of the operation when inference is made based on the logical formula shown in FIG. 2 in the embodiment shown in FIG. 1. When activated from the outside, first, the input means 1 starts operating. The input means 1 inputs a logical formula as shown in FIG. 2 and stores it in the knowledge base storage means 2. Then, the converting means 3 is notified that the input processing has been completed.

Upon receiving this notification, the conversion means 3 converts the logical formula stored in the knowledge base storage means 2 into a standard form and stores it in the work memory 4. As a result, the logical formula shown in FIG. 2 is converted into the logical formula shown in FIG. 3. For example,
The logical formula "X→b" in Figure 2 (1) is converted to the logical formula "~XVbJ" in Figure 3 (9).
The logical expressions in ) to (8) are also converted to the logical expressions in FIG. 3 (10) to (16).

Next, the input means 1 inputs a question "X?" from the user, that is, a question asking for the value of the logical expression "X", and passes it to the deductive calculation means 5 and activates it. teeth,
The following deductive operation is performed based on this logical formula "X".

■ Among the standard form logical expressions in work memory 4, the logical expression
Find all logical expressions that include the negation [~X] of '~X
” is deleted and the obtained solution is stored in the work memory 4. Third
The logical expressions “~Xvb” and “~Xvj” in Figures (9) and (14)
” includes “~X”, so delete this “~X” and get “b” as the solution.

"j" is obtained. And this logical formula "b".

“j” is stored in the work memory 4.

■ Input this logical formula rb in the work memory 4, find a logical formula that includes the negation "~b" of the logical formula rbJ among the logical formulas in the work memory 4, and delete all "~b". The obtained solution is stored in the work memory 4. Figure 3 (10
)'s logical expression ``~bVcJ'' includes ``~b'', so
By eliminating "~b", "C" is obtained as a solution. Then, it is stored in the work memory 4.

■ Input "j" stored in the work memory 4 in ■ and perform the same processing as described above. "~hJ" is obtained from the logical formula "~hv~j" in FIG. 3 (15). As a result, it can be seen that the value of logical formula X is not "h".

■ Input the logical formula "c" stored in the work memory 4 in ■ and perform the above-mentioned processing.
16) logical formula rd V ~(''.

“~d” and “~f” are obtained as solutions from “~fV~C”. As a result, the value of the logical expression "x" is rd.

It turns out that it is not an "f" either.

(2) This logical formula "~d" is input and processed, and "e" is obtained as a solution from the logical formula 'dVeJ in FIG. 3 (12).

■Input and process this logical formula "e".

From the logical formula "~evfvgvh" in Figure 3 (13), rf
VgVhJ is obtained.

■In this logical formula rfVgVhJ, we have already obtained “
~h” and process it. As a result, rfVgJ is obtained as a solution.

■This logical formula “~f which has already been obtained by ■ in fVgJ
” is input and processed. As a result, "g" is obtained.

As a result of the above processing, 'gJ is obtained as the value of the logical expression "X". The number of times of this calculation may be 8×7/228 times.

In this way, the operation of sequentially reducing the number of logical variables through derivation and searching for a solution is a deductive operation.

In this embodiment, when performing deduction, the order in which logical expressions are arranged does not matter, and logical expressions can be added or deleted whenever necessary. This is an advantage over expert systems, where changing the order of the rules can lead to different conclusions.

(Effects of the Invention) As explained above, according to the deductive calculation method based on the derivation principle of the present invention, it is possible to perform inference on N logical formulas and reduce the number of calculations to obtain a solution to N2 order. I can,
This number of operations is much smaller than that of conventional inference methods.

[Brief explanation of drawings]

Fig. 1 is a block diagram of an embodiment of the present invention, Fig. 2 is a diagram showing an example of a logical formula input to the embodiment of Fig. 1, and Fig. 3 is a diagram of the logical formula in Fig. 2 in a standard format. FIG. 4 is a diagram showing an example of the configuration of a conventional expert system. 1... Input means, 2... Knowledge base storage means, 3.
... Conversion means, 4. Work memory, 5. Deductive calculation means, 6. Output means. Agent Patent Attorney Honjo Brokerage Figure 1 ~X v b ~bVc ~dV~C Ve Figure 3 = 16 Figure 4

Claims (1)

[Claims] Input means for inputting a logical formula; conversion means for converting the format of the logical formula input by the input means into a standard form; A deductive calculation method based on a derivation principle, which has a deductive calculation means that performs deductive calculations to obtain a solution.