Research on Mathematical Dialectical Logic for Intelligent Information Processing †
Abstract
:1. The Basic Assumptions of Information and Intelligence
2. The Research Program of Universal Logics
2.1. Origin
2.2. Basic Principle
2.3. The Three Domains of the Universal Logics W = <{ ⊥ }∪ Range <α>, Domain, Model Domain>
Range | Domain | Model Domain |
---|---|---|
[0,1]n,n>0 | Multigranularity | Continuous state |
[0,1]n,n =1,2,3,… | Singlegranularity | Multistate |
[0,1] Continuous | Singlestate | |
{0,u,1} Triple | ||
{0,1} Two |
2.4. Layer up to Establish a Variety of Reasoning Model
Feature | Inference Mechanism | Inference Model |
---|---|---|
Changing information | Dynamic interaction and balance mechanism of internal and external parameters | Evolutionary reasoning |
Incomplete information | Containment and correction mechanisms for errors | Sub coordination reasoning; Non monotonic reasoning; Fault-tolerant reasoning |
Reasoning based on certain assumptions | Analogic, hypothetical deduction, case-based reasoning | |
From special premise to general hypothesis | Incompletely inductive andDiscovery reasoning | |
Complete information | From a special premise to a general conclusion | Completely inductive inference |
From general premise to special conclusion | Deductive inference | |
Fundamental theory | Propositional connectives, quantifier | Predicate calculus |
Propositional connectives | Propositional calculus |
2.5. Introduce Various Flexible Quantifier to Universal Logics
- ◆
- Threshold element quantifier signing the truth value’s deviation in proposition ♂k
- ◆
- Hypothetical quantifier indicating the Degree of trust of a proposition $k
- ◆
- Scope quantifier restricting Individual variable range ∮k
- ◆
- Position quantifier indicating Relative position of individual variables ♀k
- ◆
- Transition quantifiers changing the transition properties of truth value distributions ∫k
2.6. The Application Needs to Gradually Expand as the Establishment of Universal Logic Is a Gradual Growth Process
3. A Variety of Universal Propositional Logic Operator of Integrity Cluster (Library)
3.1. The Comparative Analysis of Four Continuous Logic Operator
3.2. Find the Breakthrough
3.3. Find the Base Model of Flexible Propositional Logic
NOT operator | x = 1 −x |
AND operator | x ∧ y = Γ[x + y − 1] |
OR operator | x ∨ y = 1 − Γ[((1 − x) + (1 − y) − 1)] |
Implication operator | x → y = Γ[1 – x + y] |
Equivalence operator | x↔y = 1 − |x − y| |
Average operator | x℗y= 1 − ((1 − x)/2 + (1 − y)/2) |
Combination operator | x©ey= Γ[x + y − e] where Γ[ ] is 0,1 limiting function. |
3.4. Find All the Uncertainties Accommodated in the Flexible Propositional Logic Operator
3.5. Find the Way and Extent of Impact from Various Uncertainties on the Logical Operator Base Model
3.6. Integrity Cluster of Flexible Logic Operator
- 0-type flexible logic has only one base model. L0(x, y), x, y∈[0,1].
- 1-type flexible logic has three subtypes, which are all complete logical spectrum.Lk(x, y, k), x, y, k∈[0,1], Lh(x, y, h), x, y, h∈[0,1], Lβ(x, y, β), x, y, β∈[0,1]
- 2-type flexible logic has three subtypes, which are all complete logical spectrum. Lk, h(x, y,k, h), x, y, k, h∈[0,1], Lhβ(x, y, h,β), x, y, h,β∈[0,1], Lkβ(x, y, k,β), x, y, k,β∈[0,1]
- 3-type flexible logic has only one base model, which is complete logical spectrum. L3(x, y, k, h, β), x, y, k, h, β∈[0,1]
3.6.1. Generalized Correlation on the Adjustment of Logic Operators
3.6.2. Generalized Correlation and Error Co-Ordination of Logical Operators
3.6.3. Generalized Correlation and Relative Weights for the Adjustment of Logical Operators
3.7. Discovery the Integrity of the Logic
- (1)
- The soundness of standard logic has been contained in reliability and completeness, and does not require special consideration.
- (2)
- After introducing all kinds of uncertainties into the Universal Logics, the reasoning result becomes indefinite. The original logical operate of isolated existence is expanded into a continuous change logical operator spectrum, which contains an infinite number of operators changes in uncertain parameters.
- (3)
- The logic operator spectrum with h parameter is the most basic logic operator spectrum. The flexible logic operator spectrum with k, h,β parameters lays the mathematical foundation for establishing the complete propositional logic.
- (4)
- Continuous value logical reasoning is the process of coexistence of symbolic calculus and numerical calculation. The result is still a continuous value. There must be an intermediate transition value x∈(0, 1) the two extremes 0 and 1. The reliability and completeness of the standard logic can only guarantee that the propositional truths do not distort at x∈{0, 1} in the process of deduction, there is no guarantee that the intermediate transition value x∈{0, 1} does not cause distortion. The anomalous results that appear in some nonstandard logic are a manifestation of this distortion. In order to avoid the possibility of distortion of the intermediate transition value x∈(0, 1) in the process of deduction, the sufficient and necessary conditions for the flexibility of the flexible logic are proposed as follows:
- (5)
- Definition: a sound flexible logic system must have the following basic properties:
- (6)
- It canbe proved h-type flexible logic spectrum is a sound logic system, it satisfies:
4. The Application on Reasoning Calculation and Online Identification
4.1. Discovering the Complete Clusters in Propositional Universal Logics
- (1)
- As to logical reasoning, we can use such clusters directly. By calculating operator-type parameters (a, b, e) and uncertainty parameters (k, h,β), we can chose the appropriate operator and get the result.
- (2)
- As to machine learning and data mining, we can use such clusters indirectly. Through on-line data processing we can distinguish operator-type parameters (a, b, e) and uncertainty parameters (k, h,β) of each logics node( like neuron). Automatically, we can identify logic behavior of processing objects.
4.2. Application in Many-Dimensions Flexible Logics and Many-Valued Logic
5. Future Work
Conflicts of Interest
References
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Input | The Boolean Algebra Can Describe the Output z of All 16 Kinds of Information Processing (Transform) Modes | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
x | y | 0① | x∧ y② | x∧ y③ | x ④ | x∧ y⑤ | y ⑥ | x≠y⑦ | x∨ y⑧ | x∧ y⑨ | x=y⑩ | y ⑪ | y→x⑫ | x ⑬ | x→y⑭ | x∨ y⑮ | 1⑯ |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Parameter | All 16 Types of Status Parameters of Information Processing (Transformation) Mode | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
① | ② | ③ | ④ | ⑤ | ⑥ | ⑦ | ⑧ | ⑨ | ⑩ | ⑪ | ⑫ | ⑬ | ⑭ | ⑮ | ⑯ | |
a | 0 | 1 | 1 | 1 | −1 | 0 | Combination implementation | 1 | −1 | Combination implementation | 0 | 1 | −1 | −1 | −1 | 1 |
b | 0 | 1 | −1 | 0 | 1 | 1 | 1 | −1 | −1 | −1 | 0 | 1 | −1 | 1 | ||
e | 0 | 1 | 0 | 0 | 0 | 0 | 0 | −1 | −1 | −1 | −1 | −1 | 2 | −1 |
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He, H.; Zhou, Y.; Chen, Z. Research on Mathematical Dialectical Logic for Intelligent Information Processing. Proceedings 2017, 1, 149. https://doi.org/10.3390/IS4SI-2017-03993
He H, Zhou Y, Chen Z. Research on Mathematical Dialectical Logic for Intelligent Information Processing. Proceedings. 2017; 1(3):149. https://doi.org/10.3390/IS4SI-2017-03993
Chicago/Turabian StyleHe, Huacan, Yanquan Zhou, and Zhicheng Chen. 2017. "Research on Mathematical Dialectical Logic for Intelligent Information Processing" Proceedings 1, no. 3: 149. https://doi.org/10.3390/IS4SI-2017-03993
APA StyleHe, H., Zhou, Y., & Chen, Z. (2017). Research on Mathematical Dialectical Logic for Intelligent Information Processing. Proceedings, 1(3), 149. https://doi.org/10.3390/IS4SI-2017-03993