阿姆斯特朗数_阿姆斯特朗的功能依赖公理数据库管理系统

阿姆斯特朗数

Armstrong axioms are a complete set of inference rules or axioms, introduced and developed by William W. Armstrong in 1974. The inference rules are sound which is used to test logical inferences of functional dependencies. The axiom which also refers to as sound is used to infer all the functional dependencies on a relational database. The Axioms are a set of rules, that when applied to a specific set, generates a closure of functional dependencies.

Armstrong公理是由William W. Armstrong在1974年引入和开发的一整套推理规则或公理。推理规则是健全的，用于测试功能依赖项的逻辑推理。也称为声音的公理用于推断关系数据库上的所有功能依赖性。公理是一组规则，当将其应用于特定集合时，会生成功能依赖关系的闭包。

Armstrong's Axioms has two different set of rules,

阿姆斯特朗公理有两个不同的规则集，

Axioms or primary rules
公理或主要规则
1. Axiom of Reflexivity
2. Axiom of Augmentation
3. Axiom of Transitivity
Additional rules or Secondary rules
附加规则或辅助规则
1. Union
2. Composition
3. Decomposition
4. Pseudo Transitivity

1) Axioms or primary rules

1)公理或主要规则

Let suppose T (k) with the set of attributes k be a relation scheme. Subsequently, we will represent subsets of k as A, B, C. The standard notation in database theory for the set of attributes is AB rather than A∪B.

让与该组属性的假设T(k)的 k为一个关系模式。随后，我们将k的子集表示为A ， B ， C 。数据库理论中属性集的标准表示法是AB而不是A∪B 。

Axiom of Reflexivity:
自反公理 ：

If a set of attributes is P and its subset is
如果一组属性为P并且其子集为

Q, then P holds Q. If Q ⊆ P, then P → Q. This property is called as Trivial functional dependency. Where P holds Q (P → Q) denote P functionally decides Q.
Q ，则P持有Q。如果Q⊆P ，则P→Q 。此属性称为琐碎功能依赖项。当P持有Q ( P→Q )时， P在功能上决定Q。
Axiom of Augmentation:
扩充公理 ：

If
如果

P holds Q (P → Q) and R is a set of attributes, then PR holds QR (PR → QR). It means that a change in attributes in dependencies does not create a change in basic dependencies. If P → Q, then PR → QR for any R.
P拥有Q ( P→Q )， R是一组属性，然后PR拥有QR ( PR→QR )。这意味着依赖项中的属性更改不会创建基本依赖项中的更改。如果P→Q ，则任何R的 PR→QR 。
Axiom of Transitivity:
传递公理 ：

If
如果

P holds Q (P → Q) and Q holds R (Q → R), then P hold R (P → R). Where P holds R (P → R) denote P functionally decides R, same with P holds Q and Q holds R.
P保持Q ( P→Q )， Q保持R ( Q→R )，然后P保持R ( P→R )。其中P持有R ( P→R )表示P在功能上决定R ，与P持有Q和Q持有R相同 。

2) Additional rules or secondary rules

2)附加规则或次要规则

These rules can be derived from the above axioms.

这些规则可以从上述公理导出。

Union:
联盟：

If
如果

P holds Q (P → Q) and P holds R (P → R), then P → QR. If X → Y and X → Z, then X → YZ.
P保持Q ( P→Q )， P保持R ( P→R )，然后P→QR 。如果X→Y和X→Z ，则X→YZ 。
Composition:
组成：

If
如果

P holds Q (P → Q) and A holds B (A → B), then PA → QB.
P保持Q ( P→Q )， A保持B ( A→B )，然后PA→QB 。

proof,
证明，
1. P → Q (Given)
2. A → B (Given)
3. PA → QA (Augmentation of 1 and A)
4. PA → Q (Decomposition of 3)
5. PA → PB (Augmentation of 2 and P)
6. PA → B (Decomposition of 5)
7. PA → QB (Union 4 and 6)
Decomposition:
分解：

This rule is contrary of union rule. If
此规则与工会规则相反。如果

P → QR, then P holds Q (P → Q) and P holds R (P → R). If X → YZ, then X → Y and X → Z.
P→QR ，则P保持Q ( P→Q )， P保持R ( P→R )。如果X→YZ ，则X→Y和X→Z 。

proof,
证明，
1. P → QR (Given)
2. QR → Q (Reflexivity)
3. P → Q (Transitivity of 1 and 2)
Pseudo Transitivity:
伪传递性 ：

If
如果

P → RQ and Q → S, then P → RS.
P→RQ和Q→S ，然后P→RS 。

proof,
证明，
1. P → RQ (Given)
2. Q → S (Given)
3. RQ → RS (Augmentation of 2 and R)
4. P → RS (Transitivity of 1 and 3)

Trivial Functional Dependency

琐碎的功能依赖

Trivial	If P holds Q (P → Q), where P is a subset of Q, then it is called a Trivial Functional Dependency. Trivial always holds Functional Dependency.
Non-Trivial	If P holds Q (P → Q), where Q is not a subset of P, then it is called as a Non-Trivial Functional Dependency.
Completely Non-Trivial	If P holds Q (P → Q), where P intersect Y = Φ, it is called as a Completely Non-Trivial Functional Dependency.

不重要的	如果P持有Q ( P→Q )，其中P是Q的子集，则称为琐碎函数依赖。平凡的总是拥有功能依赖。
非优惠	如果P持有Q ( P→Q )，其中Q不是P的子集，则称其为非私有功能依赖性。
完全非优惠	如果P保持Q ( P→Q )，其中P与Y =Φ相交，则称为完全非依赖函数依赖。