阿姆斯特朗數
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年引入和開發的一整套推理規則或公理。推理規則是健全的，用于測試功能依賴項的邏輯推理。 也稱為聲音的公理用于推斷關系數據庫上的所有功能依賴性。 公理是一組規則，當將其應用于特定集合時，會生成功能依賴關系的閉包。
Armstrong's Axioms has two different set of rules,
阿姆斯特朗公理有兩個不同的規則集，
Axioms or primary rules
公理或主要規則
- Axiom of Reflexivity
- Axiom of Augmentation
- Axiom of Transitivity
Additional rules or Secondary rules
附加規則或輔助規則
- Union
- Composition
- Decomposition
- Pseudo Transitivity
1) Axioms or primary rules
1)公理或主要規則
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.
讓與該組屬性的假設T(k)的 k為一個關系模式。 隨后，我們將k的子集表示為A ， B ， C 。 數據庫理論中屬性集的標準表示法是AB而不是A∪B 。
Axiom of Reflexivity:
自反公理 ：
If a set of attributes is P and its subset is
如果一組屬性為P并且其子集為
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 ，則P持有Q。 如果Q?P ，則P→Q 。 此屬性稱為瑣碎功能依賴項。 當P持有Q ( P→Q )時， P在功能上決定Q。
Axiom of Augmentation:
擴充公理 ：
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擁有Q ( P→Q )， R是一組屬性，然后PR擁有QR ( PR→QR )。 這意味著依賴項中的屬性更改不會創建基本依賴項中的更改。 如果P→Q ，則任何R的 PR→QR 。
Axiom of Transitivity:
傳遞公理 ：
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在功能上決定R ，與P持有Q和Q持有R相同 。
2) Additional rules or secondary rules
2)附加規則或次要規則
These rules can be derived from the above axioms.
這些規則可以從上述公理導出。
Union:
聯盟 ：
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 ，則X→YZ 。
Composition:
組成 ：
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,
證明，
- P → Q (Given)
- A → B (Given)
- PA → QA (Augmentation of 1 and A)
- PA → Q (Decomposition of 3)
- PA → PB (Augmentation of 2 and P)
- PA → B (Decomposition of 5)
- PA → QB (Union 4 and 6)
Decomposition:
分解 ：
This rule is contrary of union rule. If
此規則與工會規則相反。 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 ，則P保持Q ( P→Q )， P保持R ( P→R )。 如果X→YZ ，則X→Y和X→Z 。
proof,
證明，
- P → QR (Given)
- QR → Q (Reflexivity)
- P → Q (Transitivity of 1 and 2)
Pseudo Transitivity:
偽傳遞性 ：
If
如果
P → RQ and Q → S, then P → RS.
P→RQ和Q→S ，然后P→RS 。
proof,
證明，
- P → RQ (Given)
- Q → S (Given)
- RQ → RS (Augmentation of 2 and R)
- P → RS (Transitivity of 1 and 3)
Trivial Functional Dependency
瑣碎的功能依賴
| 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，則稱為瑣碎函數依賴。 平凡的總是擁有功能依賴。 |
| 非優惠 | 如果P持有Q ( P→Q )，其中Q不是P的子集，則稱其為非??私有功能依賴性。 |
| 完全非優惠 | 如果P保持Q ( P→Q )，其中P與Y =Φ相交，則稱為完全非依賴函數依賴。 |
翻譯自: https://www.includehelp.com/dbms/armstrongs-axioms-in-functional-dependency.aspx
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