Chapter 8 Relational Database Design¶

Abstract

1NF：字段是最小的的单元不可再分
2NF：满足1NF,表中的字段必须完全依赖于全部主键而非部分主键
3NF：满足2NF,非主键外的所有字段必须互不依赖
BCNF：主键通常为一个。如果主键为多个，不能存在部分或传递依赖。
正则分解$F_c$(找到正交基)
依赖保持判断
最小函数依赖集$F_m$

Introduction(分解)¶

Example

What about combining instructor and department?

Pitfalls of the “bad” relations

Information repetition (信息重复)
Insertion anomalies (插入异常)
Update difficulty (更新困难)

数据之间存在着隐含的函数约束关系，知道了 id 就可以决定其他元素。 e.g. id $\rightarrow$ name, salary, dept_name; dept_name $\rightarrow$ building, budget
产生冗余的原因是 dept_name 决定了部分属性，但他却不是这个表的 primary key.
好的关系：只有 candidate key 能决定其他属性。
拆表后要有重叠的属性，否则无法拼接回去。这里的公共属性必须是分拆出一个关系模式的 primary key, 这是无损（没有信息损失）连接。

lossy decomposition

employee(ID, name, street, city, salary) $\rightarrow$ employee1 (ID, name) and employee2 (name, street, city, salary)

Example of Lossless-Join Decomposition

Lossless-join Decomposition(无损分解)¶

Let $R$ be a relation schema and let $R_1$ and $R_2$ form a decomposition of $R$. That is $R = R_1 \cup R_2$.

We say that the decomposition is a lossless decomposition if there is no loss of information by replacing R with the two relation schemas $R = R_1 \cup R_2$. 如果用两个关系模式$R_1$和$R_2$去代替$R$没有信息丢失就是无损分解，用关系代数更简洁地可以表示为 $r = \prod_{R_1}(r) \bowtie \prod_{R_2}(r)$.

And, conversely a decomposition is lossy if $r\subset \prod_{R_1}(r) \bowtie \prod_{R_2}(r)$对于有损分解，我们自然连接获得了更多的元组，但实际意义上我们却拥有更少的信息。 Note: more tuples implies more uncertainty (less information).

A decomposition of $R$ into $R_1$ and $R_2$ is lossless join if at least one of the following dependencies holds: (充分条件)

Proof

$R_1\cap R_2\rightarrow R_1$
$R_1\cap R_2 \rightarrow R_2$
即公共属性是其中一个关系的 Key.

Devise a Theory for the Following(规范化理论)¶

Decide whether a particular relation R is in “good” form(确定一个关系是否是良构的).
In the case that a relation R is not in “good” form, decompose it into a set of relations $\{R_1, R_2, \ldots, R_n\}$ such that
- each relation is in good form
- the decomposition is a lossless-join decomposition 如果关系是不好的，我们希望把它无损分解成好的关系。
Our theory is based on:
- functional dependencies函数依赖
- multivalued dependencies
Normal Forms(NF): $1NF \rightarrow 2NF \rightarrow 3NF \rightarrow \textbf{BCNF} \rightarrow 4NF$ 有些函数依赖，不能在 BCNF 中得到体现，需要把几个表拼在一起才能体现，叫依赖保持。这时我们需要从 BCNF 回到 3NF.

Functional Dependencies(使用函数依赖进行分解)¶

超码:给定$r(R)$,$R$的一个子集$K$,对于所有的元组有：$t_1 \neq t_2 \rightarrow t_1[K] \neq t_2[K]$

Functional Dependencies are constraints on the set of legal relations. (来自于应用层面的规定)
Require that the value for a certain set of attributes determines uniquely the value for another set of attributes. e.g. dept_name $\rightarrow$ building
A functional dependency is a generalization of the notion of a key.

Let $R$ be a relation schema $\alpha\subseteq R$ and $\beta\subseteq R$ ($\alpha, \beta$ 是属性的集合) The functional dependency $\alpha\rightarrow \beta$ holds on $R$ if and only if for any legal relations $r(R)$, whenever any two tuples $t_1$ and $t_2$ of $r$ agree on the attributes $\alpha$, they also agree on the attributes $\beta$. That is,

\[ t_1[\alpha] = t_2 [\alpha] \Rightarrow t_1[\beta] = t_2 [\beta] \]

实例满足函数依赖 $\rightarrow$ 函数依赖在模式$r(R)$上成立

通过数据库实例可以证伪函数依赖，但不能证实。（依赖是来自应用层面的规定，先有函数依赖，再有数据库中的数据）

Example

$A\rightarrow B$ 可以证伪，但也不能因此就说 $B\rightarrow A$

K is a superkey for relation schema $R$ if and only if $K\rightarrow R$
K is a candidate key for $R$ if and only if
- $K\rightarrow R$, and
- for no $\alpha\subset K$, $\alpha\rightarrow R$

A functional dependency is trivial if it is satisfied by all relations.
全集可以决定子集。
In general, $\alpha\rightarrow \beta$ is trivial if $\beta\subseteq \alpha$

Closure(闭包)¶

Closure of a Set of Functional Dependencies¶

Given a set $F$ of functional dependencies, there are certain other functional dependencies that are logically implied by $F$.
The set of all functional dependencies logically implied by $F$ is the closure of $F$. We denote the closure of $F$ by $F^+$. (能够从给定的集合$F$推导出的所有函数依赖的集合)

e.g. $F=\{A\rightarrow B,B\rightarrow C\}$ then $F^+=\{A\rightarrow B, B\rightarrow C, A\rightarrow C, AB\rightarrow B, AB\rightarrow C,\ldots\}$

We can find $F^+$, the closure of $F$, by repeatedly applying Armstrong’s Axioms:

if $\beta\subseteq \alpha$ then $\alpha \rightarrow \beta$ (reflexivity, 自反律)
if $\alpha\rightarrow \beta$ then $\gamma \alpha \rightarrow \gamma \beta$ (augmentation, 增补律)
if $\alpha\rightarrow \beta$ and $\beta \rightarrow \gamma$ then $\alpha\rightarrow \gamma$ (transitivity, 传递律)

These rules are

Sound（正确有效的） generate only functional dependencies that actually hold
Complete（完备的） generate all functional dependencies that hold

Example

Additional rules:

If $\alpha\rightarrow \beta$ holds and $\alpha\rightarrow \gamma$ holds, then $\alpha\rightarrow \beta\gamma$ holds (union, 合并)
If $\alpha\rightarrow \beta\gamma$ holds, then $\alpha\rightarrow \beta$ holds and $\alpha\rightarrow \gamma$ holds (decomposition, 分解)
If $\alpha\rightarrow \beta$ holds and $\gamma \beta\rightarrow \delta$ holds, then $\alpha \gamma\rightarrow \delta$ holds (pseudotransitivity，伪传递)

Example

函数依赖，右边的公共属性可以去掉，使得函数双方没有交集。

Closure of Attribute Sets(属性集的闭包)¶

Given a set of attributes $a$, define the closure of a under $F$ (denoted by $a+$) as the set of attributes that are functionally determined by $a$ under $F$

Example

Uses of Attribute Closure¶

Testing for superkey:(判断是否为超码，如果是，则$a^+$中包含所有属性) To test if $\alpha$ is a superkey, we compute $\alpha+$, and check if $\alpha+$ contains all attributes of $R$.
Testing functional dependencies
- To check if a functional dependency $\alpha\rightarrow \beta$ holds (or, in other words, is in $F+$), just check if $\beta\subseteq\alpha+$.
- That is, we compute $\alpha+$ by using attribute closure, and then check if it contains $\beta$.检查某个属性是个否属于这个闭包
- Is a simple and cheap test, and very useful
Computing closure of F For each $\gamma\subseteq R$, we find the closure $\gamma+$, and for each $S \subseteq \gamma+$, we output a functional dependency $\gamma\rightarrow S$.

Example

Canonical Cover（正则覆盖）¶

函数依赖是最简单的形式，不存在冗余的函数依赖。（因为要用于检查插入的数据是否会违反函数依赖）
如果我们可以去除函数依赖的一个属性而不改变其闭包，则称该属性是无关的
从函数依赖的左侧删除一个属性可以使其成为更强的约束
从函数依赖的右侧删除一个属性可以使其成为更弱的约束

a canonical cover of F is a “minimal” set of functional dependencies equivalent to F, having no redundant dependencies or redundant parts of dependencies.

Example

A canonical cover for $F$ is a set of dependencies Fc such that

$F$ logically implies all dependencies in $F_c$
$F_c$ logically implies all dependencies in $F$
No functional dependency in $F_c$ contains an extraneous attribute
Each left side of functional dependency in $F_c$ is unique.

How to find the extraneous attribute(无关属性)

从左侧删除：如果$A \in \alpha$，且$F$逻辑蕴含$(F-\{\alpha \rightarrow \beta\}) \cup \{(\alpha-A) \rightarrow \beta \}$
从右侧删除：如果$A \in \beta$，且$F$逻辑蕴含$(F-\{\alpha \rightarrow \beta\}) \cup \{\alpha \rightarrow (\beta - A) \}$

Computing a Canonical Cover

Normal Form(范式)¶

Boyce-Codd Normal Form¶

消除了基于函数依赖能够发现的全部冗余，虽然可能还保留着其它类型的冗余

A relation schema $R$ is in BCNF with respect to a set $F$ of functional dependencies if for all functional dependencies in $F^+$ of the form where $\alpha \subseteq R$ and $\beta \subseteq R$, at least one of the following holds

$\alpha \rightarrow \beta$ is trivial
$\alpha$ is a superkey for $R$.

任何非平凡的函数依赖的左边都是一个 key.

Decomposing a Schema into BCNF¶

对于不是 key 的函数依赖，就把它分解出来作为单独的关系模式。
Suppose we have a schema $R$ and a non-trivial dependency $\alpha\rightarrow \beta$ causes a violation of BCNF. We decompose $R$ into: $(\alpha \cup \beta)$ and $(R-(\beta-\alpha))$
$\alpha$ 作为两个关系模式的公共属性，也是一个关系的 key, 这样才是无损分解。

Example

Dependency Preservation¶

依赖保持：原来的函数依赖，都可以在分解后的函数依赖中得到单独检验。否则需要把几个关系连接在一起才能检验依赖的，称为依赖不保持。

Constraints, including functional dependencies, are costly to check in practice unless they pertain to only one relation.

If it is sufficient to test only those dependencies on each individual relation of a decomposition in order to ensure that all functional dependencies hold, then that decomposition is dependency preserving (保持依赖). （如果通过检验单一关系上的函数依赖，就能确保所有的函数依赖成立，那么这样的分解是依赖保持的）（或者，原来关系R上的每一个函数依赖，都可以在分解后的单一关系上得到检验或者推导得到。） Because it is not always possible to achieve both BCNF and dependency preservation, we consider a weaker normal form, known as third normal form.

Let $F_i$ be the set of all functional dependencies in $F^+$ that include only attributes in $R_i$. ($F_i$: the restriction of $F$ on $R_i$)

A decomposition is dependency preserving, if $(F_1\cup F_2 \cup \ldots \cup F_n )^+ = F^+$
If it is not, then checking updates for violation of functional dependencies may require computing joins, which is expensive.

Example

Third Normal Form¶

任何一个非平凡函数依赖，如果左边不是一个 super key, 那么右边必须包含在一个 candidate key 里面。

A relation schema $R$ is in third normal form (3NF) if for all: $\alpha\rightarrow \beta $ in $F^+$ at least one of the following holds:

$\alpha\rightarrow \beta$ is trivial (i.e., $\beta \in \alpha$)
$\alpha$ is a superkey for R
Each attribute A in $\beta-\alpha$ is contained in a candidate key for $R$.
候选码有很多个，包含在某一个候选码即可。

Example

Goals of Normalization

In the case that a relation scheme R is not in “good” form, decompose it into a set of relation scheme $\{R_1, R_2, \ldots, R_n\}$ such that

each relation scheme is in good form (i.e., BCNF or 3NF)
the decomposition is a lossless-join decomposition
Preferably, the decomposition should be dependency preserving

E-R Modeling and Normal Forms

这里的无损分解，先指定一个路径，考虑每两个关系直接是否无损（公共属性是否为其中一个关系的 key）。

3NF的缺点：可能不得不用空值来表达数据项之间的某些可能有意义的联系，并且存在信息重复的问题。

Multivalued Dependencies¶

There are database schemas in BCNF that do not seem to be sufficiently normalized.

Example

存在两种不相关的多值依赖。老师 id 可以多值决定 child_name, 又可以多值决定 phone, 但这两个属性是不相关的，放在一个表里就会组合。
第二张图的为 Fourth Normal Form (4NF).

四范式：不存在非平凡的多值依赖。

Let R be a relation schema and let $\alpha\subset R$ and $\beta\subset R$. The multivalued dependency $\alpha\rightarrow\rightarrow\beta$ holds on $R$ if in any legal relation $r(R)$, for all pairs for tuples $t_1$ and $t_2$ in $r$ such that $t_1[\alpha] = t_2 [\alpha]$, there exist tuples $t_3$ and $t_4$ in $r$ such that:

\[ t_3[\alpha] = t_4[\alpha] = t_1[\alpha]=t_2[\alpha]\\ t_3[\beta]=t_1[\beta]\\ t_3[R-\beta]=t_2[R-\beta]\\ t_4[\beta]=t_2[\beta]\\ t_4[R-\beta]=t_1[R-\beta] \]

A relation schema $R$ is in 4NF with respect to a set $D$ of functional and multivalued dependencies if for all multivalued dependencies in $D^+$ of the form $\alpha\rightarrow \rightarrow \beta$, where $\alpha\subset R$ and $\beta\subset R$, at least one of the following hold:

$\alpha\rightarrow \rightarrow \beta$ is trivial (i.e., $\beta \subset \alpha$ or $\alpha \cup \beta= R$)
即除了 $\alpha,\beta$ 为没有其他属性。
$\alpha$ is a superkey for schema $R$

任何一个多值依赖，要么左边就是个 key, 要么这个依赖是平凡的。

E-R Modeling and Normal Forms

不是 BCNF, 因此也不是 4NF.

Overall Database Design Process¶

Denormalization for Performance

Some aspects of database design are not caught by normalization.
有时候我们需要引入冗余，来保持性能。

Example

Chapter 8 Relational Database Design¶

Introduction(分解)¶

Lossless-join Decomposition(无损分解)¶

Devise a Theory for the Following(规范化理论)¶

Functional Dependencies(使用函数依赖进行分解)¶

Closure(闭包)¶

Closure of a Set of Functional Dependencies¶

Closure of Attribute Sets(属性集的闭包)¶

Uses of Attribute Closure¶

Canonical Cover（正则覆盖）¶

Normal Form(范式)¶

Boyce-Codd Normal Form¶

Decomposing a Schema into BCNF¶

Dependency Preservation¶

Third Normal Form¶

Multivalued Dependencies¶

Overall Database Design Process¶

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