正则表达式属性（理论）
Regular expression properties (theory)
我正在阅读 Aho 和 Ullman 的著作《解析、翻译和编译理论》。在第 2 章介绍正则表达式的部分中，有一个正则表达式的属性列表。我不明白属性 2 和 8。这是属性列表：
(1) + = +
(2) ∅* =
(3) + ( + ) = ( + ) +
(4) () = ()
(5) ( + ) = +
(6) ( + ) = +
(7) = =
(8) ∅ = ∅ = ∅
(9) * = + *
(10) (* )* = *
(11) + =
(12) + ∅ =
其中∅为表示正则集的正则表达式∅,为任意正则表达式,为空串
性质 (2) 和 (8) 如何证明？
编辑：为了解释+、*等符号，这里给出一些书中给出的定义（引用）：
DEFINITION Let be a finite alphabet. We define a regular set over
recursively in the following manner:
(1) ∅ (the empty set) is a regular set over .
(2) {} is a regular set over .
(3) {} is a regular set over for all in .
(4) If and are regular sets over , then so are
(a) ∪ .
(b) .
(c) *.
(5) Nothing else is a regular set.
Thus a subset of * is regular if and only if it is ∅, {}, or {},
for some in , or can be obtained from these by a finite number of
applications of the operations union, concatenation, and closure.
.
DEFINITION Regular expressions over and the regular expressions
they denote are defined recursively, as follows:
(1) ∅ is a regular expression denoting the regular set ∅.
(2) is a regular expression denoting the regular set {}.
(3) in is a regular expression denoting the regular set {}.
(4) If and are regular expressions denoting the regular sets
and , respectively, then
(a) (+) is a regular expression denoting ∪ .
(b) () is a regular expression denoting .
(c) ()* is a regular expression denoting *.
(5) Nothing else is a regular expression.
我的猜测是 2 和 8 属性可能只是一个简单的数学运算：
属性 2
∅是空集，则∅* = 成立，∅+ = 也成立，∅{Infinity} = 也成立，因为e是空串。
一个正则表达式是一个字符串，因此一个空的正则表达式重复任意次数或任何操作，仍然是一个空的正则表达式，它又等于一个空的右边的字符串。
参考：
Why is the Kleene star of a null set is an empty string?
属性 8
∅ = ∅ = ∅ 是正确的，∅∅∅ = ∅∅∅ = ∅ 也是如此，因为空集与任何东西组合都会产生空集。
参考：
Regular expressions with empty set/empty string
What is the difference between language of empty string and empty set language?
How can concatenating empty sets (languages) result in a set containing empty string?
我正在阅读 Aho 和 Ullman 的著作《解析、翻译和编译理论》。在第 2 章介绍正则表达式的部分中，有一个正则表达式的属性列表。我不明白属性 2 和 8。这是属性列表：
(1) + = +
(2) ∅* =
(3) + ( + ) = ( + ) +
(4) () = ()
(5) ( + ) = +
(6) ( + ) = +
(7) = =
(8) ∅ = ∅ = ∅
(9) * = + *
(10) (* )* = *
(11) + =
(12) + ∅ =
其中∅为表示正则集的正则表达式∅,为任意正则表达式,为空串
性质 (2) 和 (8) 如何证明？
编辑：为了解释+、*等符号，这里给出一些书中给出的定义（引用）：
DEFINITION Let be a finite alphabet. We define a regular set over recursively in the following manner:
(1) ∅ (the empty set) is a regular set over .
(2) {} is a regular set over .
(3) {} is a regular set over for all in .
(4) If and are regular sets over , then so are
(a) ∪ .
(b) .
(c) *.
(5) Nothing else is a regular set.
Thus a subset of * is regular if and only if it is ∅, {}, or {}, for some in , or can be obtained from these by a finite number of applications of the operations union, concatenation, and closure.
.
DEFINITION Regular expressions over and the regular expressions they denote are defined recursively, as follows:
(1) ∅ is a regular expression denoting the regular set ∅.
(2) is a regular expression denoting the regular set {}.
(3) in is a regular expression denoting the regular set {}.
(4) If and are regular expressions denoting the regular sets and , respectively, then
(a) (+) is a regular expression denoting ∪ .
(b) () is a regular expression denoting .
(c) ()* is a regular expression denoting *.
(5) Nothing else is a regular expression.
我的猜测是 2 和 8 属性可能只是一个简单的数学运算：
属性 2
∅是空集，则∅* = 成立，∅+ = 也成立，∅{Infinity} = 也成立，因为e是空串。
一个正则表达式是一个字符串，因此一个空的正则表达式重复任意次数或任何操作，仍然是一个空的正则表达式，它又等于一个空的右边的字符串。
参考： Why is the Kleene star of a null set is an empty string?
属性 8
∅ = ∅ = ∅ 是正确的，∅∅∅ = ∅∅∅ = ∅ 也是如此，因为空集与任何东西组合都会产生空集。
参考：
Regular expressions with empty set/empty string
What is the difference between language of empty string and empty set language?
How can concatenating empty sets (languages) result in a set containing empty string?