定义：**Turing机**是五元组

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，其中

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

是状态的有穷集；latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

是字母表，包含空格符latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

和左端符latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，但不包含latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

和latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

是**初始状态**，latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

是**停机状态**的集合，latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

是转移函数，它是从latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

到latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

的函数，使得对所有

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，若latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，则latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

,对所有

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

和latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，若latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

，则latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

。

定义：Turing机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的

**格局**是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。状态分量属于H的格局称为停机格局。

基本机器：

写符号机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向左移带头机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向右移带头机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向右找第一个

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

：latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向左找第一个

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

：latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向右找第一个非

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

：latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

向左找第一个非

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

：latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

复制机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

左平移机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

定义：设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是Turing机，使得

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex包含两个不同的停机状态（

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex和

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex分别表示“是”和“否”）状态分量是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的任何停机格局都称为接受格局。而状态分量是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的停机格局称为拒绝格局。对输入

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex产生接受格局则我们说M

**接受**

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex,若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex产生拒绝格局则我们说M

**拒绝**

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。

设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是字母表，称为M的输入字母表；通过固定

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的子集，我们允许Turing机在计算中使用除在输入里出现的符号外的额外符号。如果

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是语言，并且对任何字符串

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，下列关系为真：若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex则M接受

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex；若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex则M拒绝

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，那么我们说

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

**判定**语言

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。

若存在Turing机判定语言L则L称为**递归**的。即Turing机判定语言L的条件是，当在输入

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex上启动时它总是停机并且停机状态是对输入的正确回答：若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex则是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，若

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex则是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。

定义：设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是Turing机，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是字母表，并设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。假设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex在输入

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex上停机，而且对某个

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，则

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex称为

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex在

**输入**

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex上的

**输出**，并表示成

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。

设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是从

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex到

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的任意函数。若对所有

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，则我们说M

**计算**函数f。即对所有

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，M在输入

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex上最终停机，并且当它确实停机时带上包含字符串

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。若存在Turing机计算函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，则

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex称为

**递归**的。

定义：设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是Turing机使得

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，并设对某个

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是从

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex到

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的函数。若对所有

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，则我们说

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

**计算**函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。若存在计算函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的Turing机M，则

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex称为

**递归**的。

定义：

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是Turing机，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，是字母表，并设

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是语言。若对任意字符串

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，下列关系为真：

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex当且仅当M在输入

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex上停机，则我们说

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex

**半判定**

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。语言

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是

**递归可枚举**的当且仅当存在Turing机

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex半判定

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex。

定理：若语言是递归的，则它是递归可枚举的。

定理：若L是递归语言，则它的补

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex也是递归的。

定理：如果非确定型Turing机M半判定或判定语言L或者计算函数f，则存在标准型Turing机M'半判定或者判定语言L或者计算函数f。

定义：**文法**是四元组

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex，其中V是字母表；

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是终结符集，

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex称为非终结符集；

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是起始符；并且规则集R是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex的有穷子集。

任何上下文无关文法是文法。

定理：语言被文法生成当且仅当它是递归可枚举的。

定义：文法计算。函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是文法可计算当且仅当存在计算它的文法G。

定理：函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是递归的当且仅当他是文法可计算的。

定义：基本函数

定义：原始递归函数：

定义：极小化。可极小化的。

定义：

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex递归

定理：函数

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex是

latex error! exitcode was 2 (signal 0), transscript follows: failed to exec() latex递归的当且仅当它是递归的。