定义:Turing机是五元组

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

failed to exec() latex
,其中

定义:Turing机

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

failed to exec() latex
格局
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:

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
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

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
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
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

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
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
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
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
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。

图灵机 (last edited 2008-02-23 15:34:54 by localhost)

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