Computability 5: Decidability and R.E. Sets (II)

  

1. Listing Theorem

　　A non-empty set is R.E. iff it is the range of a unary total computable funciton.

　　That means the elements of an R.E. set can be effectively generated.

　　We can also prove that a set is R.E. iff it is the range of a computable function.

 

　　Corollary. $Tot = \{x\text{ | }\phi_x\text{ is total}\}$ is not R.E.

　　Otherwise suppose $Tot = Ran(f)$, where $f$ is a total unary computable function, we can construct total computable function $\phi_{f(x)}(x)+1$, which differs from every total computable function.

　　This proposition can also be proved by Rice-Shapiro Theorem, which will be introduced later.

　　Corollary. An infinite set is recursive iff it is the range of a total increasing computable function.

　　Corollary. An infinite R.E. set has an infinite recursive subset.

 

2. Rice-Shapiro Theorem

　　Suppose that $A$ is a set of unary computable functions such that the set $\{x\text{ | }\phi_x\in A\}$ is an R.E. set, then for any unary computable function $f$, $f \in A$ iff there is a finite function $\phi\subseteq f$ with $\phi\in A$ . (The proof of this one is more hermetic than the previous version of Rice's Theorem in decidability.)

　　According to this theorem, following sets are not recursively enumerable:

　　　　

P.S　　

　　　证明 Rice 定理和 Quasi-Rice 定理构造的都是形如 $\phi_{f(x)}(y)\simeq\begin{cases}\text{some computable function} & x\in W_x \\ \text{undefined} & \text{otherwise}\end{cases}$ 的 computable function。

　　　证明 Rice-Shapiro 定理，设 $\phi_e(x)\simeq\psi_U(x,x)$，构造的是 $\phi_{f(x)}(y)\simeq\begin{cases}\text{some computable function} & \neg H(e,x,y) \\ \text{undefined} & \text{otherwise}\end{cases}$。

　　　前者保证了 $x\in W_x\Leftrightarrow\phi_{f(x)}\downarrow\Rightarrow y\in W_{f(x)}$，后者使得 $x\in W_x\Leftrightarrow\phi_{f(x)}\text{ is finite}\Rightarrow f(x)\notin Tot$。

 

3. Special Sets

　　A set $A$ is a productive set if there is a total computable function $g$ such that whenever $W_x\subseteq A$, then $g(x)\in A\backslash W_x$. The function $g$ is called a productive function for $A$. For example, $\overline{K}$ is productive with productive function $g(x) = x$.

 

　　Reduction Theorem.

　　Suppose that set $A$ is productive, and there is a total computable function $f$ such that $x\in A$ iff $f(x)\in B$; then $B$ is productive.

 

　　Quasi-Rice Theorem.

　　If $\beta$ is a non-trivial set of unary computable functions who contains $f_\varnothing$, then the set $\{x\text{ | }\phi_x\in\beta\}$ is productive.

 

　　A set $A$ is creative if it is R.E. and $\overline{A}$ is productive.

 

　　Corollary of Quasi-Rice Theorem.

　　Suppose α is a set of unary computable functions and let $A = \{x\text{ | }\phi_x\in\alpha\}$; if $A$ is R.E but neither $\varnothing$ nor $\mathbb{N}$, then $A$ is creative.

 

　　Subset Theorem.

　　A productive set must contain an infinite R.E. subset, which can be proved by construction based on the following lemma:

　　For any total computable function $g$, there is a total computable function $k$ such that for all $x$, $W_{k(x)}=W_x\cup{g(x)}$.

 

　　A simple set is an R.E. set whose complement is infinite but does nnot contain infinite R.E. subset.

　　We can prove that simple sets exist by constructing $Ran(\phi_x(\mu z(\phi_x(z)>2x)))$.

 

　　Characteristic Theorem.

　　A simple set can never be recursive or creative.

　　

　　Furthermore, suppose $A$ is a simple set, then there exists a set $\{\pi(x,y)\text{ | }x\in A\}$, which is r.e but neither recursive, creative nor simple.

 

 

P.S.

　　(1) 欲证一个集合是 R.E. set，有以下几种思路：

　　i. 证明对应问题是partially decidable的，前面总结过方法。

　　ii. 构造一个computable function，证明该集合是其定义域。

　　iii. 构造一个computable function，证明该集合是其值域。

　　(2) 欲证一个集合不是 R.E. set，一种简便方法是用 Rice-Shapiro定理进行归谬：

　　i. 要么构造一个 $f\in A$，然后证明不存在定理中描述的 $\theta$；

　　ii. 要么同时构造一个 $f\notin A$ 以及定理中描述的 $\theta$。

　　(3) 证明一个集合是productive set，通常有三种思路：

　　i. 最底层的，by definition, 构造一个productive function就好了。例如现在已知 $A$ 是 productive set，欲证 $B$ 是 productive set，通常做法是构造 a pair of total computable functions $S$ and $K$ such that $$W_x\subseteq B\Rightarrow W_{S(x)}\subseteq A\Rightarrow g(S(x))\in A\backslash W_{S(x)}\Rightarrow K(g(S(x)))\in B\backslash W_x$$

　　ii. 比较高级的工具是 reduction theorem，其中规约映射通常用s-m-n定理构造。

　　iii. 更省事的方法是用 Quasi-Rice定理一步到位。

 

 

References:

　　1. Cutland, Nigel. Computability: an introduction to recursive function theory[M]. Cambridge: Cambridge University Press, 1980