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. $Tot = {x ext{ | }phi_x ext{ is total}}$ 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 ${x ext{ | }phi_xin A}$ is an R.E. set, then for any unary computable function $f$, $f in A$ iff there is a finite function $phisubseteq f$ with $phiin A$ . (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 $phi_{f(x)}(y)simeqegin{cases} ext{some computable function} & xin W_x \ ext{undefined} & ext{otherwise}end{cases}$ 的 computable function。
证明 Rice-Shapiro 定理，设 $phi_e(x)simeqpsi_U(x,x)$，构造的是 $phi_{f(x)}(y)simeqegin{cases} ext{some computable function} & eg H(e,x,y) \ ext{undefined} & ext{otherwise}end{cases}$。
前者保证了 $xin W_xLeftrightarrowphi_{f(x)}downarrowRightarrow yin W_{f(x)}$，后者使得 $xin W_xLeftrightarrowphi_{f(x)} ext{ is finite}Rightarrow f(x) otin Tot$。
3. Special Sets
A set $A$ is a productive set if there is a total computable function $g$ such that whenever $W_xsubseteq A$, then $g(x)in Aackslash W_x$. 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 $xin A$ iff $f(x)in B$; then $B$ is productive.
Quasi-Rice Theorem.
If $eta$ is a non-trivial set of unary computable functions who contains $f_varnothing$, then the set ${x ext{ | }phi_xineta}$ 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 $A = {x ext{ | }phi_xinalpha}$; 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 $x$, $W_{k(x)}=W_xcup{g(x)}$.
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 ${pi(x,y) ext{ | }xin A}$, 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 $fin A$，然后证明不存在定理中描述的 $ heta$；
ii. 要么同时构造一个 $f otin A$ 以及定理中描述的 $ heta$。
(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 $$W_xsubseteq BRightarrow W_{S(x)}subseteq ARightarrow g(S(x))in Aackslash W_{S(x)}Rightarrow K(g(S(x)))in Backslash W_x$$
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