Computability 3: Universal Programs

 

　　The Universal Function for n-ary computable functions is the (n+1)-ary function:

　　　　$\psi_U^{(n)}(e,x_1,...,x_n)\simeq\phi_e^{(n)}(x_1,...,x_n)$

 

　　Theorem. For each $n$, the universal function $\psi_U$ is computable.

 

　　To avoid using Church's Thesis, we prove this theorem step by step in a top-down way as the following sketch.

　　Step 1. The configuration of the registers can be coded as $c=\prod_{i>0}p_i^{r_i}$^_i, where $p_i$ is the ith prime, and given the number of the next instruction $j$ we can encode the current computation state as $\sigma=\pi(c,j)$. To prove the universal functions are computable, we should prove no further than $c_n(e,\vec{x},t)$ and $j_n(e,\vec{x},t)$ are computable since:

　　　　

　　And we can prove that by proving the following function is a primitive recursive function (a stronger conclusion):

　　　　

　　Step 2. From another perspective, $\sigma_n$ can be defined as follow:

　　　　

　　Hence, we need to prove config and next are primitive recursive functions.

　　Step 3. We further define the following auxillary functions:

　　　　　　

　　And we should prove the functions above are primitive recursive due to the following facts:

　　　　　　　

　　　　　　　　

　　Step 4. To prove $ln$, $gn$, $ch$ and $v$ are primitive recursive functions, we should resort to the definitions of program code nunber and instruction code number. Here, I shall omit further details of the proof of this step.

　　

 

　　Corollary 1. For each $n$, the following predicates are decidable:

　　(1) $S_n(e,\vec{x},y,t)$ defined as $P_e(\vec{x})$ converges to $y$ in $t$ or fewer steps;

　　(2) $H_n(e,\vec{x},t)$ defined as $P_e(\vec{x})$ converges in $t$ or fewer steps.

 

　　Corollary 2. (Kleene's normal form theorem) There is a total computable function $U(\vec{x})$ and a decidable predicate $T_n(e,\vec{x},z)$ such that:

　　

 

　　As a matter of fact, every computable function can be obtained from primitive recursive functions by using at most one application of the unbounded minimalisation.

 

 

　　Application 1. The problem '$\phi_x$ is total' is undecidable.

　　To see this, we can construct a unary function defined by cases based on this predicate, which is other than any $\phi_e(x)$ for any $e$ by using Cantor's Diagonal Argument.

　　This can also be deduced from the Rice's Theorem. A further conclusion would be whether a function is extensible to a total function is undecidable. For example, you cannot decide whether $f(x)=\psi_U(x,x)+1$ is extensible, otherwise $x\in W_x$ is decidable (see more about decidability).

　　As a matter of fact, we shall also see that this preidcate is not partially decidable in the near future.

 

　　Application 2. There is a total computable function that is not primitive recursive.

　　Here, we provide a proof sketch:

　　

 

　　Application 3. (Effectiveness of Production)

　　There is a total computable function $s(x,y)$ such that $\phi_{s(x,y)}=\phi_x\phi_y$ for all $x$ and $y$.

 

　　Application 4. (Effectiveness of Inverse)

　　There is a computable function such that:

　　1. $g(x,y)$ is defined if and only if $y \in E_x$, and

　　2. if $y\in E_x$, then $g(x,y)\in W_x$ and $\phi_x(g(x,y))=y$.

 

　　Application 5. (Effectiveness of Recursion)

　　Consider function $f$ defined by the following recursion:

　　　　

　　　　

　　There is a total computable function $r(e_1,e_2)$ such that:

　　　　

 

 

References:

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