Universal Programs

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

　　Theorem. For each n, the universal function Ψ_U is computable.

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　　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 = Π p_i^r_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 σ = π(c,j). To prove the universal functions are computable, we should prove no further than c_n(e,x,t) and j_n(e,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, σ_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 In, 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.

　　P.S. 虽然讲了整整一节课，但这个证明跟当年的递归定理比起来简直弱爆了～

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

　　(1) S_n(e,x,y,t) defined as P_e(x) converges to y in t or fewer steps;

　　(2) H_n(e,x,t) defined as P_e(x) converges in t or fewer steps.

　　Corollary 2. (Kleene‘s normal form theorem) There is a total computable function U(x) and a decidable predicate T_n(e,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.

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　　Application 1. The problem ‘Φ_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 Φ_e(x) for any e by using Cantor‘s Diagonal Argument.

　　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 Φ_s(x,y) = Φ_x Φ_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 ∈ E_x, and

　　　2. if y ∈ E_x, then g(x,y)∈ W_x and Φ_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(e₁,e₂) such that:

References:

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

Universal Programs

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原文地址：http://www.cnblogs.com/DevinZ/p/4430229.html