# day23 - Click to edit Master subtitle style S-m-n Theorem...

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Unformatted text preview: Click to edit Master subtitle style 10/4/11 S-m-n Theorem 10/4/11 Dec 2, © UCF (Charles E. 22 Parameter (S-m-n) Theorem • Theorem: For each n,m>0, there is a prf Smn(u1,…,un,y) such that ¡(m+n)(x1,…,xm, u1,…,un, y) = &(m)(x1,…, xm, Smn(u1,…,un,y)) • The proof of this is highly dependent on the system in which you proved universality and the encoding you chose. 10/4/11 Dec 2, © UCF (Charles E. 33 S-m-n for FRS • We would need to create a new FRS, from an existing one F, that fixes the value of ui as the exponent of the prime pm+i. • Sketch of proof: Assume we normally start with p1x1 … pmxm p1u1 … pm+nun & Here the first m are variable; the next n are fixed; & denotes prime factors used to trigger first phase of computation. Assume that we use fixed point as convergence. We start with just p1x1 … pmxm, with q the first unused prime. q ¡ x & q t x replaces F x¢ & x in F q x & q x ensures we loop at end x & q pm+1u1 … pm+nun & x adds fixed input, start state and q this is selected once and never again Dec 2,...
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day23 - Click to edit Master subtitle style S-m-n Theorem...

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