This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Click to edit Master subtitle style 10/4/11 Smn Theorem 10/4/11 Dec 2, © UCF (Charles E. 22 Parameter (Smn) 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 Smn 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,...
View
Full Document
 Spring '08
 Staff
 Logic, Quantification, Universal quantification, Charles E, (Charles E.

Click to edit the document details