1. Show that prfs are closed under three-way mutual induction. Three-way mutual induction means that each induction step after calculating the base is computed using the previous value of the other function. The formal hypothesis is:
Assume g1, g2, g3, h1, h2, and h3are already known to be prf, then so are f1, f2, and f3, where
f1(x,0) = g1(x); f1(x,y+1) = h1(f2(x,y),f3(x,y));
f2(x,0) = g2(x); f2(x,y+1) = h2(f3(x,y),f1(x,y))
f3(x,0) = g3(x); f3(x,y+1) = h3(f1(x,y),f2(x,y))
2. Let S be an arbitrary non-empty, re set. Furthermore, let S be the range of some partial recursive function fs. Show that S is the range of some primitive recursive function, call it hs.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Denote by <a, b, c> := <<a, b>,<c>> where <x,y>=pair(x,y) and denote a=p1(z)=<<z>1>1, b=p2(z)= <<z>1>2, c=p3(z)=<z>2, where z=<a, b, c>. Then both <a, b, c> = pair(pair(a,b),c) as well as p1, p2 and p3 are prfs.

Denote T(x,y) = <f1(x,y), f2(x,y), f3(x,y)>. Then also f1(x,y)=p1(T(x,y)), f2(x,y)=p2(T(x,y)), f3(x,y)=p3(T(x,y)).

Moreover, T(x, 0) = <g1(x), g2(x), g3(x)> is prf. (as a composition of prfs) and

T(x, y+1) = <f1(x,y), f2(x,y), f3(x,y)> =
= < h1(f2(x,y),f3(x,y)), h2(f3(x,y),f1(x,y)), h3(f1(x,y),f2(x,y)) >
= <h1(p2...
$15.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

Find A Tutor

View available Computer Science - Other Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

Upload a file
Continue without uploading

We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

  • 1
  • 2
  • 3
Live Chats