CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #2

David Mix Barrington

Assignment posted Fri 26 Sept 2003

Make sure you have read the statement on academic honesty on the homework assignment index page.

• Question 1 (20): Let HANG be the language {(M,w): w in H(M)} where H(M) is the hanging language from Question 5 of HW#1. Prove that HANG is r.e.-complete.

• Question 2 (20): Here are two functions from N to N. In each case either prove the function to be primitive recursive or prove it to be not primitive recursive:

  • (a,10) the "superfactorial" function SF(n), defined by SF(0) = 1, SF(n+1) = (SF(n)+1)!

  • (b,10) the "hyperfactorial" function HF(n), defined by HF(0) = 1, HF(n+1) = SF(HF(n)+1)

• Question 3 (10): A K-oracle TM is a Turing machine with an extra oracle tape on which it may write a string and (in one step) answer the question of whether that string is in the language K. A language is said to be K-r.e. if it is equal to L(M) where M is a K-oracle TM. The language KK is defined to be {M: M is a K-oracle TM and "M" is in L(M)}. Prove that KK is K-r.e. complete under K-computable reductions. (A function is K-computable if it is computed by some K-oracle TM that always halts.)

• Question 4 (40): A total order on the set Σ^* is a relation on that set that is anti-reflexive, antisymmetric, transitive, and total. ("R is total" means that if x and y are two different strings, either R(x,y) or R(y,x) is true.) A total order R is computable if there exists an always-halting TM that decides whether R(x,y) is true on input (x,y).

  • (a,10) Define L(x,y) ("the lexicographic order") to be true iff either x is shorter than y or x and y are the same length and there exist strings t, u, and v (possibly empty) such that x = t0u and y = t1v. Define D(x,y) "the dictionary order" to be true iff either x is a prefix of y (y = xz for some nonempty z) or there exist strings t, u, and v (possibly empty) with x = t0u and y = t1v. Prove that both L and D are computable total orders.

  • (b,10) We say that y is the L-successor of x if L(x,y) and there does not exist a string z such that L(x,z) and L(z,y). We define "y is the D-successor of x" similarly. Prove that the languages {(x,y): y is the L-successor of x} and {(x,y): y is the D-successor of x} are each recursive.

  • (c,20) Let T(x,y) be any computable total order on strings. Prove that the language {(x,y): y is the T-successor of x} is a co-r.e. language. Can you find a T such that T-successorship is not recursive?

Last modified 27 September 2003