PROBLEMS

Complexity Problems 41. Show that the set A above is reducible to the sets D and F bymappings which can be computed by finite automata.2. Provide a mapping from set D above to set E. Analyze the spacerequirements of the mapping.3. Show that set A above is reducible to set C. Take a Turing machineM c which decides membership in C, combine it with the machinewhich maps A into C, and produce a machine which decidesmembership in A. Further, do this so that the space requirementsare only O(log 2 n).4. Precisely state and prove theorems 1 and 2 for time complexity.5. Show that if the set A is a member of the class DTIME(B) thenDTIME(A) is a subclass of DTIME(B). Be sure to consider the caseswhere the sets A and B have speedup.6. Prove theorem 4 for nondeterministic time classes.7. Prove that complexity classes named by sets with speedup cannot benamed by recursive functions. (If S has speedup then there is norecursive f such that DSPACE(A) = DSPACE(f).)The Classes P and NP1. Estimate the time (in seconds, years, etc.) that it would take to solve aproblem which takes O(2 n ) steps on a typical computer. Do this forvarious values of n.2. Prove that any set in P can be reduced to any other set in P via somepolynomial time mapping.3. Verify theorem 2.4. Show that the existence of a set in NP whose complement was notalso in NP would lead to a proof that P ≠ NP.5. Demonstrate that the entire satisfiability problem is indeed in NP.6. Present an algorithm for converting a Turing machine instructioninto the set of clauses needed in the proof that SAT is NP-complete.