ECS 120: Theory of Computation Handout 9
UC Davis — Phillip Rogaway February 28, 2002
Problem Set 9 — Due Thusday, March 7, 2002
Problem 1 Prove that Lis decidable iff Lis listable in lexicographic order. (A language is
listable in lexicographic order if some program outputs x1]x2]x3]· · ·,L={x1, x2, x3, . . .},
and x1< x2< x3<· · · where “<” denotes the usual lexicographic ordering on strings.)
Problem 2 (Counts as 20 points, same as 2 ordinary problems.)
Part A. Let L={hMi:Mis a TM that accepts some string of prime length}. Prove that L
is not decidable.
Part B. Let L={hGi:Gis a CFG and Gaccepts an odd-length string}. Prove that Lis not
decidable.
Part C. Let L={hMi:Mis a TM and L(M) = L(M)∗}. Prove that Lis not r.e.
Part D. Let L={hMi:Mis a TM and L(M) = L(M)∗}. Prove that Lis not r.e.
Part E. Let L={hMi:Mis a TM and L(M) = L(M)∗}. Prove that Lis not co-r.e.
Part F. Let L={hG1, G2i:G1and G2are CFGs and L(G1) = L(G2)}. Prove that Lis not
decidable. You may use the fact that A={hGi:Gis a CFG and L(G) = Σ∗}is undecidable.