K = { i | M_{i} is a TM that halts on blank input }.Note: do not use Rice's Theorem.
E_{TM} = { ⟨ M ⟩ | M is a TM such that L(M) = ∅ }and
FINITE = { ⟨ M ⟩ | M is a TM such that L(M) is a finite set }.
COFINITE = { ⟨ M ⟩ | the complement of L(M) is finite }
SUBSET = { ⟨ M_{1}, M_{2} ⟩ | L(M_{1}) ⊆ L(M_{2}) }Use as few quantifiers as you can and place all of the quantifiers at the beginning. In particular, none of the quantifiers should appear after a negation.
Let
MIN_LBA = { ⟨ £_{i} ⟩ | £_{i} is minimal }.Show that MIN_LBA is Turing recognizable.
INFINITE = { ⟨ M ⟩ | M is a TM and L(M) is an infinite set }and
ALL_{TM} = { ⟨ M ⟩ | M is a TM and L(M) = Σ* }.Note: You must construct a computable function f such that for all TM's M
⟨M⟩ ∈ INFINITE if and only if f (⟨M⟩) ∈ ALL_{TM}.Carefully argue that the f you constructed is indeed computable and that both directions of the "if and only if" hold.
A = C / D = { x ∈ Σ^{*} | w = xy for some w ∈ C and y ∈ D }for some C and D in SPACE[ log n ].
(1,2), (2,8713470183741023847120).where the length of 8713470183741023847120 might be much larger than log n. (It might be √n, for example.)
You may ignore isolated vertices (vertices without any edges attached). So, in the example above, you can generate an adjacency matrix for the graph
(1,2), (2,3).I.e., if the vertices numbered between 2 and 8713470183741023847120 do not appear on the adjacency list, you can assume that they don't exist. However, you still have to deal with the fact that 8713470183741023847120 is a rather large number and might have length bigger than log n.
Show that if a language L has polynomial-sized circuits, then there exists a sparse set S such that L ∈ P^{ S}.
Hint: you can stuff the circuit C_{n} in the sparse set S, but how do you get it out?
Hint: mimic Karp-Lipton-Sipser, but be careful how you check if the sparse set you guessed "works".
x ∈ L ⇒ Prob _{r} [ M(x, r) accepts ] > ½
x ∉ L ⇒ Prob _{r} [ M(x, r) accepts ] < ½where the probability is taken over r ∈ {0,1}^{p(n)} and n = | x | .
Show that NP ⊆ PP.
We say that a function f is in the class #P if there exists a nondeterministic polynomial-time Turing machine N such that for all x ∈ Σ^{*}, f (x) = #acc_{N}(x).
Show that the class #P is closed under addition and multiplication. That is, for any two functions f and g in #P, the functions
h_{1}(x) = f (x) + g(x)and
h_{2}(x) = f (x) ⋅ g(x)must also be in #P.