CSE596, Fall 2015
Problem Set 6
Due Wed. 10/28
Reading. For next week, read the short Chapter 4 and start to grapple with the technical content
of sections 5.1–5.3. Note that lectures have already introduced the concept of running time and (less
formally) space, so Theorem 5.5 in particular says nothing new. (Whereas, Theorem 5.6 is deep and
complicated and should be skim/skipped while I will literally “dance around it” in lecture.) Theorem
5.3 may also be skimmed; in lecture I will state only the case T (n) = O(n) and contrast it with
deterministic linear time. At the same time, please read the Allender-Loui-Regan (ALR) notes which
were given out in hardcopy last month: up through section 2.3. Lectures will follow ALR on giving less
detail about constructibility; that and O-notation will be shortcut by stating a sweeping theorem that
all functions f given by formulas built up using +, −, ∗, /, log, exp only (in particular no sin, cos, log∗ )
are:
• fully time constructible provided f (n) ≥ n + 1,
• fully space constructible provided f (n) ≥ log2 (n), and
• totally ordered by asymptotic growth: for all such functions f and f 0 , either f = o(f 0 ), f = Θ(f 0 ),
or f = ω(f 0 ) (i.e., f 0 = o(f )).
I don’t know if anyone has actually proved this theorem (well, G.H. Hardy proved the third part
decades before Turing machines were invented), but it is true and cuts a lot of the text’s technical
“mustard.”
This is a longer assignment. Instead of a discussion problem intended to discuss a related topic,
I’m implementing a request to list more practice problems for reductions. Note that last week’s second
problem included the text’s “Homework 3.19” on page 71. Homeworks 3.20, 3.26, and 3.27 are shortand-sweet once you get the pattern; I’ve assigned 3.28 below. Homework 3.25 is a change to a decidable
problem, but now suppose we change it from being an “L1 ”-type problem to “L2 ”-type with regard
to the “triad” in lectures and problem set 5:
L2 = {e : (∃w ∈ Σ∗ )[Me (w) makes a left-move after copying w to tape 2 and rewinding both tapes]}.
It is a real challenge to show that now the problem is undecidable—whereas with just 1 tape and no
initial copy stage it is decidable.
(1) Prove that the following decision problem Crash is undecidable, by reduction from the Acceptance
Problem or Halting Problem:
Instance: A Turing machine M 0 with a single one-way-infinite tape, as in the text;
Question: Is there an input x such that M 0 (x) ‘crashes’ by attempting to move its head off
the left end of the tape?
Also answer whether the language of this decision problem is computably enumerable or not and
justify your answer. (12 + 3 + 3 = 18 pts.)
(2) Text, “Homework 3.28” on page 72. (9+9 = 18 pts.)
(3) Show that the language L1 = { e : L(Me ) = { λ } } is neither c.e. nor co-c.e. (36 pts. total, for 72
on the set)
(For 12 points extra credit, let R(x, y, z) be a three-place decidable predicate and let the language
A be defined by: for all x, x ∈ A ⇐⇒ (∀y)(∃z)R(x, y, z). Show that A ≤m P T .)