Types of variables: Unless otherwise indicated, a variable
A or B denotes a language, D denotes a DFA,
G denotes a context-free grammar,
M denotes a Turing machine, n denotes a (non-negative)
number, and w denotes a string.
Remember that N is the set of all numbers,
which we may also think
of as the set Σ^{*} of all strings.
Directions: Each of the first two questions
is a statement that may be true
or false. Please state whether it is true or false -- you will
get five points for a correct boolean answer and there is no penalty
for a wrong guess. Then justify your answer -- the remaining
points will depend on the quality and validity of your justification.
- Question 1 (10):
(true/false with justification)
If A and B are both r.e. sets, then A intersect B must be r.e. as well.
- Question 2 (10):
(true/false with justification)
The set {M: L(M) = overline(L(M))} is r.e.-complete. (Recall that
if X is any language, overline(X) is the complement of X.)
- Question 3 (15):
(true/false with justification)
The set {(N_{1},N_{2}):
N_{1} and N_{2} are NFA's and
L(N_{1}) = L(N_{2})} is
primitive recursive.
- Question 4 (10):
(true/false with justification)
If A is r.e.-complete and B ≤ overline{A}, then B is not
r.e.-complete.
- Question 5 (15):
(true/false with justification)
The set EASY-THEOREM = {φ: φ has length n and φ has a proof
of length at most n^{2}} is recursive.
- Question 6 (20):
(true/false with justification)
The language {G: L(G) ≠ ∅} is in the class P.
- Question 7 (20):
(true/false with justification)
The formula
[∃x:∀y: Sphere(x) implies Small(y)]
implies
[(∃x: Sphere(x)) implies (∀y: Small(y))]
is valid for Tarski's World.
Crib Sheet:
- L(M) is the set of numbers n such that M halts on input
n with output 1. Note that the set {M_{n}: n is in N}
contains all
possible Turing machines in some standard encoding and that each M_{n}
is a valid Turing machine.
- L(D) is the set of strings w such that D running on w
finishes in an accepting state.
- L(G) is the set of strings w such that w
can be generated from G's start symbol according to G's rules.
- A language is r.e. if it is equal to L(M)
for some Turing machine M.
- A language is recursive if it is equal to L(M)
for some Turing machine M that always halts.
- A function f is total recursive if there exists a
Turing machine that always halts and gives output f(n) on any input n.
- The language A is reducible to B, written ``A ≤ B'',
if there exists a total recursive function f such
that for any n, n is in A iff f(n) is in B.
- The language A is r.e.-complete iff A is r.e. and for any
language B, if B is r.e. then B ≤ A.
- The language K = {n: n is in L(M_{n})} is r.e.-complete.
- The Bloop programming language is defined in terms of basic operations
and bounded loops. The Floop programming language also has free loops.
- The class P is the set of all languages whose decision problem
can be solved by a Turing machine that runs in a time polynomial in the input
size.
- The language FO-THEOREM is the set of statements of first-order logic
theory that are provable from the first-order axioms.
- As a first-order structure, N is the set N of numbers
together with the standard definitions of equality, order, addition,
multiplication, and exponentiation of numbers.
- In Tarski's World, an object is never larger than
itself. ``Larger(a,b)'' means ``a is larger than b''.
Last modified 20 November 2003