CS 220 Winter 2011, Assignment 3, Due: February 17
1. Show that the function
f
(
x, y
) =
x
*
y
cannot be computed by any
L
1
program.
2. Consider a programming language
Q
whose only nonI/O instructions are of the forms:
x
←
0,
x
←
1,
x
←
x
+
y
,
x
←
x

y
(monus),
do x ... end
. Denote by
Q
i
the set of all programs with
do
loop depth
i
. Show that a function is computable by a
G
program with running time bounded
by an elementary function if and only if it is computable by a
Q
1
program.
3. Show that for
i
≥
1, a function is computable by a
Q
i
program if and only if it is computable by
an
L
i
+1
program.
4. From the definition of partial recursive functions, show that every partial recursive function can
be computed by a
G
program.
5. Consider a DFA (deterministic finite automaton)
M
operating on the upperright quadrant of
the plane, filled with
λ
’s, with the boundaries delimited by $’s.
M
starts in its intital state on
the $ in the cell at the origin (location (0
,
0)).
M
has a finite number of states and can only read
and not write. Note that
M
has now four directions of moves. Show that the halting problem
for such DFAs is undecidable.
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 Winter '08
 Ibarra,O
 Recursion, Halting problem, TM, computable function

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