Comp 411 - Fall 2006 - 1 - Problem Set #8
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Comp 411 Computer Organization
Spring 2006
Problem Set #8
Issued Thursday,10/25/06; Due Thursday, 11/1/06
Homework Information: Some of the problems are probably too long to be done the night
before the due date, so plan accordingly. Late homework will not be accepted. Feel free to get
help from others, but the work you hand in should be your own.
Problem 1. “How many 1s could a TM write, if a TM could write 1s?”
In the following problem consider Turing machines with the following specifications. Each
Turing machine has n states labeled {S1, S2, … Sn}, the Turing machine begins in state S1and
halts by transitioning to the special state S0, each cell of the Turing machine’s tape can contain
either a “1” or a “0”, and each move of the Turing machine is based solely on its current state
and the value of the tape cell under the tape head. A move consists of first modifying the
contents of the current tape cell under the head of the Turing machine, moving the tape either
left, L, or right, R, followed by a transition to the next state.
The following truth table defines the behavior of a Turing machine. Note that in this FSM the
outputs are a function of both the current state and the tape value.
Current
State
Tape
Value
Write
Tape
Move
Next
State
S1 0 1 L S3
S1 1 1 R S2
S2 0 1 R S3
S2 1 1 R S0
S3 0 1 R S1
S3 1 1 L S3
(A) How large a ROM is required to implement an n-state Turing machine that adheres to the
given specification (give the number of words and the bits-per-word)? Ignoring the
possibility of equivalent machines, in general how many n-state Turing machines are
there?
(B) Given an infinite tape (in both directions), with “0”s in every cell. What will the Turing
machine described by the truth table above leave on the tape when it halts? Show the
status of the tape in the region around the head after each move.
(C) Design a 2-state Turing Machine that writes as many “1”s as possible onto an all zero (in
both directions) tape and then halts. Hint: you can write four “1”s.
(D) Suppose that we choose to ignore the value of the cell under the read head, thus turning
our Turing machine into a finite-state machine. In this case, how many “1”s can we write
if the FSM has n states?
