COMS W3261
Computer Science Theory
Homework Assignment #3
October 29, 2012
Due in class 1:10pm, November 14, 2012
Instructions
- You may consult with others but your answers must be written in your own words.
- Problems (1)-(5) are each worth 20 points.
- This assignment may be handed in November 19, 2012 for 50% credit.
Problems
- Informally describe a Turing machine that accepts all strings of the form
{ a^{n}b^{n}c^{n} | n ≥ 1 }.
Show the sequence of ID's that your TM goes through starting
with the input aabbcc.
- Consider the following Turing machine
- M =
({A, B, C, D}, {a}, {a, X, 0, 1, #}, δ, A, #, {D})
.
- Here, we are using # for the blank symbol.
- The transition function δ is given by the following table:
State |
Symbol |
a |
X |
0 |
1 |
# |
A |
BXL |
AXR |
A0R |
A1R |
C#L |
B |
BaL |
BXL |
A1R |
B0L |
A1R |
C |
- |
C#L |
D0R |
D1R |
- |
D |
- |
- |
- |
- |
- |
- Show the sequence of ID's that M goes through starting with the input
aaaa
.
- Starting with an input consisting of n
a
's, n > 0,
what string will this Turing machine have on its tape after it has halted?
- Briefly explain how the Turing machine does this computation and characterize the
role of each state.
- Using big-O notation, how many moves will this Turing machine make
on an input consisting of n
a
's before halting? Briefly justify
your answer.
- Is it decidable given a Turing machine M and an input string w whether
w is not in L(M)? Prove your answer.
- Post's Correspondence Problems.
- Is PCP with a single-symbol alphabet decidable? Briefly justify your answer.
- Is PCP with a two-symbol alphabet decidable? Briefly justify your answer.
- What class of languages can a Turing machine recognize if it
- Has only two working states, and one accepting state from which it
never makes any transitions?
- Never overprints a different symbol on the input tape?
That is, if in the transition function for the Turing machine
whenever (q, Y, D) is in δ(p, X), then Y = X.
- Has only {0, 1, B} as tape symbols?
- Never moves its input head left?
- Give a brief one or two-sentence justification for each of your answers.
aho@cs.columbia.edu