# COMS W3261 Computer Science Theory Homework Assignment #3 October 29, 2012 Due in class 1:10pm, November 14, 2012

## Instructions

• Problems (1)-(5) are each worth 20 points.
• This assignment may be handed in November 19, 2012 for 50% credit.

## Problems

1. Informally describe a Turing machine that accepts all strings of the form { anbncn | n ≥ 1 }. Show the sequence of ID's that your TM goes through starting with the input aabbcc.

2. Consider the following Turing machine
3. 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` ` - ` ` - ` ` - ` ` - ` ` - `

1. Show the sequence of ID's that M goes through starting with the input `aaaa`.
2. 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?
3. Briefly explain how the Turing machine does this computation and characterize the role of each state.
4. 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.

4. Is it decidable given a Turing machine M and an input string w whether w is not in L(M)? Prove your answer.

5. Post's Correspondence Problems.
1. Is PCP with a single-symbol alphabet decidable? Briefly justify your answer.
2. Is PCP with a two-symbol alphabet decidable? Briefly justify your answer.

6. What class of languages can a Turing machine recognize if it
1. Has only two working states, and one accepting state from which it never makes any transitions?
2. 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.
3. Has only {0, 1, B} as tape symbols?
4. Never moves its input head left?
Give a brief one or two-sentence justification for each of your answers.

aho@cs.columbia.edu