Due Dates: Oral presentations of material begin on 20 October.

Written descriptions of material presented due one week after presentation.

**Topics:** Pushdown Automata (Chapter 2).

An integral part of this class is understanding and presenting the problems
assigned as homework. Everyone is expected to do all the problems,
but we will take turns on who presents the problem solutions to the class
(every 2-3 weeks, depending on the number of students in the class). Within
a week of presenting a problem solution to the class, you must submit a written
description of it, via the Blackboard system. The written solutions
will be posted on the Blackboard website for the class, so, they can used
by everyone to study for the exams. Since it's hard to write
down answers that are concise and are easily readable by all, if you wish
to improve a grade on any problem, you may resubmit it for grading.

## Undergraduate Problems

All students enrolled should complete the following:

- Give a string that can be derived in two different ways from the grammar:
E → E + E | E x E | (E) | a

- Convert the following CFG into an equivalent CFG in Chomsky normal form, using the procedure given in Theorem 2.6.
E → E + E | E x E | (E) | a

- Let G be the context free grammar:
E → E + T | T

T → T x F | F

F → (E) | `a`

Using the procedure in Theorem 2.12, convert G into an equivalent pushdown automata.

- Give a context free grammar that generates the language
A = {a^{i}b^{j}c^{k} | i,j,k ≥ 0 and i=j or j=k}

Is your grammar ambiguous? Why or why not?

- Describe a Pushdown Automata that accepts the language:
L = {w | w contains more 1's than 0's}

- Convert the following CFG into an equivalent CFG in Chomsky normal form, using the procedure given in Theorem 2.6.
A → BAB | B| ε

B → 00 | ε

## Graduate Problems

Students enrolled for graduate credit, should complete all the undergraduate problems, as well as:

- Let G be the context free grammar:
R → XRX | S

S → `a`T`b` | `b`T`a`

T → XTX | X | ε

X → `a` | `b`

Using the procedure in Theorem 2.12, convert G into an equivalent pushdown automata.

- Show that the collection of context free languages is closed under the union operation.