Midterm problem set (PDF)

Mid-Term 1 Due Fri., 14 Oct.
CMPSC 230
Theory of Computing
Fall 2005
Home page: http://cs.allegheny.edu/~rroos/cs230f2005/
Weblog: http://cs230f2005.blogspot.com
Mid-Term Take-Home Problem Set
All input alphabets are Σ = {a, b} unless otherwise noted.
1. Design an NFA that accepts the language of the regular expression ((ab) ∗ ∪ b ∗ (a ∪ b))(ab ∪ ba)
2. Design a DFA that accepts the same language as the following NFA:
a,b
a
1
a
2
a
b
ε
3
3. Find the regular expression that describes the language accepted by the following DFA:
b
1
a, b
2
a
b
a
4. Let Σ = {u, d, l, r} (“up,” “down,” “left,” “right). Then any string over Σ can be interpreted
as a drawing consisting of a sequence of unit-length line segments drawn in the indicated
direction, where each line starts where the previous line ended. For instance, the strings
urdl, drul, ruld, etc., all represent squares with side length equal to one unit. Since we
can “back up” or retrace a line segment, the following strings also represent unit squares:
udulrldr, ldrlurd, etc. The string uudrdl is not a square — (it has an extra line sticking
up at the top left corner).
Handed out on 7 October 2005 Handout # 5

Due Fri., 14 Oct. 2 Mid-Term
(a) Describe the set of all strings over Σ that represent a unit-length line. Is it a regular
language?
(b) Describe the set of all strings over Σ that represent the left and bottom sides of a unit
square — in other words, a right-angle with unit sides and the “opening” facing up and
to the right — dr and lu are two examples.
(c) Let L be the set of all strings over Σ that represent rectangles of any dimensions. (For
example, lurrdl represents a 2-by-1 rectangle.) Prove that L is not a regular language.
(Hint — you only need to find a single element of L that violates the pumping lemma,
so try to choose one that has a nice form you can use.)
5. We know that every regular language is also a context-free language. For each of the regular
languages described below, construct a context-free grammar that generates the language.
(a) The regular expression: ((a ∪ b)a) ∗ ∪ b ∗ ab ∗
(b) {w | |w| a is a multiple of 3} (assume Σ = {a, b})
(c) The language accepted by
S
a
a
A
b
b
B
b
a
C
a,b
HINT: can you figure out a way to construct the grammar
simply by looking at the transitions in the DFA?
6. Give an informal description of a nondeterministic pushdown machine that accepts the language
of all strings w over {a, b, c} such that either w is of the form ucu r , where u ∈ {a, b} ∗ , or
w is of the form b i c i for some i ≥ 0. You don’t have to draw the state machine, just describe
how it works. Be sure to indicate where the machine needs to make any nondeterministic
choices.
Handout # 5 Handed out on 7 October 2005