This preview shows pages 1–3. Sign up to view the full content.
CS373 – Fall 2003 – NMG
*************Solutions ********************+
October 27, 2003
Homework Assignment 7 for chapter 7
Due 10/31/03 (Friday) at end of class
Problems related to chapter 7.
Unless otherwise noted you are to assume the conventions and definitions used
in our text by Linz
for push down automata and related topics.
If you must write the
δ
rules for a pushdown
automaton, write concise, clear comments to the right of the transition rules (as needed) explaining the meaning
and intent of the rules.
1.
Construct an NPDA that
accepts L = {w : n
a
(w) + n
b
(w) = n
c
(w)}.
2.
Construct an NPDA that
accepts L = {a
3
b
n
c
n
: n
≥
0}.
3.
Construct an NPDA that
accepts L = {a
n
b
n+m
c
m
: n
≥
0, m > 0 }.
4.
Consider
the following “mystery” NPDA:
δ
(s, a, z) = {s, cz}
δ
(s, a, c) = {s, cc}
δ
(s, b, z) = {(s, cz), (r, z)}
δ
(s, b, c) = {(s, cc), (r, c)}
δ
(r, a, c) = {r,
λ
}
δ
(r, b, c) = {r,
λ
}
δ
(r,
λ
, z) = {f, z}
where Q = {s, r, f},
∑
= {a,b},
Γ
= {c, z},
F (final state) = {f}
Determine if the following strings are accepted or not.
Show the
sequences of instantaneous
transitions for accepted strings, or give a convincing argument for those not accepted.
From this exercise or analysis of the rules, state the language accepted by this NPDA in set
theoretic notation; give a convincing reason for your choice of language.
a
b
aababaaab
baaabbaab
aabbabbaa
abbbabbbb
State the language accepted by this NPDA in set theoretic notation.
5.
An alternate definition of an NPDA is a modification of the definition given by Linz.
The alternate
definition allows a
λ
to appear as the third argument of
δ
(top of the stack).
The meaning of
λ
in
this position is that the stack top is ignored as an input to
δ
and consequently nothing is popped off.
Also only one symbol at a time may be pushed on the stack as opposed to a string of symbols in the
Linz definition.
In addition, there is no end of stack symbol (z) as in Linz.
All computations start
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentwith an empty stack.
Finally, language acceptance requires that the stack be empty as well as being
in a final state when the input string is completely consumed (see problem 2). The formal definition
would be
M = {Q,
∑
,
Γ
,
δ
, q
0
, F} … note there are 6 things instead of 7
Where Q is a finite set of states
∑
is the input alphabet
Γ
is the stack alphabet
δ
: Q X (
∑
∪
{
λ
}) X (
Γ
∪
{
λ
})
→
finite subsets of Q X
Γ
* , the transition function
q
0
is the initial state
F
⊆
Q is the set of final states
The Linz solution to the even palindrome problem given in Linz’s book, example 7.4, page 186 is a
bit ugly.
Using the above definitions, show a more elegant solution.
You should be able to do it in
two states and 5 transitions functions.
6.
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '10
 Icamarra

Click to edit the document details