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S4 g, Theory ofComputer Science
(b) Which vertices are the sons of 57(c) Which vertex is the
father of 57(d) \Xlhat is the length of the path from 1 to 97(e)
What is the left-right order of leaves?(f) What is the height of
the tree?
2
10
7 8
oFig. 2.13 The directed tree for Example 2.18.
Solutions(a) 10, 4, 9, 8, 6 are leaves. 1, 2. 3, 5, 7 are
internal vertices.(bi 7 and 8 are the sons of 5.(c) 3 is the father
of 5.(d) Four (the path is 1 ~ 3 ~ 5 ~ 7 ~ 9).(e) 10 - 4 - 9 - 8 -
6.(f) Four (1 ~ 3 ~ 5 ~ 7 ~ 9 is the longest path).
2.3 STRINGS AND THEIR PROPERTIESA string over an alphabet set 2:
is a finite sequence of symbols from 2:.
NOTATION: 2:* denotes the set of all strings (including A, the
empty string)over the alphabet set 2:. That is, 2:+ = 2:* -
{A}.
2.3.1 OPERATIONS ON STRINGSThe basic operation for strings is
the binary concatenation operation. Wedefine this operation as
follows: Let x and y be two strings in 2:*. Let us forma new string
:: by placing y after x, i.e. z = xy. The string z is said to
beobtained by concatenation of x and y.
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Chapter 2: Maihematical Preliminaries ~ 55
EXAMPLE 2.1 9
Find J:::V and yx, where(a) x = 010,(b) x = a1\.,
y = 1y = ALGOL
Solution(a) X)' = 0101, yx = 1010.(b) xy =a /\ i\LGOL
yx = ALGOL aA.We give below some basic properties of
concatenation.
Property 1 Concatenation on a set 12* is associative since for
each x, y, ::: in12*, x(y:::) = (xy):::.Property 2 Identity
element. The set 12* has an identity element A W.r.t.the binary
operation of concatenation as
x1\. = Ax = x for every x in 12*
Property 3 12* has left and right cancellations. For x, y, Z in
12*,z:x- = ;:;y implies x = y (left cancellation)x:: = yz implies x
= y (right cancellation)
Property 4 For x, y in 12*, we have
Ixyl = Ixl + Iylwhere Ixi, Iy , . 'xv i denote the lengths of
the strings x, y, xy, respectively.
We introduce below some more operations on strings.
Transpose OperationWe extend the concatenation operation to
define the transpose operation asfollows:
For any x in 12* and a in 12,(xayT = a(x)T
For example. (aaabab;T is babaaa.Palindrome. A palindrome is a
string which is the same whether written
forward or backward, e.g. Malayalam. A palindrome of even length
can beobtained by concatenation of a string and its transpose.
Prefix and suffix of a string. A prefix of a string is a
substring of leadingsymbols of that string. For example, w is a
prefix of y if there exists y' in 12*such that y = Hy'. Tuen we
write w < y. For example, the string 123 has fourprefixes, i.e.
A. L 12, 123.
Similarly, a suffix of a string is a substring of trailing
symbols of thatstring, i.e. w is a suffix of y if there exists y' E
12* such that y = y'w. Forexample, the string 123 has four
suffixes, i.e. 1\., 3, 23, 123.
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56 l;;\ Theory ofComputer Science
Theorem 2.6 (Levi's theorem) Let v, W, x and Y E 1:* and vw =Ay.
Then:(i) there exists a unique string z in 1:* such that v = xz and
y = zw if
11'1> Ixl;(ii) v = x, Y = w, i.e. z = A if Iv I = 1x I;
(iii) there exists a unique stling z in 1:* such that x = vz,
and = zy ifII'I < 14
Proof We shall give a very simple proof by representing the
strings by adiagram (see Fig. 2.14).
t==x--...------y-----~Case 1: Ivl > I.xl v=xZ y =zw
I1-_:===:===:::.==;==::1Case 2: Ivl =!xi v=x W =Y
--
Case 3: Ivl < Ixl x = vz w = zyFig. 2.14 Illustration of
Levi's theorem.
2.3.2 TERMINAL AND NONTERMINAL SYMBOLSThe definitions in this
section will be used in subsequent chapters.
A tenninal symbol is a unique indivisible object used in the
generation ofstrings.
A nonterminal symbol is a unique object but divisible, used in
thegeneration of strings. A nonterminal symbol will be constructed
from theterminal symbols: the number of terminal symbols in a
nontenninal symbolmay vary; it is also called a variable. In a
natural language, e.g. English, theletters a, b, A, B, etc. are
terminals and the words boy, cat, dog, go arenonterrninal symbols.
In programming languages, A, B, C, ... , Z, :, =, begin,and. if,
then, etc. are terminal symbols.
The following will be a variable in Pascal:
< For statement > .~ for < control variable > =<
for list > do < statement >
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