Automata Theory CS 3313

1
Automata TheoryCS 3313

• Chapter 2
• Revision of Mathematical Notations and Techniques

2
String Alphabets Language A SYMBOL is an
abstract entity. Letters (a-z) and digits
(0-9) are examples of frequently used
symbols. A STRING (or word) is a finite
sequence of symbols juxtaposed
(adjacent). For example a, b and c are
the symbols and abcb is a string. The
LENGTH of string w, denoted w, is the number
of symbols composing the string. For
example abcb has length 4.
3
The empty string, denoted by ? (in some
other books it denoted by ?), is the
string consisting of zero symbols. Thus ?
0. (length of empty is equal to zero) A
PREFIX of a string is any number of leading
symbols of that string and a SUFFIX is any
number of trailing
symbols. For example, string abc has
prefixes ?, a, ab and abc its suffixes are
?, c, bc and abc. Note A prefix or suffix of
a string, other than the string itself is
called PROPER PREFIX or SUFFIX.
4
The CONCATENATION of the two strings is the
string formed by writing the first, followed
by the second, with no space between
them. For example, the concatenation of dog
and house is doghouse. Juxtaposition
is used as the concatenation operator. That
is, if s1 and s2 are strings, then s1s2 is
the concatenation of these two
strings. The empty string is the identity for
the concatenation operator.   That is,
?w w? w.
5
An ALPHABET is the finite set of symbols.
A certain specified set of string of characters
from the alphabet will be called the
LANGUAGE. In other words, A LANGUAGE is a set
of string of symbols from some one
alphabet. The empty set, ?, and the set
consisting of the empty string ? are
languages. Note they are distinct the
latter has a member while the former does
not. The set of Palindromes (string that read
the same forward and backward) over the
alphabet 0, 1 is an infinite
language. Some members of this language are
?, 0, 1, 00, 11, 010, 101, , 1101011 etc.
6
Note that the set of all palindromes over an
infinite collection of symbols is
technically not a language because its
string are not collectively built from an
alphabet.              Another language is the
set of all string over a fixed alphabet
?. We denotes this language by ?. For
example, ? a, then ? ?, a, aa, aaa,
. If ? 0, 1, then ? ?, 0, 1, 00, 01,
10, 11, 100, 000, . Some examples - X ?
a, b X ? 8 ?, a, b, aa, ab, ba, bb, ,
bbbbbbbb - X ? a, b X is odd
?, a, b, aaa, aab, , bbbbbbb
7
An alphabet is a finite, non-empty set of
elements, called as symbols or letters. Let
X be an alphabet. A word over the alphabet X
is a finite sequence of symbols from X. The
empty word is L, is the empty
sequence. Note The empty word L is a word
over any alphabet. The universal language of
all words over an alphabet X is denoted by
X. X is an infinite set. A language L
over an alphabet X is a subset of X. L Î X
- This means that language is a set of words.
8
Note F, which is empty set, is a language
over any alphabet. Similarly L , the set
containing only one word, that is empty word,
is a language over any alphabet. But F ¹
L
9
Graphs Trees       A GRAPH denoted G
(V, E), consists of a finite set vertices (or
nodes) V and a set of pairs of vertices E called
edges. An example graph is shown in following
Figure
10
Here V 1, 2, 3, 4, 5 and E (n, m)
nm 4 A PATH in a graph is sequence of
vertices V1, V2, V3, , Vk, K ? 1, such
that there is an edge (Vi , Vi1) for each i,
1 ? i ? k.        E (1, 3), (1, 4), (3,
4), (2, 5)        The length of the path is
k-1. For example, 1, 3, 4 is a path in the
graph of Figure above so is 5 by
itself.             If V1 Vk the path is a
CYCLE.
11
Directed Graph A directed graph (or
digraph), also denoted G V, E, consists of a
finite set of vertices V and a set of ordered
pairs of vertices E called arcs. We denote an
arc from V to W by V ? W. A PATH in a digraph
is sequence of vertices V1, V2, V3, , Vk,
K ? 1, such that Vi ? Vi1 is an arc for each
i, 1 ? i ? k. If V ? W is an arc we say V
is a predecessor of W and W is a successor of V.
12
Trees A TREE (strictly speaking, an ordered,
directed tree) is a digraph with the following
properties. There is one vertex, called the
root, that has no predecessors and from which
there is a path to every vertex. Each vertex
other than the root has exactly one
predecessor. The successors of each vertex are
ordered from left to right. We shall draw
trees with the root at the top and all arcs
pointing downward. The arrow on the arcs are
therefore not needed to indicate direction, and
they will not be shown.
13
    The successors of each vertex will be drawn
in left-to-right order. There is a special
terminology for trees that differs from the
general terminology for arbitrary graphs. A
successor of a vertex is called a SON, and the
predecessor is called the FATHER. If there is
a path from vertex V1 to vertex V2, then V1 is
said to be ANCESTOR of V2 and V2 is said to be
a DESCENDANT of V1. Note that the case V1
V2 is not ruled out any vertex is an ancestor
and a descendant of itself. A vertex with no
sons is called a LEAF, and the other vertices
are called INTERIOR vertices.
14
Inductive Proof Many theorems will be proved
by mathematical induction. Suppose we have a
statement P(n) about a nonnegative integer
n. A commonly chosen example is to take P(n)
to be
15
The principle of mathematical induction is
that P(n) follows from a)               
P(0), and b)                P (n-1) implies
P(n) for n ? 1. Condition (a) in an inductive
proof is called the BASIS, and condition (b) is
called the inductive step. The left-hand side
of (b), that is P(n-1), is called the inductive
hypothesis. Example let us prove eq. by
mathematical induction. We establish (a) by
substituting 0 for n in (eq.) and observing that
both are 0. To prove (b), we substitute n-1
for n in (eq.) and try to prove (eq.) from the
result. That is, we must show for n ? 1 that
Proof.
16
Set Notation A collection of the objects
(members of the set) without repetition. Finit
e sets may be specified by listing their members
between brackets. For example, we used 0, 1
to denote the alphabet of symbols 0 and 1. If
every member of A is a member of B, then we write
A ? B and say A is contained in B. A ? B is
synonymous with B ? A. If A ? B but A ? B,
that is, every member of A is in B and there is
some member of B that is not in A, then we write
A ? B. Sets A and B are equal if they have
the same members. That is A B if and only if
A ? B and B ? A.
17
Operations on sets The usual operations
defined on sets are 1) A U B, the UNION defined
on sets are                X / X is in A or X
is in B 2) A B, the INTERSECTION of A and B,
is                X / X is in A and X is in
B 3) A B, the difference of A and B, is
         X / X is in A and is not in B 4)  A
x B, the Cartesian product of A and B, is the set
of ordered pair (a, b) such that a is in A and b
is in B. 5)   2A, the power set of A, is the set
of all subsets of A
18
Example, A 1, 2 and B 2, 3     AUB
1, 2, 3     A B 2     A-B 1    
B-A 3     AxB (1,2), (1,3), (2,2),
(2,3)     BxA (2,1), (2,2), (3,1),
(3,2)     2A ?, 1, 2, 1,2    If A
and B have n and m members, respectively, then
AxB has nxm members and 2A has 2n members.   
Cardinality number of members
19
Relations A (binary) relation is a set of
pairs. The first component of each pair is
chosen form a set called the domain and the
second component of each pair is chosen form a
(possibly different) set called the range. We
shall use primarily relations in which the domain
and range are the same set S. In that case we
say the relation is on S. If R is a relation
and (a, b) is a pair in R, then we often write
aRb.
20
Properties of relations We say a relation R
on set S is 1)   Reflexive if aRa for all a in
S 2) Irreflexive if aRa is false for all a in
S 3) Transitive if aRb and bRc imply
aRc 4) Symmetric if aRb implies
bRa 5) Asymmetric if aRb implies that bRa is
false.      Note that any asymmetric relation
must be irreflexive. Example the relation lt
on the set of integers is transitive because a lt
b and b lt c implies a lt c. It is asymmetric
and hence irreflexive because a lt b implies b lt a
false. A relation R that is reflexive,
symmetric, and transitive is said to be an
equivalence relation.
21
Closures of Relations Suppose P is a set of
properties of relations. The P-closure of a
relation R is the smallest relation R that
includes all the pairs of R and possesses the
properties in P. For example, the transitive
closure of R, denoted R is defined by 1) If
(a, b) is in R, then (a, b) is in R. 2) If (a,
b) is in R and (b, c) is in R, then (a, c) is in
R. 3) Noting is in R unless it so follows from
(1) and (2).  
22
   It should be evident that any pair placed in
R by rule (1) and (2) belongs there, else R
would either not include R or not be
transitive. Also an easy inductive proof
shows that R is in fact transitive. Thus R
includes R, is transitive, and contains as few
pairs as any relation that includes R and is
transitive. The reflexive and transitive
closure of R, denoted R. For example let
R (1, 2), (2, 2), (2, 3) be a relation on the
set of 1, 2, 3, then R (1, 2), (2, 2),
(2, 3), (1, 3), and R (1, 1), (1, 2), (1,
3), (2, 2), (2, 3), (3, 3)
23
   let R (aRb), (bRb), (bRc) be a relation
on the set of a, b, c, then R (aRb),
(bRb), (bRc), (aRc), and - - aRc also R
(aRa), (aRb), (aRc), (bRb), (bRc), (cRc) - -
all cases