Automata TheoryCS 3313

Chapter 2

Revision of Mathematical Notations and Techniques

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.

* 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.

* 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.

* 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.

* 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}

* 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 ‘’, is the empty sequence.

* Note: The empty word 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 X* - This means that language is a set of words.

* Note: , which is empty set, is a language over any alphabet.

* Similarly {}, the set containing only one word, that is empty word, is a language over any alphabet.

* But { }

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:

Here V = {1, 2, 3, 4, 5} and E = {(n, m) | n+m = 4}

* A PATH in a graph is sequence of vertices V1, V2, V3, …,

Vk, K 1, such that there is an edge (Vi , Vi+1) 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.

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 Vi+1 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.

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.

    * 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.

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:

6

)12)(1(2

0

nnni

n

i

* 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.

Set Notation

* A collection of the objects (members of the set) without repetition.

* Finite sets may be specified by listing their members betweenbrackets.

* 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.

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

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

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 andrange 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.

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 < on the set of integers is transitive because a < b and b < c implies a < c.

* It is asymmetric and hence irreflexive because a < b implies b < a false.

* A relation R that is reflexive, symmetric, and transitive is said to be an equivalence relation.

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).

 

   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)}

  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