automata - chap1+introduction

8/10/2019 Automata - chap1+introduction

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WELCOME TO SSK5204

AUTOMATA THEORY AND FORMAL LANGUAGE

DR. NOR FAZLIDA MOHD SANI

DEPT. OF COMPUTER SCIENCE,

& INFORMATION SECURITY RESEARCH GROUP

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CHAPTER 1: INTRODUCTION

Why study automata?

Terminology and mathematical concepts

Formal proof

Concepts of automata theory

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WHYSTUDYAUTOMATA?

Automata theory is the study of abstract computing

devices, or machines 1930s, before there were computers, Alan Turing

studied an abstract machine that had all the

capabilities of todays computer Turings goal was to describe what computer could do and

could not

1940s and 1950s, simpler kinds of machine, which

today call finite automata studied by a number ofresearchers.

Originally proposed to model brain function, turned out to

be extremely useful for a variety of other purposes 3


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CONT. Late 1950s, the linguist N. Chomsky study of formal

grammars Not strictly machine, these grammars have close relationship

to abstract automata and serve today as the basis of someimportant software components, including parts of compilers

In 1969, C. Cook extended Turings study Cook able to separate those problems that can be solvedefficiently by computer from those problems that can inprinciple be solved, but in practice take so much timeproblem called intractable or NP-hard

All these theoretical developments bear directly on what

computer scientists do today Finite automata & certain formal grammarsused in the

design and construction of important kinds of software

Turing machinehelp understand what we can expect fromour software

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CONT.

Regular expressions are used in many systems.

E.g., UNIX a.*b.

E.g., document-type definition, DTDs describe XML tags witha RE format like person (name, addr, child*).

Finite automata model protocols, electronic circuits. Theory is used in model-checking.

Context-free grammars are used to describe the syntax

of essentially every programming language.

Not to forget their important role in describing natural

languages.

And DTDs taken as a whole, are really CFGs.

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CONT.

When developing solutions to real problems, weoften confront the limitations of what software cando.

Undecidable thingsno program whatever can do it.

Intractable thingsthere are programs, but no fastprograms. Well learn how to deal formally with discrete

systems.

Proofs: You never really prove a program correct, but

you need to be thinking of why a tricky technique reallyworks.

Well gain experience with abstract models andconstructions.

Models layered software architectures.6


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AUTOMATATHEORY

Automata theory deals with the definitions and properties of

mathematical models of computation.

These models play a role in several applied areas of computer

science, such as:

Finite automataused in text processing, compiler, andhardware design Context-free grammarused in programming languages

and artificial intelligence

Excellent place to begin study of the theory of computation.

Allows practice with formal definitions of computation as itintroduces concepts relevant to other nontheoretical areas of

computer science.

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TERMINOLOGYANDMATHEMATICALCONCEPTS

- SETS

A group of object represented as a unit

May contain any type of object, incl. numbers, symbols, and

even other sets.

The objects in a set are called its elementsor members.

Thus, set {7, 21, 57} contains the element 7, 21, and 57

Symbol and denote set membership and nonmembership

7 {7, 21, 57} and 8 {7, 21, 57}

A is a subsetof B, written AB, if every member of A also is

a member of B

A is a proper subset of B, written A B, if A is a subset of B

and not equal to B.

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SETSCONT.

An infinite set contains infinitely many elements

The set of natural numbers as {1,2,3,} The set of integer is written {,-2,-1,0,1,2,} The set with 0 members is called empty set, written

To describe a set containing elements according to some rule,write {n|rule about n}

{n|n= m2 for some mN} means the set of perfect squares.

Two sets A and B, the un ionof A and B, writtenAB,

combining all the elements in A and B into a single set

Intersectionof A and B, writtenA B, is the set of elementsthat are in both A and B.

The compliment of A, , is the set of all elements underconsideration that are not in A.

Venn diagram examples9


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SEQUENCESANDTUPLES

A sequence of objects is a list of these objects in some order

Examples: sequence 7, 21, 57 written (7,21,57)

In sequence the order and repetition does matter.

Sequences may be finite or infinite.

Finite sequences often are called tup les

A sequence with kelements is a k-tuple. Thus (7,21,57) is a 3-

tuple. A 2-tuple is also called pair.

Sets and sequences may appear as elements of other sets

and sequences.

Power set of A is the set of all subsets of A

If A is the set {0,1}, the power set of A is the set {, {0}, {1}, {0,1}}.

The set of all pairs whose elements are 0s and 1s is {(0,0), (0,1),

(1,0), (1,1)}10


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SEQUENCESANDTUPLESCONT.

IfAand Bare two sets, the Cartesian Prod uct or cross produc t of

Aand B, writtenA B

Is the set of all pairs wherein the first element is a member ofAand

the second element is a member of B. Example: If A = {1,2} and B = {x,y,z}

A B= {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)}.

Also can take the Cartesian product of ksets,A1,A2, ,Ak, writtenA1 A2 Ak, It is the set consisting of all k-tuples (a1,a2,,ak)where aiAi. Example: If A and B are as above example,

A B A= {(1,x,1), (1,x,2), (1,y,1), (1,y,2), (1,z,1), (1,z,2), (2,x,1), (2,x,2), (2,y,1),

(2,y,2), (2,z,1), (2,z,2)}.

If we have the Cartesian product of a set with itself, we use the

shorthand AA A =Ak

Example: The set N2equals N N . It consists of all pairs of natural numbers. May

also write as {(i,j)|i,j 1}.

11k


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FUNCTIONSANDRELATIONS

A function is an object that sets up an input-output relationship.

If fis a function whose output value is bwhen the input value is a,

write as

f(a) = b.

Function also called a mapping

The set of possible inputs to the function called its domain.

The outputs of a function come from a set called its range.

The notation for saying that fis a function with domain Dand range R

is

f: DR

Describe a specific function in several ways:

Procedure for computing an output from a specified input

Table that list all possible inputs and gives the output for each input.

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FUNCTIONSANDRELATIONSCONT.

Example: Consider the function f: {0,1,2,3,4} {0,1,2,3,4}.

This function adds 1 to its input and then outputs the result modulo 5.

A number modulo m is the remainder after division by m. Forexample, the minute hand on a clock face counts modulo 60. When

we do modular arithmetic we define Zm = {0,1,2,,m-1}.With thisnotation, the aforementioned function f has the form f: Z5Z5.

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n f(n)

0 1

1 2

2 3

3 4

4 0


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Example: Two-dimensional table is used if the domain of

function is the Cartesian product of two sets. Function, g: Z4

Z4Z4. The entry at the row labeled iand the column labeled

jin the table is the value of g(i,j).

The function gis the addition function modulo 4.

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g 0 1 2 30 0 1 2 3

1 1 2 3 0

2 2 3 0 1

3 3 0 1 2


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When domain of a function f isA1 Akfor some setsA1,,Ak,the input to f is a k-tuple (a1,a2,,ak) and we call aithe arguments tof.

A function with k arguments is called a k-ary function, and k is the

arity of the function.

If k is 1, fhas single argument and f is called a unary function.

If k is 2, fis a binary function.

A predicate or property is a function whose range is {TRUE, FALSE}.

A property whose domain is a set of k-tuplesA Ais called arelat ion, a k-ary relation, or a k-ary relation on A.

A common case is a 2-ary relation, called binary relat ion.

If Ris a binary relation, the statement aRbmeans that Arb =

TRUE.

Similarly if Ris a k-ary relation, the statement R(a1,,ak) meansthat R(a1,,ak) = TRUE.

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FUNCTIONSANDRELATIONSCONT. Example: in childrens game called Scissor-Paper-Stone, the two player simultaneously

select a member of the set {SCISSORS, PAPER, STONE} and indicate their selection

with hand signals. If the two selections are the same, the game starts over. If the

selections differ, one player wins, according to the relation beats.

From table can determine that SCISSORS beatsPAPER is TRUE and that PAPER beats

SCISSORS is FALSE.

Describing predicates with sets instead of functions is more convenient. The predicate P :

D {TRUE, FALSE} may be written (D,S), where S = {a D| P(a) = TRUE}, or simply S

if the domain D is obvious from the context. Hence the relation beats may written

{(SCISSORS, PAPER), (PAPER, STONE), (STONE SCISSORS)}.

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beats SCCISSORS PAPER STONE

SCCISSORS FALSE TRUE FALSE

PAPER FALSE FALSE TRUE

STONE TRUE FALSE FALSE


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Special type of binary relation, called an

equivalence relat ion, captures the notion of two

objects being equal in some feature.

A binary relation Ris an equivalence relation if R

satisfies three condition:

1. Ris ref lexiveif for everyx,xRx;

2. R is symmetr icif for everyxand y,xRyimplies yRx;

and

3. R is t ransi t iveif for everyx, y, and z,xRyand yRzimpliesxRz.

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Example: Define an equivalence relation on the

natural numbers, written 7. For i,j, Nsay that i7j, if i-jis a multiple of 7. This is an equivalence

relation because it satisfies the three conditions.

First, it is reflexive, as ii = 0, which is a multiple of 7. Second, it is symmetric, as ijis a multiple of 7 ifjiis

a multiple of 7.

Third, it is transitive, as whenever i-jis a multiple of 7

andj-kis multiple of 7, then i-k= (i-j) (j-k) is the sum oftwo multiples of 7 and hence a multiple of 7, too.

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GRAPHS

An und irected graph, or simply graph, is a set of points with

lines connecting some of the points.

The points are called nodesor vertices, and the lines are

called edges, as shown in following figure.

(a) Degree = (b) Degree =

The number of edges at a particular node is the degreeof that

node.

No more than one edge is allowed between any two nodes.19

1

2

3 4

5

1 2

3 4


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GRAPHSCONT.

In graph Gthat contains nodes iandj, the pair (i,j) represents

the edge that connects i andj.

The order of i andjdoesnt matter in an undirected graph, sothe pairs (i,j) and (j, i) represent the same edge.

If Vis the set of nodes of Gand Eis the set of edges, we sayG= (V, E).

Graph can be describe with diagram or more formally by

specifying Vand E.

Example:

Formal description for graph (a) is

({1,2,3,4,5}, {(1,2), (2,3), (3,4), (4,5), (5,1)})

Formal description for graph (b)

({1,2,3,4}, {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)})20

1

2

3 4

5

1 2

3 4

(a)

(b)


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GRAPHSCONT.

Graphs frequently are used to represent data.

For convenience, we label the nodes and/or edges of a graph,

which then called a labeled graph.

Graph Gis a subgraph of graph Hif the nodes of Gare a

subset of the nodes of H, and the edges of Gare the edges ofHon the corresponding nodes.

Figure shows a graph H and

a subgraph G (shown darker)

Pathis a sequence of nodes connected by edges.

Simple path is a path that doesnt repeat any nodes. A graph is connectedif every two nodes have a path

between them.

A path is a cyc leif it starts and ends in the same node.

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GRAPHSCONT.

Simple cycle is one that contains at least three nodes and

repeats only the first and last nodes.

A graph is a treeif it is connected and has no simple cycles.

Tree may contain a specially designated node called the

roo t. The nodes of degree 1 in a tree, other than the root, are

called the leaves.

If it has arrows instead of lines, the graph is a directed graph.

The number of arrows pointing froma particular node is the

outdegreeof that node, and The number of arrows pointing to a particular node is the

indegree.

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GRAPHSCONT.

In directed graph, edge from itojrepresented as a pair (i,j).

Formal description of directed graph Gis (V, E) where Vis the set

of nodes and Eis the set of edges.

Formal description for graph below:

({1,2,3,4,5,6}, {(1,2), (1,5), (2,1), (2,4), (5,4), (5,6), (6,1), (6,3)}).

A path in which all the arrows point in the same direction as its stepsis called a directed path.

A directed graph is st rongly con nected if a directed path connects

every two nodes.23

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3

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STRINGSANDLANGUAGES

Strings and characters are fundamental building

blocks in computer science.

Alphabetto be any nonempty finite set.

The members of the alphabet are thesymbols

ofthe alphabet.

Generally use capital Greek letters and to

designate alphabets and a typewriter font for

symbols from an alphabet.

Example of alphabets:

1= {0,1};

2= {a,b,c,d,e,f,g,h,i,j,k,l,m,o,p,q,r,s,t,u,v,w,x,y,z};

= {0,1,x,y,z}.24


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STRINGSANDLANGUAGES CONT.

A str ing over an alphabet is a finite sequence of symbols

from that alphabet, usually written next to one another and not

separated by commas.

If 1= {0,1}, then 01001 is a string over 1.

If wis a string over , the lengthof w, written |w|, is thenumber of symbols that it contains.

The string of length zero is called the empty string and written

,plays the role of 0 in a number system

If w has length n, we can write w= w1w2wn where eachwi

. The reverseof w, written w , is the string obtained by writing

w in the opposite order (i.e., wnwn-1w1)

Stringz is a subst r ingof wif zappears consecutively within

w.25


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STRINGSANDLANGUAGES CONT.

If we have stringxof length mand string yof length n, the

concatenation ofxand y, writtenxy.

String obtained by appending yto the end ofx.

Superscript notation is used to concatenate a string with

itself many times.

The lex icographic order ing of strings is the same as the

familiar dictionary ordering, except that shorter strings

precede longer strings.

Thus the lexicographic ordering of all strings over the alphabet{0,1}is

(, 0,1,00,01,10,11,000,)

Alanguage

is a set of strings.

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BOOLEANLOGIC

Boo lean logic is a mathematical system built around the two

values TRUE and FALSE (Boolean values) always

represented by values 1 and 0.

Boolean values can be manipulated with special designed

operations, called Boolean operat ions, such as:

Negationor NOT, with symbol , the opposite value

Conjunct ion, or AND, with symbol , the conjunction of

two Boolean values is 1 if both of those values are 1

Disjunct ion, or OR, with symbol , the disjunction of two

Boolean values is 1 if either of those values is 1.0 0 = 0 0 0 = 0 0 = 1

0 1 = 0 0 1 = 1 1 = 0

1 0 = 0 1 0 = 1

1 1 = 1 1 1 = 127


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BOOLEANLOGICCONT.

Other Boolean operations:

Exclusive or, or XOR, symbol , is 1 if either but not

both of its two operands are 1

Equali ty, symbol , is 1 if both of its operands have

the same value Impl icat ion, symbol , is 0 if its first operand is 1 and

its second is 0; otherwise is 1.

0

0 = 0 0

0 = 1 0

0 = 10 1 = 1 0 1 = 0 0 1 = 1

1 0 = 1 1 0 = 0 1 0 = 0

1 1 = 0 1 1 = 1 1 1 = 128


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BOOLEANLOGICCONT.

Distr ibu t ive law for AND and OR, similar for

addition and multiplication, which states that a (b

+ c) = (a b) + (a c). The Boolean version comes

in two forms:

P(QR) equals (PQ) (PR), and its dual

P(QR) equals (PQ) (PR)

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FORMALPROOF

Proof is something that every computer scientist needs to

understand

Formal proof of the correctness of a program should go hand-

in-hand with writing of the program itself.

Recursion or iterationmight unlikely write the code correctlywhen testing tells the code is incorrect, we still need to get itright

To make recursion or iteration correctneed to set up an inductivehypothesis.

The process of understanding the workings of a correct program,

same as the process of proving theorems by induction.

Automata theory cover methodologies of formal proof:

1. Deductive (sequence of justified steps), and

2. Inductive (recursive proofs of a parameterized statement that use

the statement itself with lower values of the parameter)30


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DEDUCTIVEPROOFS

Consists of a sequence of statements whose truth

leads from some initial statement, called hypothesis

or the given statement(s), to a conclusion

statement.

Hypothesis may be true or false, typically consistsof several independent statements connected by

logical AND

Theorem is proved when go from a hypothesis Hto

a conclusion C, the statement is if Hthen C, saysthat Cis deduced from H.

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DEDUCTIVEPROOFSCONT.

Example:

Theorem 1.3: Ifx4, then 2xx2. Can convince informally that Theorem 1.3 is true, with H

is x4, has parameterx, thus neither true nor false.

Its true depends on value of parameterx; e.g., H is trueforx= 6 and false forx= 2.

The C is 2xx2 uses parameterxand true for certainvalues ofx. C is false forx = 3, since 23= 8, which is

not as large as 32 = 9. on the other hand, C is true forx

= 4, since 24= 42 = 16. Forx= 5, the statement is alsotrue, since 25= 32 and 52 = 25.

We have completed an informal but accurate proof. (we

shall return to the proof and make it more precise in

inductive proofs)32


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REDUCTIONTODEFINITIONS

Many in automata theory, the terms used in the statement

may less obvious.

If not sure how to start a proof, convert all terms in the

hypothesis to their definitions.

Example: Theorem 1.5: Let Sbe a finite subset of some infinite set U. Let T

be the complement of Swith respect to U. Then Tis infinite.

Restating the facts into definitions:

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Original Statement New Statement

Sis finite There is a integer nsuch

that S=n

Uis infinite For no integerpis U=p

Tis complement of S S

T= Uand S

T=


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REDUCTIONTODEFINITIONSCONT.

Need to use a common proof technique called proof bycontradiction, which assume that the conclusion is false. Thenuse the assumption, together with parts of the hypothesis, to

prove the opposite of one of the given statements of the

hypothesis.

The contradiction of conclusion is Tis finite. Restate theassumption that T is finite as T=mfor some integer m. One of the given statement, ST= Uand ST= . Element

of Uare exactly the elements of Sand T. Thus, there must be n+

melements of U. Since n+ mis an integer, we have shown that

U=n + m, follows that Uis finite. But the statement that Uisfinite contradicts the given statement that Uis infinite.

By the principle of proof by contradiction we may conclude thetheorem is true.

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REDUCTIONTODEFINITIONSCONT.

Proofs do not have to be so wordy.

The reprove of theorem in a few lines:

PROOF: (of Theorem 1.5) We know that ST= Uand S

and T are disjoint, so S + T= U. Since S is finite,

S = nfor some integer n, and since U is infinite, there isno integerpsuch that U=p. So assume that T is finite;that is; T = mfor some integer m. Then U=S +T= n+ m, which contradicts the given statement thatthere is no integerpequal to U.

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OTHERTHEOREMSFORMS

The if-then form of theorem is most common in typical areasof mathematics.

However, there are other kinds of statement proved as

theorems also.

Ways of Saying If-Then Some other ways in which if Hthen C might appear:

Himplies C, Honly if C, Cif H, or Whenever Hholds, Cfollows(and with other variants form).

If-And-Only-If Statements

Form of A if and only if B, other form A iff B, A is equivalent to B,or A exactly when B.

These statements are actually two if-then statements: if A then Band if B then A

To prove A if and only if B by proving two statements:1. The if part: if B then A, and

2. The only-if part: if A then B, which often stated in equivalent form Aonly if B

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ADDITIONALFORMSOFPROOF

Proving Equivalence About Sets

In automata theory, we are frequently asked to prove a theorem

which says that the sets constructed in two different ways are the

same sets.

Often this sets are sets of character strings, and the sets are

called languages. If E and F are two expressions representing sets, the statement

E=F means that the two sets represented the same.

Commutative law of union says that we can take the union of two

sets R and S in either order., R S = S R

Contrapositive The contrapositive of the statement if H then C is if not C then

not H. A statement and its contrapositive are either both true or both

false, so we can prove either to prove the other.37


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ADDITIONALFORMSOFPROOFCONT.

Counterexample

A strategy for implementing a program for exampleandneed to decide whether or not the theorem is true.

The resolve the question, we may alternately try to prove

the theorem, and if cannot, try to prove that the statementis false.

Proof by Contradiction

Another way to prove a statement of form if H then C is toprove the statement H and not C implies falsehood.

Start by assume hypothesis H and the negation of theconclusion C.

Complete the proof by showing something known to be

false. (example Theorem 1.5)38


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INDUCTIVEPROOFS

Form of proof that is essential when dealing with

recursively defined objects or concepts such as

trees and expressions of various sorts.

Inductions on Integers

Given statement S(n), ninteger to prove. Common

approach is to prove:

1. Basis show S(i) for a particular integer i. Usually i=0 or i=1.

(or maybe higher, idepends on S)

2. Induction step assuming ni, wherei is the basis integer,

and show that if S(n) then S(n+1). The Induction Principle: If we prove S(i) and we prove that for all n

i, S(n) implies S(n+1), then we may conclude S(n) for all ni.

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INDUCTIVEPROOFSCONT.

Example: Theorem 1.3 states that Ifx4, then 2xx2.

BASIS: If x =4, then 2xandx2 are both 16. Thus 2442

holds.

INDUCTION: Suppose for somex4 that 2xx2. We

need to prove the same statement with x+1 in place of x,

that is 2[x+1][x+1]2.

In this case, we can write 2[x+1]as 2 2x . Since S(x) tells us

that 2xx2, we can conclude that 2x+1= 2 2x2x2.

But we need to show that 2x+1(x+1)2. One way to prove this

statement is to prove that 2x2 (x+1)2and then use the

transitivity of to show 2x+12x2 (x+1)2. In our proof that

2x2 (x+1)2 (1.1)

we may use the assumption that x 4. Begin by simplifying

(1.1):

x2 2x+1 (1.2)40


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INDUCTIVEPROOFSCONT.

Divide (1.2) byx, to get:

x2+ (1.3)

Since x 4, we know 1/ x 1/4. thus, left side of (1.3) is at

least 4, and the right side is at most 2.25. We have thus

proved the truth of (1.3). Therefore, Equations (1.1) and (1.2)are also true. Equation (1.3) in turn gives us 2x2 [x+1]2for x

4 and let us prove statement S(x+1), which we recall was 2x+1

(x+1)2.

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CONCEPTSOFAUTOMATATHEORY

The concepts include the alphabet ( a set ofsymbols), strings (a list of symbols from analphabet), and language (a set of strings from thesame alphabet).

Languages

If is an alphabet, and L*, then Lis a language

over .

Problems in automata is the question of deciding

whether a given string is a member of someparticular language

The problem Lis : Given a string win *, decide

whether or not wis in L. 42

automata - chap1+introduction

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