St. Paul University Manila

(St. Paul University System) 680 Pedro Gil St., Malate, Manila

College of Business and Management Computer Science and Information Technology Programs

LABORATORY MANUAL

in

Theory of Computation

Maria Regina G. Lim – Cachero October, 2010

Preface

This laboratory manual includes a collection of experiments and exercises

suitable for learning the topics contained in Theory of Computation. It starts with a discussion and application of Mathematical Notations and Terminologies, an essential part to being introduced to the Computational Theory.

The concepts discussed in this course focuses on Automata and Languages, as

well as, Computability Theory. Hence, most, if not all, experiments are directed towards applying the theories of these concepts in various kinds of computing problems.

The manual intends to make learning Theory of Computation more meaningful. Thus, a learner’s success in accomplishing the experiments and exercises in this manual will lead to better results in the quizzes and examinations for this course.

Any error or modification in this manual may be reported to my email address so

that the material can be improved accordingly.

Special Note to Students

To better understand the different domains of Theory of Computation, familiarity

to mathematical notations and terminologies will work in your favor. Hence, please do

your own refresher of basic mathematical and algebraic concepts, such as Sets,

Algebraic expressions. This will facilitate your absorption of the concepts in this course.

Maria Regina G. Lim-Cachero rhejlim@yahoo.com

Table of Contents

MATHEMATICAL NOTATIONS AND TERMINOLOGIES

Sets ……………………………………………………………….. 1

Functions .…………………………………………………………. 5

Relations …………………………………………………………... 6

AUTOMATA AND LANGUAGES

Deterministic Finite Automata …………………………………….. 7

Regular Expressions ……………………………………………… 8

Context Free Grammars ..………………………………………… 10

Pushdown Automata….. ..………………………………………… 12

COMPUTABILITY THEORY

Turing Machine……. ………………………………………………. 14

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MATHEMATICAL NOTIONS AND TERMINOLOGY

OBJECTIVES

1. Develop a good understanding of the basic mathematical objects, tools, and notations used in learning theory of computation;

2. Solve problems on set operations, Venn diagramming, and relations and functions;

3. Be familiar with how these notions and terminologies are used in computational theory.

TOPIC : SETS

The branch of mathematics that studies sets, which are collections of objects.

It is used in the definitions of nearly all mathematical objects.

Set theory deals with operations between, relations among, and statements about sets.

x is a member of A is written xA

A member of a set is also known as its element

The basic binary relation between two sets is the subset relation, also called set inclusion. Subset means every element of S is also an element of T.

Subset Principles

o A B if and only if whenever xA it follows that x∈B

o AB if and only if AB and AB

o For any set A, AU

o AA

o If AB and BA then A=B

o If AB and BC then AC

o A∩BA and AAB

Basic Set Operations

o Union - of the sets A and B, denoted , is the set whose members are members of at least one of A or B.

o Intersection - of the sets A and B, denoted , is the set whose members are members of both A and B.

o Complement - of set A relative to set U, denoted Ac, is the set of all members of U that are not members of A. This is also known as set difference.

o Symmetric Difference - of sets A and B is the set whose members are members of exactly one of A and B.

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o Cartesian Product - of A and B, denoted X, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B.

o Power Set - of a set A is the set whose members are all possible subsets of A.

EXPERIMENTS:

I. True or False?

_________1. {a,a,b} {a,b}

_________2. {c, a, b} = {c, b, a}

_________3. {x,y,z} =

_________4. {1,2,3,4} = {1,2,3}

_________5. {e,f,g} {h,i,j} = {j,h,i} {g,f,e}

_________6. {a,b,c,c,d} {a,b,c,d,e,f,g}

_________7. {a,b,b} {a,b,c}

II. What is the resulting set?

__________1. {2,3,5} {3,5,7}

__________2. {a,b,c} {2,3}

__________3. {a,b,c} {2,3}

__________4. {2,4,6} {3,4,5}

__________5. {1,2,3,4,5,6} {2,3,5,7,9,11}

__________6. P({1,2})

III. Answer the following:

__________1. Let A={a, b, {c, d}, e}. How many elements does A contain?

__________2. Let A = {2, {4, 5}, 4}. Which statement is correct?

a) 5 is an element of A. b) {5} is an element of A. c) {4, 5} is an element of A. d) {5} is a subset of A.

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__________3. Which of these sets is finite?

a) {x | x is even} b) {x | x < 5} c) {1, 2, 3,...} d) {1, 2, 3,...,999,1000}

__________4. Let S={1, 2, 3}. How many subsets does S contain?

__________5. Let D E. Suppose a D and b E. Which of the following statements must be true?

a) c D b) b D c) a E d) a D

__________6. Let A = {x | x is even}, B = {1, 2, 3,..., 99, 100}, C = {3, 5, 7, 9}, D = {101, 102} and E = {101, 103, 105}. Which of these sets can equal S if S A and S and B are disjoint?

a) A b) B c) C d) D e) E

__________7. Let S = {a, b}. How many elements does the power set 2S contain?

__________8. Which set S does the power set 2S = { , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} come from?

a) {{1},{2},{3}} b) {1, 2, 3} c) {{1, 2}, {2, 3}, {1, 3}} d) {{1, 2, 3}}

__________9. Let A = {x, y, z}, B = {v, w, x}. Which of the following statements is correct?

a) A B = {v, w, x, y, z} b) A B = {v, w, y, z} c) A B = {v, w, x, y} d) A B = {x, w, x, y, z}

__________10. Let A = {1, 2, 3, ..., 8, 9} and B = {3, 5, 7, 9}. Which of the following statements is correct?.

a) A B = {2, 4, 6} b) A B = {1, 2, 3, 4, 5, 6, 7, 8, 9} c) A B = {1, 2, 4, 6, 8} d) A B = {2, 4, 6, 8}

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__________11. Let C = {1, 2, 3, 4} and D = {1, 3, 5, 7, 9}. How many elements does the set C D contain?

How many elements does the set C D contain?

__________12. Let A = {2, 3, 4}, B = {3} and C = {x | x is even}. Which statement is correct?

a) C A = B b) C B = A c) A C d) C / A = B

__________13. Let A B, B C and D A = C. Which statement is always false?

a) B D b) A C c) A = B b) B D = and B A

__________14. Let U = {1, 2, 3, ..., 8, 9} and A = {1, 3, 5, 7}. Find A'.

a) A' = {2, 4, 6, 8} b) A' = {2, 4, 6, 8, 9} c) A' = {2, 4, 6} d) A' = {9}

__________15. Let U = {1, 2, 3,..., 8, 9}, B = {1, 3, 5, 7} and C = {2, 3, 4, 5, 6}. How many elements does the set (B C)' contain?

How many elements does the set (C - B)' contain?

IV. Draw the Venn Diagram for the following:

1. A = {2, 4, 5, 3, 1, 6, 8} B = {3, 5, 4, 7, 9, 12} C = {4, 7, 10} D = {11,14, 9, 7} E = {1, 23, 25}

2. In an interview of 50 students,

12 liked Proposition 8 and Proposition 13

18 liked Proposition 8, but not Proposition 5

4 like Proposition 8, Proposition 13 and Proposition 5

25 liked Proposition 8

15 liked Proposition 13

10 liked Proposition 5, but not Proposition 8 or Proposition 13

1 liked Proposition 13 and Proposition 5, but not Proposition 8

3. In a class of 50 college students, 30 are studying VB .Net, 25 studying Java, and 10 are studying both. How many students are not studying either computer language?

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TOPIC : FUNCTIONS

it is an object that sets up an input-output relationship.

A function takes an input and produces an output; in every function, the same input always produces the same output.

also called mapping, i.e. f(a) – b can be read as f maps a to b.

Parts of a Function

o DOMAIN - the set of possible inputs to the function.

o RANGE – the set of outputs of a function

o The notation for saying that f is a function with domain D and range R is:

f : D R.

EXPERIMENT:

Answer the following:

1. B (x) = x2 – x3 Evaluate B(5)

2. s (x) = 2x2 – 2x Evaluate s(3)

3. f (x) = 5x + 2 Evaluate f(2)

4. h(x) = 3x3 Evaluate h(2)

5. k(x) = 3x – 1 Evaluate k(-1)

6. Let f 1, f 2, f 3, f 4, f 5 be functions of R into R and let f 1(x) = x2 + 3x - 4. Which of these functions are equal to f 1? a) f 2(x) = x2 b) f 3(y) = y2 - 4 c) f 4(z) = z2 + 3z - 4 d) f 5(x) = x2 + 3x

7. Let the functions f, g, and h be defined by :

(a) f (x) = x3 where x R (b) g (y) = y3 where 2 < y < 9 (c) h (z) = z3 where 1 < z < 10 Which of these functions are equal?

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TOPIC : RELATIONS

A set of ordered pairs

The domain is the set of all the first numbers of the ordered pairs; all of the x-values.

The range is the set of the second numbers in each pair; or the y-values.

e.g. in a relation {(1,3), (4,7), (2,8)} The domain is {1,4,2} and the range is {3,7,9}

EXPERIMENT:

1. What is the domain and range of the following relation? { (-1,2), (2, 51), (1, 3), (8, 22), (9, 51) }

2. The relation R is given by the set of ordered pairs, R = {(2, 4), (3, 4), (1,3), (3, 5), (2, 3) }. Which of the following is the domain of R?

a) {2, 3, 1, 5} b) {1, 2, 3} c) {1, 2, 3, 4, 5} d) {2, 4, 3, 5}

3. The relation R is given by the set of ordered pairs, R = {(2, 4), (3, 4), (1,3),

(3, 5), (2, 3) }. What should the range of R be? a) {3, 4, 5} b) {1, 2, 3} c) {1, 2, 3, 4, 5} d) {4, 4, 2, 5}

4. Find the inverse of the relation

R = {(2, 4), (3, 4), (1,3), (3, 5), (2, 3)} a) {(2, 4), (3, 4), (1, 3), (3, 5), (2, 3)} b) {(4, 2), (4, 3), (3, 1), (5, 3), (3, 2)} c) {5, 4, 3, 2, 1} d) {(4, 4), (3, 3), (1, 1), (5, 5), (2, 2)}

5. Two sets are given by E = {2, 4, 5, 6} and F = {1, 3, 6}. Let R be a relation from set E to set F, which is defined by the open sentence "x is greater than y". How many elements does the solution set of R contain?

6. Which relations below are functions? Relation #1 { (3,4), (4,5), (6,7), (8,9) } Relation #2 { (3,4), (4,5), (6,7), (3,9) } Relation #3 { (-3,4), (4,-5), (0,0), (8,9) } Relation #4 { (8, 11), (34,5), (6,17), (8,19) }

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AUTOMATA and LANGUAGES OBJECTIVES

1. Be familiar with the study of abstract computing devices or “machines”;

2. Appreciate automata theory as useful models for designing computing hardware and software.

3. Create automata that will implement specific problems;

4. Evaluate regular expressions and context free grammars.

TOPIC : FINITE AUTOMATA

Finite automata is an important way to describe certain simple, but highly useful languages called “regular languages."

A graph with a finite number of nodes, called states.

Arcs are labeled with one or more symbols from some alphabet.

One state is designated the start state or initial state.

Some states are final states or accepting states.

The language of the FA is the set of strings that label paths that go from the start state to some accepting state.

A finite (deterministic) acceptor A is specified by five (5) pieces of information or tuples:

A = (S, A, i, , T) ;

Where:

S is a finite set called the set of states

A is the finite input alphabet

i is a fixed element of S called the initial state

is a function : S X A → S, called the transition function

T is a subset of S (possibly empty!) called the set of terminal or final states.

There are two ways of providing the 5 tuples to specify an acceptor: transition diagrams and transition tables.

A transition diagram is a special kind of directed labeled graph: the vertices are labeled by the states S of A; there is an arrow labeled a from the vertex

labeled s to the vertex labeled t precisely when (s, a) = t in A.

That is to say, the input a causes the acceptor A to change from state s to state t.

Example:

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EXPERIMENT:

Create a Deterministic Finite Automata (DFA) for the following problems. State the 5 Tuples of the DFA. Construct the Transition Diagram for the DFA as well.

Create DFAs accepting 0 and 1 as input and recognizes all strings that:

1. ends in a 1

2. have odd number of 1’s

3. contain even number of 0’s and any number of 1’s

4. have a sequence ’11’ somewhere in the string

5. contain at least one 1 and an even number of 0’s follow the last 1

6. contain even number of zeroes and even number of ones

7. that start and end with the same number (either 1 or 0).

8. starts with “101” and succeeded by any number of 1’s and 0’s

9. is preceded by any number of 1’s and 0’s but ends with “101”

10. contain two consecutive 0’s followed by two consecutive 1’s.

TOPIC : REGULAR EXPRESSIONS

Symbols and alphabet

o A symbol is a valid character in a language

o Alphabet is set of legal symbols; Typically denoted as

Metacharacters/metasymbols that have special meanings

o Defining reg-ex operations (e.g. |, (, ), *, +, etc.)

o Escape character (\) to turn off special meanings

o Empty string and empty set ={}

Basic regular expressions

o Single characters (a={a} for any character a )

o Empty string ={}

o Empty set

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Basic regular expression operations

o Union/Alternation, denoted by a | b or a + b

o Concatenation, denoted by ab

o Repetition (Kleen closure, 0 or more times) a*

EXPERIMENTS

Simplify the following Regular Expressions:

1. (A + B + B) + + 1

2. N + M + K + N + M +

3. (AB) (B+B)

4. C(A+A+A+B)

5. C(A+A+A+)

6. ( + D) + D

7. (L + M) + N

8. B + B + B +

9. C + C + D + D +

10. R + L

11. Z + Z + X

12. S + S + R

13. (B + B)C

14. (E + D + E + D)

15. (AB)C+C

Evaluate the following Regular Expressions based on their elements:

A = {2, 4, 6}, B = {3, 8, 10}, C = {1, 9}, D = { }, E = { }

1. D + C + B = ?

2. BE = ?

3. DC = ?

4. AB = ?

5. AD = ?

6. B + B = ?

7. (CD) B =?

8. (C + D) + (C + D) =?

9. D(C + B) = ?

10. E(A + B) = ?

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TOPIC : CONTEXT FREE GRAMMARS

Contest free grammars (CFGs) are formal notations for expressing recursive definition of languages.

There is a finite set of symbols that form the strings of the language being defined. The alphabet are called TERMINALS or TERMINAL SYMBOLS.

There is a finite set of variables also called NONTERMINALS or SYNTACTIC CATEGORIES.

One of the variables represents the language being defined, it is called the START SYMBOL. Other variables represent auxiliary classes of strings that are used to help define the language of the start symbol.

There is a finite set of productions or rules that represent the recursive definition of a language.

Production Composition: Each production consists of:

A. a variable that is being defined by the production, called the HEAD.

B. the production symbol →

C. a string of zero or more terminals called the BODY.

CFG = (V,T,P,S) where

V = set of variables

T = the terminals

P = set of productions

S = start symbol

EXPERIMENT:

Using the grammar, show the derivations of the following sentences

E → I I → Ib

E → E + E I → I0

E → E * E I → I1

E → ( E )

I → a

I → b

I → Ia

1. (ab * aa1) + bb0

2. aaa + bbb + a

3. ab0 + bb1 * b1

4. a * b + a

5. 11 + ab

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Using the grammar, construct the parse tree of the following sentences

E E + E | E – E |

E * E | E / E |

- E

E ( E )

E id

id a | b | c

1. (a + b) * (b – c)

2. c * (a + b)

3. a + b * c – a / c + b

4. (c / a) + a + b

5. a + c / a – b + (c * a + b)

Using the grammar, do a bottom-up parsing of the following sentences:

R R / R

R RR

R R *

R ( R )

R a

R b

1 . a / b * a

2 . ( a / b b / a )

3 . a / b / ( c * a )

4 . a * ( a / c ) * c

5 . ( a b ) / ( a b )

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TOPIC : PUSHDOWN AUTOMATA

a pushdown automaton (PDA) is a finite automaton that can make use of a stack containing data.

They can use the top of the stack to decide which transition to take.

They can manipulate the stack as part of performing a transition.

A PDA is formally defined as a 7-tuple:

Where :

Q is a finite set of states

is a finite set known as input alphabet

is a finite set known as stack alphabet

is the transition function, that denotes the power set of S.

qo Q is the start state Z is the initial stack symbol

F Q is the set of accepting states

Transitions:

Let ((p, a, β), (q, γ)) be a transition.

It means that we o Move from state p. o Read a from the tape, o Pop the string β from the stack, o Move to state q, o Push string γ onto the stack.

The first three (p, a, β), are “input.”

The last two (q, γ) are “output.”

((p, a, β), (q, γ)) is drawn as:

When we push β, we push the symbols of β as we read them right to left. o When we push the string abc, we push c, then push b, then push a.

When we pop γ, we pop the symbols of γ as we read them from left to right (reverse order).

o When we pop the string abc, we pop a, then pop b, then pop c.

p q

a, ;

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EXPERIMENT:

State the language that is accepted by the following PDAs. Give at least five strings that can be accepted and five strings that are not.

PDA 1:

PDA 2:

PDA 3

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COMPUTABILITY THEORY OBJECTIVES

1. Be familiar with the application of Turing Machine in the design of computing resources;

2. Appreciate computability theory as part of the foundations of computer science.

3. Create Turing machines that will implement specific problems;

TOPIC: TURING MACHINE

First proposed by Alan Turing in 1936

Turing machines are not physical objects but mathematical ones.

It is similar to a finite automaton but with an unlimited and unrestricted memory.

A Turing machine is a much more accurate model of a general purpose computer. A Turing machine can do everything that a real computer can do.

The Turing machine model uses an infinite tape as its unlimited memory.

It has a tape head that can read and write symbols and move around on the tape.

Initially the tape contains only the input string and is blank everywhere else.

If the machine needs to store information, it may write this information on the tape.

To read the information that it has written, the machine can move its head back over it. The machine continues computing until it decides to produce an output.

The outputs accept and reject are obtained by entering designated accepting and rejecting states.

If it doesn't enter an accepting or a rejecting state, it will go on forever, never halting.

A TM consists of o A TAPE which is divided into cells, one next to the other. o A HEAD that can read and write symbols on the tape and move the

tape left and right one (and only one) cell at a time. o A finite TABLE ("action table", or transition function) of instructions

(usually quintuples [5-tuples] o A STATE REGISTER that stores the state of the Turing table, one of

finitely many.

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EXPERIMENT:

Using the following Turing Machines, show the configuration of the following strings. Likewise, state the language that is accepted by the Turing machine.:

1. 0

2. 00

3. 000

4. 0000

5. 000000

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Consider the following Turing Machines, then state the language that they accept:

TM 1:

(q0,0) = (q0,B,R)

(q0,1) = (q1,B,R)

(q1,1) = (q1,B,R)

(q1,B) = (qf,B,R)

TM 2:

(q0,0) = (q1, 1, R)

(q1,1) = (q0, 0, R)

(q1,B) = (qf, B, R)

TM 3:

(q0,0) = (q1,1,R)

(q1,1) = (q2,0,L)

(q2,1) = (q0,1,R)

(q1,B) = (qf,B,R)

Using the following transition table, construct the transition diagram for the following Turing Machines:

Table No. 1

State 0 1 B

q0 (q1, B, R) (q5, B, R) -

q1 (q1, 0, R) (q2,1, R) -

q2 (q3, 1, L) (q2, 1, R) (q4, B, L)

q3 (q3, 0, L) (q3, 1, L) (q0, B, R)

q4 (q4, 0, L) (q4, B, L) (q6, 0, R)

q5 (q5, B, R) (q5, B, R) (q6, B, R)

q6 - -

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Table No. 2

State 0 1 X Y B

q0 (q1, X,R) - - (q3,Y,R) -

q1 (q1, 0,R) (q2,Y,L) - (q1,Y,R) -

q2 (q2, 0, L) - (q0,X,R) (q2,Y,L) -

q3 - - - (q3, Y,R) (q4,B,)

q4 - - - -