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Automata Theory

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2008- 2009

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iq.edu.uomustansiriyah.wwwcom.mustansiriya.www

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:علومات مهمة م Instructor: Hassan Kassim Mohammad Objective Theory of computation is the theoretical study of capabilities and limitations of computers(Theoretical models of computation) Develop skills of precise and formal reasoning in computer science.

REFERENCES: 1. Introduction to Computer Theory 2nd Edition

Daniel I. A. Cohen John Wiley & Sons, Inc 1997. ISBN 0-471-13772-3 2. Introduction to Automata Theory, Languages, and Computation, 2/E,

John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, Addison-Wesley 2001. ISBN 0-201-44124-1.

Prerequisites

• Discrete Mathematics • Introduction to Programming

Units: 3 Grading Policy 1st exam: 5% attendance: 5% homework & projects & Assignments & Quizzes: 5% 2nd exam: 5% attendance: 5% homework & projects & Assignments & Quizzes: 5% Final exam: 70%

Notes • Student must attend at least 80% of the total class sessions to pass the course. • Any kind of cheating/plagiarism may result in an Fail grade in the course. • No labs. But you should write some programs with any language you may know. • There will be 30 lectures 100 minutes each.

(enforced by surprise quizzes – hence no makeup for a quiz). • Late homework submissions will be penalized according to the formula: 100 –10*x2 where x: the number

of days of late submission. • Cheating on homework #i will cause you to get zero grade for the homework.

Office Hours

Sunday 10:00 - 1:00 1:00 – 3:35 3:35 – 5:00 Monday 8:30 – 3:35 3:35 – 5:00 Wednesday 10:00 -1:00 1:00 – 2:15 2:15 – 5:00

Contact Information Office: 3rd Floor, computer science dept. room no. 67 E-mail: world_nora@yahoo.com

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Syllabus

1. Introduction, terminology, definitions 2. LANGUAGE 3. Introduction to Language 4. Regular Expressions 5. Finite Automata 6. Deterministic Finite Automaton (DFA) 7. NON Deterministic Finite Automaton (NFA) 8. Language Accepted by DFA 9. Language Accepted by NFA 10. FA with Epsilon moves 11. Construct regular expression from NFA 12. Regular Expression to NFA 13. Moore and Mealy machines 14. Pumping lemma for regular languages 15. Kleene Theorem 16. Regular Grammar 17. Myhill-Nerode Theorem Minimization of DFA 18. --------------------------EXAM 19. Context-free Languages 20. Pushdown Automata 21. CFG/CFL to NPDA 22. NPDA to CFG/CFL 23. CFG derivation trees Parsing 24. Chomsky normal form 25. Greibach normal form 26. Ambiguous CFL's 27. ---------------------------EXAM 28. TURING MACHINES TM 29. COMPUTABILITY and COMPLEXITY 30. Unsolvable Problems 31. Time Complexity 32. CYK algorithm for CFG's 33. CFL pumping lemma and properties 34. Halting Problem, language, construction 35. Church Turing Thesis 36. --------------------------FINAL EXAM

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1ملخص محاضرة -حسن قاسم . أ-النظرية االحتسابية-قسم علوم الحاسبات-آلية العلوم-الجامعة المستنصرية

As a computer IT, you must study the following: 1- Automata and formal language. Which answers - What are computers (Or what are models of computers) 2- Compatibility. Which answers - What can be computed by computers? 3- Complexity. Which answers - What can be efficiently computed? Automaton & automata In automata we will simulates parts of computers. Or we will make mathematical models of computers Automata are more powerful than any real computer because we can design any machine on papers that can do everything we want. Kinds of languages: 1- Talking language: (e.g.: English language.): - It has alphabetic: ∑={a,b,c,….z , A,B,C,…..Z ,….} - From these alphabetic we make sentences that belong to English language. Now we want to know is this sentence is true or false so - We need grammar in English. Example: Ali is a good boy. (It is a sentence Є English language.) It's semantic (meaning) & syntax (structure) is true according to English grammar. Mona eats papers. (It is a sentence Є English language.) It's semantic (meaning) is false but it's syntax (structure) is true according to English grammar. 2- Programming language: (e.g.: c++ language.): - It has alphabetic: ∑={a,b,c,….z , A,B,C,…..Z , ?, /, - ,\.}. - From these alphabetic we make sentences that belong to programming language. Now we want to know is this sentence is true or false so - We need a compiler to make sure that syntax is true and we don't need to check the meaning because we talk to computer that doesn't know any thing. 3- Formal language: (any language we want.) - It has strings. - From these strings we make sentences that belong to this formal language. Now we want to know is this sentence is true or false so - We need rules.

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Example: Alphabetic: ∑={0,1}. Sentences: 0000001. 1010101. Rules: Accept any sentence start with zero and refuse sentences that start with one. So we accept: 0000001. As a sentence satisfy the rules. And refuse: 1010101. As a sentence doesn't satisfy the rules. so we can make our formal language but we must take in mind it's: - ∑ alphabetic. (Arguments.) - rules. From which we can get the language elements and vice versa. Example 1: L = {yellow, black, red}. // this is a language elements. The language alphabetic belongs to English. Its rules: accept any element in this set and refuse others elements: So we accept yellow. // as it satisfies the language rules. And refuse gray. // as it doesn't satisfies the language rules. Yes input No Automaton Example 2: L = {x}. // this is a language element. The language alphabetic belongs to English. Its rules: set of all x over the alphabetic ∑. So we accept x, xx, xxx, xxxx. // as it satisfies the language rules. Example 3: L = {x}. // this is a language element. The language alphabetic belongs to English. Its rules: set of all words over the alphabetic ∑ with odd length. So we accept { x, xxx, xxxxx, xxxxxxx,……}. // as it satisfies the language rules. And refuse: xx // because it is not odd length. Note from this example: We can write our language as following: L= { x , x³, x5…..}. And we can write the rule description as mathematical expression: Accept: { x^2n+1: n ≥0}. Now let us see the empty word: λ - It is a word of length zero.

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- Its notation is: λ or ε. - Some language accept empty words others not. 1- Concatenation:

الثاني منها أول عنصر والجزء) المجموعة األولي(ولي األاللغة األول منها أول عنصر في آل الكلمات التي يكون الجزء).الثانية المجموعة(الثانية اللغةفي

- L1 concatenated L2 if: {W| w= xy, x belongs to L1 & y belongs to L2}. - Notation to the concatenation: . (The dot.): So if L1 = {x, xxx}. //language (set) one. And L2 = {xx}. // language (set) two. So (L1.L2) means L1 concatenated L2 and it is equal = {xxx, xxxxx}. // according to the definition of concatenation. Examples on concatenations: Ex1: L1 = {a, b}. L2 = {c, d}. L1.L2 = {ac, ad, bC, bd} Note: ab differ from ba. Ex2: ∑= {x}. L1 = {set of all odd words over ∑ with odd length}. L1 = {set of all even words over ∑ with odd length}. L1= {x, xxx, xxxxx, xxxxxxx……}. L2= {λ, xx, xxxx, xxxxxx…}. L1.L2 = {x, xxx, xxxxx, xxxxxxx…}. Note:

.المجموعةالتكرار غير مسموح داخل Ex3: L1 = {x, xxx}. L2 = {xx}. L1.L2 = {xxx, xxxxx}. Some rules on concatenation: λ.x = x L1.L2 = {set of elements}

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2ملخص محاضرة -حسن قاسم . أ-النظرية االحتسابية-قسم علوم الحاسبات-آلية العلوم-الجامعة المستنصرية

)RE(lar expression ReguRegular languages are set of words, or strings, over an alphabet of characters valid for the language. λ is the null string (there are no symbols in this string) * is the set of all strings of length greater than or equal to 0 Example: A = {a,b} // the alphabet is composed of a and b a* = {λ, a,b,aa,ab,ba,bb,aaa,aab,…} The symbol * is called the Kleene star. ∅ (empty set) λ (empty string) ( ) delimiter , ∪ + union (selection) concatenation ∑ = {a, b, c, · · · } The language does not include every possible combination of characters, so we need a way to define which combinations are valid. Forming Expressions Given regular expressions x and y, x + y is a regular expression representing the set of all strings in either x or y (set union) x = {a, b}, y = {c, d}, x + y = {a, b, c, d} Regular expressions give us a method for finding recognized words in strings, making them crucial for lexing Regular expressions are good for describing lexemes (words) in a programming language: Identifier = (a+b+· · ·+z)(a+b+· · ·+z+A+B+· · ·+Z+0+· · ·+9)_ Digit = (0 + · · · + 9) Number = (1 + · · · + 9)Digit_ + −(1 + · · · + 9)Digit_ Keywords: if = if,while = while, · · · Mark Hills CS421 Lecture 9: Regular Expressions and Finite Automata More Examples of Concatenation Operation • Example 1 Let A={0,1}, W1 = 00110011, W2 = 00000 W1 • W2 = 0011001100000 W2 • W1 = 0000000110011 W1 • λ = W1 = 00110011 λ• W2 = W2 = 00000 x = {a, b}, y = {c, d}, xy = {ac, ad, bc, bd}

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Examples of regular expressions // NB. star-closure precedes concatenation; concatenation precedes union Describe the language (what is the output of the following RE ) a ∪ b ∪ c =={a,b,c} r = λ r = a r = r1 + r2 r = r1 r2 r = r1* r = (r1)

λ == {λ} λ* == {λ} a == {a} (a)2 == (a)(a) = {aa} (a)* == {λ, a, aa, aaa, ….} (a)+ == {a, aa, aaa, ….} (a + a) == {a} (a + b) == {a, b} (a + b)2 == (a + b)(a + b) == {aa, ab, ba, bb} (a + b + c) == {a, b, c} (a + b)* == {λ, a, b, aa, bb, ab, ba, aaa, bbb, aab, bba, ….} (abc) == {abc} (λ + a) bc == {bc, abc} ab* == {a, ab, abb, abbb, …} (ab)* == {λ, ab, abab, ababab, …} a + b* == {a, λ, b, bb, bbb, …} a (a + b)* == {a, aa, ab, aaa, abb, aba, abaa, … } (a + b)* a (a + b)* == {a, aaa, aab, baa, bab, …} (a + λ)* == (a)* = {λ, a, aa, aaa, ….} x* (a + b) + (a + b) = x* (a + b) x* y + y = x* y (x + λ)x* = x* (x + λ) = x* (x + λ)( x + λ)* (x + λ) = x* start with _ == a (a + b)* end with _ == (a + b)* b contains exactly 2 a's == (b)* a (b)* a (b)* contains at least 2 a's == (a + b)* a (a + b)* a (a + b)* contains 2 a's or 2 b's == [(b)* a (b)* a (b)*] + [(a)* b (a)* b (a)*)] even no of a == [ (b)* a (b)* a (b)* ]*

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Exercise

Ex. 1: Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that contain exactly three a's. Ex. 2: Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that end with ab. Ex. 3: Find a regular expression corresponding to the language of strings of even lengths over the alphabet of { b }.

Ex. 4: Find a regular expression corresponding to the language of all strings over the alphabet { a, b } that contain exactly two successive a's. (متتابعة) Ex. 4: Find a regular expression corresponding to the following expressions : (λ)* = (x)* + (λ) = aa* b = bba*a = (a + b)* ba=

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3ملخص محاضرة -حسن قاسم . أ-النظرية االحتسابية-قسم علوم الحاسبات-آلية العلوم-الجامعة المستنصرية

Finite Automata

The language can be defined by three forms regular expression RE, transition graph TG, finite automata FA

A finite automaton can be thought of as the device shown below consisting of a tape and a control circuit

which satisfy the following conditions:

1. The tape has the left end and extends to the right without an end.

2. The tape is divide into squares in each of which a symbol.

3. The tape has a read only head. 4. The head is always at the leftmost square at the

beginning of the operation. 5. The head moves to the right one square every time

it reads a symbol. It never moves to the left. When it sees no symbol, it stops and the automaton terminates its operation.

6. There is a control determines the state of the automaton and also controls the movement of the head.

سيطر عليها قواعد عن طريق شريط من المدخالتتشكل رسومي يمثل تنقالت بين مجموعة حاالت

A DFA represents a finite state machine that recognizes a RE.

For example, the following FA: recognize (accept) string ab

A finite automaton consists of a finite set of states, a set of transitions (moves), one start state, and a set of final states (accepting states). In addition, a DFA has a unique transition for every state-character combination. it is a set of states, and its “control” moves from state to state in response to external “inputs” . A finite automaton, FA, provides the simplest model of a computing device. It has a central processor of finite capacity and it is based on the concept of state. finite state machine is a 5 tuple M = (Q, A, T, S ,F), where

o Q --set of states = {q0, q1, q2, ….} o A -- set of input symbols ={a,b, …, 0, 1, …} o T --set of transitions or rules T

o S -- an initial state o F -- the final state -- could be more than one final state

Designing (drawing) FA State with numbers or any name

Start - or small arrow

Final + or double circle

Transition (only one input or symbol on the edge) a,b allowed means (a or b)

loop

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Example1: Q = { 0, 1, 2 }, ∑= { a }, A = { 1 }, the initial state is 0 and T is shown in the following table.

trace string aaaa on this machine (if it is accepted or not)

State (q) Input (a) 0 1 1 2 2 2

the string is accepted if and only if we reached final state and the string is finished then Example2: Q = { 0, 1, 2 }, A= { a, b }, F = { 1 }, the initial state is 0 and T is shown in the following table.

State (q) Input (a) Input (b) 0 1 2 1 2 2 2 2 2

Example 3: Q = { 0, 1 }, A= { a, b }, f = { 0 }, the initial state is 0 and T is shown in the following table.

State (q) Input (a) Input (b) 0 0 1 1 1 1

Transition diagram:

TG has many inputs on the edge ab FA has only one input on the edge a

Deterministic Finite Automata DFA and Non Deterministic Finite Automata NFA

DFA: different input from state to different states NFA: one input from state to different states

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Language accepted by FA string is accepted by a DFA if and only if the DFA starting at the initial state and ends in an accepting state after reading the string. lets start with simplest FA.

Q)) Find if the string aababa accepted or not on the above FA

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Examples of languages accepted by FA FA RE

λ

a

aa

a+ = aa*

a*

a+b

(a+b)*

a*b

b(a+b)*

(a+b)*b

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a(a+b)*b

(a+b)* b(a+b)*

ab(a+b)*

a*babb*

(aa)*ba

contains 3 a's b*ab*ab*ab*

contains even number of a = (b*ab*ab*)+

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Converting a Regular Expression into a Finite Automaton

RE FA λ

a

aa

a+ = aa*

NFA

a*

a+b

(a+b)*

a*b

b(a+b)*

NFA

(a+b)*b

NFA

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a(a+b)*b

NFA (a+b)* b(a+b)*

NFA ab(a+b)*

NFA a*babb*

NFA (aa)*ba

contains 3 a's b*ab*ab*ab*

contains even number of a = (b*ab*ab*)+

dividable by 3

all bit strings that begin with 0 and end with 1

all bit strings whose number of 0's is a multiple of 5

all bit strings with more 1's than 0's

all bit strings with no consecutive 1's

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Converting NFA into DFA

:توجد ثالثة خطوات للتحويل

ول التنقالت عمل جدأوال

رسم الشكل الجديدثانيا

إليها التي ال يمكن الوصول الحاالت الزائدةإزالة ثالثا

1)) transition table

state a b q0 q1q2 q0 q1 q2 q3 q2 q2q3 q2 q3 q3 - q1q2 q2q3 q2q3 q2q3 q2q3 q3

Any state contains final mark it as a final state

2)) draw the new FA from transition table

3)) remove unreachable states (marked by dashed circle – state q1 and state q3 ) because we can not reach it.

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HW convert the following NFA into DFA