# THEORY of COMPUTATION

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1 THEORY of COMPUTATION ATHASIT SURARERKS ELITE Athasit Surarerks ELITE Engineering Laboratory in Theoretical Enumerable System Computer Engineering, Faculty of Engineering Chulalongkorn University 254 Phayathai road, Patumwan, Bangkok Tel : , Fax : Webpage :

2 DESCRIPTION Computable functions decidable predicates and solvable problems; computational complexity; NP-complete problems; automata theory; formal language; lambda calculus. EVALUATION Mid-Term examination 50 % Final examination 50 %

3 TEXTBOOK Essentials of Theoretical Computer Science F. D. Lewis REFERENCES Mc Graw Hill Addison Wesley W IE PRINTICE HALL Mc Graw Hill Introduction to Languages and Theory of Computation(3 rd ed.) John C. Martin Introduction to Automata Theory, Languages, and Computation, J.E. Hopcroft, R. Motwani, J.D. Ullman Introduction to Computer Theory (2 nd ed.) Daniel I. A. Cohen Languages and Machines: An Introduction to the Theory of Computer Science (2 nd ed.) Thomas A. Sudkamp Topology (2 nd ed.) James R. Munkres Discrete Mathematics and Its Applications (4 th ed.)

4 BACKGROUND SYLLOGISTIC REASONING Aristotle ( B.C.) Euclid of Alexandria ( B.C.) Chrysippus of Soli ( B.C.) DEDUCTIVE REASONING MODAL LOGIC George Boole ( A.D.) PROPOSITIONAL LOGIC Augustus De Morgan ( A.D.) DE MORGAN s LAWs Charles Babbage Created the first difference engine (producing the members of the sequence n 2 + n + 41 at the rate of about 60 every 5 minutes) The 1 st drawings of the analytical engine (describes five logical components, the store, the mill, the control, the input and the output) The construction of modern computers, logically similar to Babbage's design

5 Kurt Gödel Proved that there was no algorithm to provide proofs for all the true statements in mathematics. Universal model for all algorithms. VARIOUS VERSIONS OF A UNIVERSAL ALGORITHM MACHINE Andrei Andreevich Markov Emil Post Alonzo Church Stephen Kleene John von Neumann Alain Turing

6 Alain Turing studied problems which today lie at the heart of artificial intelligence. proposed the Turing Test which is still today the test people apply in attempting to answer whether a computer can be intelligent. Computing machinery and intelligence Warren McCulloch & Walter Pitts neurophysiologists Constructed for a neural net was a theoretical machine of the same nature as the one Turing invented. Modern linguists Investigated a very similar subject What is language in general? How could primitive humans have developed language? How do people understand it? How do they learn it as children? What ideas can be expressed, and in what way? How do people construct sentences from the idea s in their minds?

7 Noam Chomsky Massachusetts Institute of Technology Created the subject of mathematical models for the description of languages to answer these questions. MAIN TOPIC We shall study different types of theoretical machines that are mathematical models for actual physical processes.

8 MACHINE MODEL input output output machine input machine input MACHINE machine output input output machine? MAIN CONCLUSIONS this can be done or it can never be done.

9 LANGUAGES LANGUAGES Different entities (in English) letters words sentences paragraphs coherent stories Not all collections of letters form a valid sentence. Humans agree on which sequences are valid or which are not. COLLECTION & SEQUENTIAL How do they do that?

10 COMPUTER LANGUAGES Different entities letters words commands programs It is very hard to state all the rules for the language spoken English. systems Commands can be recognized by certain sequences of words. Language structure is based on explicitly rules. LANGUAGE Language means simply a set of strings involving symbols from alphabet.

11 THEORY OF FORMAL LANGUAGES Formal refers explicitly rules What sequences of symbols can occur? No liberties are tolerated. No reference to any deeper understanding is required. the form of the sequences of symbols not the meaning THEORY OF FORMAL LANGUAGES STRUCTURE One finite set of fundamental units, called alphabet, denoted. An element of alphabet is called character. A certain specified set of strings of characters will be called language denoted L. Those strings that are permissible in the language we call words. The string without letter is called empty string or null string, denoted by Λ. The language that has no word is denoted by. specified

12 SYMBOLS Union operation + Different operation Alphabet Empty string Λ ε Language L Γ Empty language LANGUAGES LANGUAGE DEFINING IMPLICITLY DEFINING Given an alphabet = { a b c z - }. We can now specify a language L as { all words in a standard dictionary }, named ENGLISH-WORDS. We define a language Γ as { all words in a standard dictionary, blank space, the usually punctuation marks }, named ENGLISH-SENTENCES.

13 LANGUAGES INFINITE LANGUAGE DEFINING The trick of defining the language Γ, By listing all rules of grammar. This allows us to give a finite description of an infinite language. Consider this sentence I eat three Sundays. RIDICULOUS This is grammatically correct. LANGUAGE LANGUAGES LANGUAGE DEFINING METHOD OF EXHAUSTION Let = {x} be an alphabet. Language L can be defined by L = { x xx xxx xxxx } L = { x n for n = }. Language L 2 = { x xxx xxxxx xxxxxxx } L 2 = { x odd } L 2 = { x 2n-1 for n = }.

14 LANGUAGES SOME DEFINITIONS We define the function length of a string to be the number of letters in the string. For example, if a word a = xxxx in L, then length(a)=4. In any language that includes Λ, we have length(λ)=0. The function reverse is defined by if a is a word in L, then reverse(a) is the same string of letters spelled backward, called the reverse of a. For example, reverse(123)=321. Remark: The reverse(a) is not necessary in the language of a. LANGUAGES SOME DEFINITIONS We define the function n a (w) of a w to be the number of letter a in the string w. For example, if a word w = aabbac in L, then n a (w)=3. Concatenation of two strings means that two strings are written down side by side. For example, x n concatenated with x m is x n+m

15 LANGUAGES SOME DEFINITIONS Language is called PALINDROME over the alphabet if Language = { Λ and all strings x such that reserve(x)=x }. For example, let ={ a, b }, and PALINDROME={ Λ a b aa bb aaa aba bab bbb }. Remark: Sometimes, we obtain another word in PALINDROME when we concatenate two words in PALINDROME. We shall see the interesting properties of this language later. LANGUAGES SOME DEFINITIONS Consider the language PALINDROME={ Λ a b aa bb aaa aba bab bbb }. We usually put words in size order and then listed all the words of the same length alphabetically. This order is called lexicographic order.

16 LANGUAGES KLEENE CLOSURE Given an alphabet, the language that any string of characters in are in this language is called the closure of the alphabet. It is denoted by *. This notation is sometimes known as the Kleene star. Kleene star can be considered as an operation that makes an infinite language. When we say infinite language, we mean infinitely many words, each of finite length. LANGUAGES KLEENE CLOSURE More general, if S is a set of words, then by S* we mean the set of all finite strings formed by concatenating words from S and from S*. Example: If S = { a ab }then S* = { Λ and any word composed of factors of a and ab }. { Λ and all strings of a and b except strings with double b }. { Λ a aa ab aaa aab aba aaaa aaab aaba }.

17 LANGUAGES KLEENE CLOSURE Example: If S = { a ab }then S* = { Λ and any word composed of factors of a and ab }. { Λ and all strings of a and b except strings with double b }. { Λ a aa ab aaa aab aba aaaa aaab aaba }. To prove that a certain word is in the closure language S*, we must show how it can be written as a concatenation of words from the base set S. Example: abaaba can be factored as (ab)(a)(ab)(a) and it is unique. LANGUAGES KLEENE CLOSURE Example: If S = { xx xxxxx }then S* ={ Λ xx xxxx xxxxx xxxxxx xxxxxxx xxxxxxxx }. { Λ and xx and x n for n = }. How can we prove this statement? Hence: proof by constructive algorithm (showing how to create it).

18 LANGUAGES KLEENE CLOSURE Example: If S = { a b ab } and T = { a b ba }, then S* = T* = { a b }*. Proof: It is clear that { a b }* S* and { a b }* T*. We have to show that S* and T* { a b }*. For x S*, in the case that x is composed of ab. Replace ab in x by a, b which are in { a b }*. Then S* { a b }*. The proof of T* { a b }* is similarity. QED LANGUAGES POSITIVE CLOSURE Given an alphabet, the language that any string (not zero) of characters in are in this language is called the positive closure of the alphabet. It is denoted by +. Example: Let Γ={ ab }. Then Γ + = { ab abab ababab }.

19 LANGUAGES Given an alphabet ={ aa bbb }. Then Σ* is the set of all strings where a s occur in even clumps and b s in groups of 3, 6, 9. Some words in Σ* are bbb aabbbaaaa bbbaa If we concatenate these three elements of Σ*, we get one big word in Σ**, which is again in Σ*. bbbaabbbaaaabbbaa = (bbb)(aa)(bbb)(aa)(aa)(bbb)(aa) Note : Σ** means (Σ*)*. TRIVIAL REMARK LANGUAGES THEOREM Theorem For any set S of strings, we have S*= S**. Proof: Every words in S** is made up of factors from S*. Every words in S* is made up of factors from S. Therefore every words in S** is made up of factors from S. We can write this S** S*. In general, it is true that S S*. So S* S**. Then S*= S**. QED

20 RECURSIVELY DEFINING LANGUAGES RECURSIVE DEFINITIONS LANGUAGE DEFINING EVEN language EVEN is the set of all positive whole numbers divisible by 2. EVEN is the set of all 2n where n = Another way we might try this: The set is defined by these three rules: Rule1: 2 is in EVEN. Rule2: if x is in EVEN, then so is x+2. Rule3: The only elements in the set EVEN are those that can be produced from the two rules above. The last rule above is completely redundant.

21 RECURSIVE DEFINITIONS LANGUAGE DEFINING EVEN language The set is defined by these three rules: Rule1: 2 is in EVEN. Rule2: if x is in EVEN, then so is x+2. Rule3: The only elements in the set EVEN are those that can be produced from the two rules above. PROBLEM: Show that 10 is in this language. By Rule1, 2 is in EVEN. By Rule2, 2+2=4 is in EVEN. By Rule2, 4+2=6 is in EVEN. By Rule2, 6+2=8 is in EVEN. By Rule2, 8+2=10 is in EVEN. PRETTY HORRIBLE! RECURSIVE DEFINITIONS LANGUAGE DEFINING EVEN language The set is defined by these three rules: Rule1: 2 is in EVEN. Rule2: if x,y are in EVEN, then so is x+y. Rule3: The only elements in the set EVEN are those that can be produced from the two rules above. PROBLEM: Show that 10 is in this language. By Rule1, 2 is in EVEN. By Rule2, 2+2=4 is in EVEN. By Rule2, 4+4=8 is in EVEN. By Rule2, 8+2=10 is in EVEN. DECIDEDLY HARD

22 RECURSIVE DEFINITIONS LANGUAGE DEFINING POSITIVE language The set is defined by these three rules: Rule1: 1 is in POSITIVE. Rule2: if x,y are in POSITIVE, then so is x+y, x-y, x yand x/y where y is not zero. Rule3: The only elements in the set POSITIVE are those that can be produced from the two rules above. PROBLEM: What is POSITIVE language? RECURSIVE DEFINITIONS LANGUAGE DEFINING POLYNOMIAL language The set is defined by these four rules: Rule1: Any number is in POLYNOMIAL Rule2: Any variable x is in POLYNOMIAL. Rule3: if x,y are in POLYNOMIAL, then so is x+y, x-y, x y and (x). Rule4: The only elements in the set POLYNOMIAL are those that can be produced from the three rules above. PROBLEM: Show that 3x 2 +2x-5 is in POLYNOMIAL. Proof: Rule1: 2, 3, 5 are in POLYNOMIAL, Rule2: x is in POLYNOMIAL, Rule3: 3x, 2x are in POLYNOMIAL, Rule3: 3xx is in POLYNOMIAL, Rule3: 3xxx+2x, 3x 2 +2x-5 are in POLYNOMIAL. QED.

23 RECURSIVE DEFINITIONS Language: Problems: ARITHMETIC EXPRESSIONS Let Σ be an alphabet for AE language. Σ = { * / ( ) }. Define rules for this language. Show that the language does not contain substring //. Show that ((3+4)-(2*6))/5 is in this language. L 1 ={ x n for n = } REMARK DEFINING LANGUAGES L 2 ={ x n for n = } L 3 ={ x n for n = } Languages can be defined by L 4 ={ x n for n = }. More precision and less guesswork are required.

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