Introduction to Automata Theory

1 1 Introduction to Automata TheoryReading: Chapter 12 What is Automata Theory ?nStudy of abstract computing devices, or machines nAutomaton = an abstract computing devicenNote:A device need not even be a physical hardware!nA fundamental question in computer science: nFind out what different models of machines can do and cannot donThe Theory of computationnComputability vs. Complexity3 Alan Turing (1912-1954)nFather of Modern Computer SciencenEnglish mathematiciannStudied abstract machines called Turing machineseven before computers existednHeard of the Turing test?(A pioneer of Automata Theory )4 Theory of Computation: A Historical Perspective1930s Alan Turing studies Turing machines Decidability Halting problem1940-1950s Finite Automata machines studied Noam Chomsky proposes the Chomsky Hierarchy for formal languages1969 Cook introduces intractable problemsor NP-Hard problems1970-Modern computer science: compilers, computational & complexity theoryevolve5 Languages & GrammarsOr words Image source: Nowak et al.

2 Nature, vol 417, 2002 nLanguages: A language is a collection of sentences of finite length all constructed from a finite alphabet of symbols nGrammars: A grammar can be regarded as a device that enumerates the sentences of a language -nothing more, nothing lessnN. Chomsky, Information and Control, Vol 2, 19596 The Chomsky HierachyRegular(DFA)Context-free(PDA)Con text-sensitive (LBA)Recursively-enumerable (TM) A containment hierarchy of classes of formal languages7 The Central Concepts of Automata Theory8 AlphabetAn alphabet is a finite, non-empty set of symbolsnWe use the symbol (sigma) to denote an alphabetnExamples:nBinary: = {0,1} nAll lower case letters: = {a,b,c,..z}nAlphanumeric: = {a-z, A-Z, 0-9}nDNA molecule letters: = {a,c,g,t} string or word is a finite sequence of symbols chosen from nEmpty string is e(or epsilon )nLength of a string w,denoted by |w| , is equal to the number of (non-e) characters in the , x = 010100 |x| = 6nx = 01e0e1e00e|x| = ?

3 Nxy= concatentationof two strings x and y 10 Powers of an alphabet Let be an k= the set of all strings of length kn * = 0U 1U 2U ..n += 1U 2U 3U ..11 LanguagesL is a said to be a language over alphabet , only if L * this is because * is the set of all strings (of all possible length including 0) over the given alphabet L be thelanguage of all strings consisting of n 0 s followed by n1 s: L = {e, 01, 0011, 000111,..} L be the language of all strings of with equal number of 0 s and 1 s: L = {e, 01, 10, 0011, 1100, 0101, 1010, 1001,..}Definition: denotes the Empty languagenLet L = {e}; Is L= ?NOCanonical ordering of strings in the language12 The Membership ProblemGiven a string w *and a language L over , decide whether or not w :Let w = 100011Q) Is w the language of strings with equal number of 0s and 1s?

4 13 Finite AutomatanSome ApplicationsnSoftware for designing and checking the behavior of digital circuitsnLexical analyzer of a typical compilernSoftware for scanning large bodies of text ( , web pages) for pattern findingnSoftware for verifying systems of all types that have a finite number of states ( , stock market transaction, communication/network protocol)14 Finite Automata : ExamplesnOn/Off switchnModeling recognition of the word then Start stateFinal stateTransitionIntermediate stateactionstate15 Structural expressionsnGrammarsnRegular , unix style to capture city names such as Palo Alto CA :n[A-Z][a-z]*([ ][A-Z][a-z]*)*[ ][A-Z][A-Z]Start with a letterA string of other letters (possiblyempty)Other space delimited words(part of city name)Should end w/ 2-letter state code16 Formal Proofs17 Deductive ProofsFrom the given statement(s) to a conclusion statement (what we want to prove)nLogical progression by direct implicationsExample for parsing a statement:n If y 4, then 2y y2.

5 (there are other ways of writing this).givenconclusion18 Example: Deductive proof Let Claim 1:If y 4, then 2y y2. Let x be any number which is obtained by adding the squares of 4 positive 2:Given x and assuming that Claim 1 is true, prove that 2x x2nProof:1)Given: x = a2+ b2+ c2+ d22)Given: a 1, b 1, c 1, d 13) a2 1, b2 1, c2 1, d2 1(by 2)4) x 4(by 1 & 3)5) 2x x2(by 4 and Claim 1) implies or follows On Theorems, Lemmas and CorollariesWe typically refer to: nA major result as a theorem nAn intermediate result that we show to prove a larger result as a lemma nA result that follows from an already proven result as a corollary 19An example:Theorem:The height of an n-node binary tree is at least floor(lgn)Lemma:Level iof a perfect binary tree has.

6 A perfect binary tree of height h has 2h+1-1 For all or For every nUniversal proofsnNotation= There exists nUsed in existential proofsnNotation=Implication is denoted by => , IF A THEN B can also be written as A=>B 21 Proving techniquesnBy contradictionnStart with the statement contradictory to the given , To prove (A => B), we start with:n(A and ~B) and then show that could never happenWhat if you want to prove that (A and B => C or D) ?nBy inductionn(3 steps) Basis, inductive hypothesis, inductive stepnBy contrapositive statementnIf Athen B If ~Bthen ~A22 Proving counter-examplenShow an example that disproves the claimnNote: There is no such thing called a proof by example !

7 NSo when asked to prove a claim, an example that satisfied that claim is nota proof 23 Different ways of saying the same thing n IfH then C => if only H holds, C follows24 If-and-Only-If statementsn A if and only if B (A <==> B)n(if part)if B then A ( <= )n(only if part)A only if B ( => )(same as if A then B )n If and only if is abbreviated as iff , A iff B nExample:nTheorem:Let x be a real number. Then floor of x = ceiling of x if and only ifx is an for iff have two parts nOne for the if part & another for the only if part 25 SummarynAutomata Theory & a historical perspectivenChomsky hierarchy nFinite automatanAlphabets, strings/words/sentences, languagesnMembership problemnProofs:nDeductive, induction, contrapositive, contradiction, counterexamplenIf and only ifnRead chapter 1 for more examples and exercises