![Structural expressions
15
Grammars
Regular expressions
E.g., unix style to capture city names such as “Palo Alto CA”:
[A-Z][a-z]*([ ][A-Z][a-z]*)*[ ][A-Z][A-Z]
Start with a letter
A string of other
letters (possibly
empty)
Other space delimited words
(part of city name)
Should end w/ 2-letter state code](https://image.slidesharecdn.com/3-160118100913/85/Intro-automata-theory-15-320.jpg)
Intro automata theory
- 1. 1 Introduction to Automata Theory
- 2. What is Automata Theory? 2 Study of abstract computing devices, or “machines” Automaton = an abstract computing device Note: A “device” need not even be a physical hardware! A fundamental question in computer science: Find out what different models of machines can do and cannot do The theory of computation Computability vs. Complexity
- 3. Alan Turing (1912-1954) 3 Father of Modern Computer Science English mathematician Studied abstract machines called Turing machines even before computers existed Heard of the Turing test? (A pioneer of automata theory)
- 4. Theory of Computation: A Historical Perspective 4 1930s • Alan Turing studies Turing machines • Decidability • Halting problem 1940-1950s • “Finite automata” machines studied • Noam Chomsky proposes the “Chomsky Hierarchy” for formal languages 1969 Cook introduces “intractable” problems or “NP-Hard” problems 1970- Modern computer science: compilers, computational & complexity theory evolve
- 5. Languages & Grammars Languages: “A language is a collection of sentences of finite length all constructed from a finite alphabet of symbols” Grammars: “A grammar can be regarded as a device that enumerates the sentences of a language” - nothing more, nothing less N. Chomsky, Information and Control, Vol 2, 1959 5 Or “words” Image source: Nowak et al. Nature, vol 417, 2002
- 6. The Chomsky Hierachy 6 Regular (DFA) Context- free (PDA) Context- sensitive (LBA) Recursively- enumerable (TM) • A containment hierarchy of classes of formal languages
- 7. 7 The Central Concepts of Automata Theory
- 8. Alphabet 8 An alphabet is a finite, non-empty set of symbols We use the symbol ∑ (sigma) to denote an alphabet Examples: Binary: ∑ = {0,1} All lower case letters: ∑ = {a,b,c,..z} Alphanumeric: ∑ = {a-z, A-Z, 0-9} DNA molecule letters: ∑ = {a,c,g,t} …
- 9. Strings 9 A string or word is a finite sequence of symbols chosen from ∑ Empty string is ε (or “epsilon”) Length of a string w, denoted by “|w|”, is equal to the number of (non- ε) characters in the string E.g., x = 010100 |x| = 6 x = 01 ε 0 ε 1 ε 00 ε |x| = ? xy = concatentation of two strings x and y
- 10. Powers of an alphabet 10 Let ∑ be an alphabet. ∑k = the set of all strings of length k ∑* = ∑0 U ∑1 U ∑2 U … ∑+ = ∑1 U ∑2 U ∑3 U …
- 11. Languages 11 L 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 ∑ Examples: 1. Let L be the language of all strings consisting of n 0’s followed by n 1’s: L = {ε,01,0011,000111,…} 2. Let L be the language of all strings of with equal number of 0’s and 1’s: L = {ε,01,10,0011,1100,0101,1010,1001,…} Definition: Ø denotes the Empty language Let L = {ε}; Is L=Ø? NO Canonical ordering of strings in the language
- 12. The Membership Problem 12 Given a string w ∈∑*and a language L over ∑, decide whether or not w ∈L. Example: Let w = 100011 Q) Is w ∈ the language of strings with equal number of 0s and 1s?
- 13. Finite Automata 13 Some Applications Software for designing and checking the behavior of digital circuits Lexical analyzer of a typical compiler Software for scanning large bodies of text (e.g., web pages) for pattern finding Software for verifying systems of all types that have a finite number of states (e.g., stock market transaction, communication/network protocol)
- 14. Finite Automata : Examples 14 On/Off switch Modeling recognition of the word “then” Start state Final stateTransition Intermediate state action state
- 15. Structural expressions 15 Grammars Regular expressions E.g., unix style to capture city names such as “Palo Alto CA”: [A-Z][a-z]*([ ][A-Z][a-z]*)*[ ][A-Z][A-Z] Start with a letter A string of other letters (possibly empty) Other space delimited words (part of city name) Should end w/ 2-letter state code
- 16. 16 Formal Proofs
- 17. Deductive Proofs 17 From the given statement(s) to a conclusion statement (what we want to prove) Logical progression by direct implications Example for parsing a statement: “If y≥4, then 2y ≥y2 .” (there are other ways of writing this). given conclusion
- 18. Example: Deductive proof 18 Let Claim 1: If y≥4, then 2y ≥y2 . Let x be any number which is obtained by adding the squares of 4 positive integers. Claim 2: Given x and assuming that Claim 1 is true, prove that 2x ≥x2 Proof: Given: x = a2 + b2 + c2 + d2 Given: a≥1, b≥1, c≥1, d≥1 a2 ≥1, b2 ≥1, c2 ≥1, d2 ≥1 (by 2) x ≥ 4 (by 1 & 3) 2x ≥ x2 (by 4 and Claim 1) “implies” or “follows”
- 19. On Theorems, Lemmas and Corollaries 19 We typically refer to: A major result as a “theorem” An intermediate result that we show to prove a larger result as a “lemma” A result that follows from an already proven result as a “corollary” An example: Theorem: The height of an n-node binary tree is at least floor(lg n) Lemma: Level i of a perfect binary tree has 2i nodes. Corollary: A perfect binary tree of height h has 2h+1 -1 nodes.
- 20. Quantifiers 20 “For all” or “For every” Universal proofs Notation* =? “There exists” Used in existential proofs Notation* =? Implication is denoted by => E.g., “IF A THEN B” can also be written as “A=>B” * I wasn’t able to locate the symbol for these notation in powerpoint. Sorry! Please follow the standard notation for these quantifiers. These will be presented in class.
- 21. Proving techniques 21 By contradiction Start with the statement contradictory to the given statement E.g., To prove (A => B), we start with: (A and ~B) … and then show that could never happen What if you want to prove that “(A and B => C or D)”? By induction (3 steps) Basis, inductive hypothesis, inductive step By contrapositive statement If A then B ≡ If ~B then ~A
- 22. Proving techniques… 22 By counter-example Show an example that disproves the claim Note: There is no such thing called a “proof by example”! So when asked to prove a claim, an example that satisfied that claim is not a proof
- 23. Different ways of saying the same thing 23 “If H then C”: i. H implies C ii. H => C iii. C if H iv. H only if C v. Whenever H holds, C follows
- 24. “If-and-Only-If” statements 24 “A if and only if B” (A <==> B) (if part) if B then A ( <= ) (only if part) A only if B ( => ) (same as “if A then B”) “If and only if” is abbreviated as “iff” i.e., “A iff B” Example: Theorem: Let x be a real number. Then floor of x = ceiling of x if and only if x is an integer. Proofs for iff have two parts One for the “if part” & another for the “only if part”
- 25. Summary 25 Automata theory & a historical perspective Chomsky hierarchy Finite automata Alphabets, strings/words/sentences, languages Membership problem Proofs: Deductive, induction, contrapositive, contradiction, counterexample If and only if Read chapter 1 for more examples and exercises