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Theory of Computation - Lectures 4 and 5


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TOC lectures 4 and 5:
Formal definition of computation, regular languages, regular expressions, Designing Finite Automata, Regular operations

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Theory of Computation - Lectures 4 and 5

  1. 1. Formal Definition of Computation Let M = (Q, ∑ , δ, q0, F) be a finite automaton and let w = w1w2.....wn be a string where each wi is a member of the alphabet ∑. Then M accepts w if a sequence of states r0, r1, ….,rn In Q exists with three conditions: 1- r0= q0, 2- δ(ri, wi+1)=ri+1, for i=0,...,n-1, and 3- rn∈ F
  2. 2. Cont.    Condition 1 says that the machine starts in the start state. Condition 2 says that the machine goes from state to another according to the transition function Condition 3 says that the machine accepts its input if it ends up in an accept state. We say that M recognizes language A if A = {w | M accepts w}
  3. 3. Regular Language  Definition: A language is called a regular language if some finite automaton recognizes it. See example 1.17 p. 41
  4. 4. Designing Finite Automata    Just like artwork, design is a creative process... Suppose you are given some language, and you want to design a FA that recognizes it...pretend that you are the machine! Take symbols, one by one, and and after each symbol, you should decide whether the string so far is in the language or not.
  5. 5. Designing Finite Automata –Cont.   What if the string is from earth to moon long? And your memory (as a FA) is like a sheet of paper? Worry-less, you will have to memorize the crucial info only, like: “What's my status now?” Example: suppose: alphabet = {0,1}, and that the language consists of all strings with an odd number of 1s. You want to construct a FA E1 to recognize this language
  6. 6. Designing Finite Automata –Cont. By pretending that you're the FA, get the symbols (1 by 1), do you need to remember all the string? No! Just remembering the number of 1s so far is even or not is enough, so that if next is 0, don't change, if 1, change!  Therefore, you have 2 states, say: q odd, qeven.
  7. 7. Designing Finite Automata –Cont.   Next, assign transitions. Assign a start state (qeven) in our example, since no symbol has been entered so far (0) and 0 is even.  What about Final state (accept)?  See example 1.21 - P.43
  8. 8. Regular Operations  We use them to study properties of regular languages, those properties will help us develop a toolbox that will include ways of proving certain other languages are nonregular (i.e. beyond the capability of FA)
  9. 9. Regular Operations  If A and B are languages, then:  Union: A∪B = {x|x ∈ A or x ∈ B}    Concatenation: A ○ B = {xy | x ∈ A and y ∈ B} Star: A* = {x1x2....xk | k>=0 and each xi∈A} The star operation is unary operation (on 1 language) while the union and conc. are binary operations
  10. 10. Regular Operations – cont. Let ∑ = {a,b,....,z}, if A = {good, bad}, and B = {boy, girl} the: A ∪ B = {good, bad, boy, girl} A ○ B = {goodboy, goodgirl, badboy, badgirl} A*= {ε, good, bad, goodgood, badbad, badgood, goodgoodgood, goodgoodbad, etc.....}
  11. 11. Regular Operations Cont.  Closed under multiplication: example, if x and y are two numbers from N, then x*y results a number also from N.   But N is not closed under division, because x/y may result a number which isn't member of N, ex. X=1, y=2, x/y=1/2 which isn't member of N Please read theorems 1.25, 1.26 p.45, 46
  12. 12. Nondeterminism  DFA vs. NFA: DFA has exactly one exiting transition arrow for each symbol in the alphabet, NFA violets that.   DFA uses symbols from alphabet only for arrow labels, NFA may use others, such as ε. What will happen if a state has two arrows for same input? Or if input is ε?
  13. 13. NFA 0,1 q1 0,1 1 q2 0,ε q3 1 q q4 4
  14. 14.  Examples: 1.30, 1.33, 1.35 P: 51-53
  15. 15. Formal definition of NFA    Similar to DFA, the NFA has the 5-tuple, yet the difference is in δ (The Transition Function) In DFA, δ takes a state and an input symbol and produces the next state. In NFA, δ takes a state and an an input symbol OR the empty string and produces the set of possible next states.
  16. 16. Cont. Therefore, point 3 of the tuple would be as: δ : Q X ∑ε → P(Q), where ∑ε is the result of ∑ ∪ ε
  17. 17. Example The formal definition of N1 (slide 13): 1- Q = {q1,q2,q3,q4} 2- ∑ = {0,1} 3- δ is given as: 4- q1 =start state 5- F ={q4} 0 1 ε q1 {q1} {q1,q2} Ф q2 {q3} Ф {q3} q3 Ф {q4} Ф q4 {q4} {q4} Ф
  18. 18. Reading  Every NFA has an equivalent DFA... To prove, read P.54- P.58  Based on your previous reading (P.45), read Closure under the regular operations (P.58-63). Which is similar to, yet easier than example P.45.
  19. 19. Regular Expressions   In math, we use + and x to build up expressions like: (5+3)x4 (Gives 32) Similarly, we use∪ and * to build up expressions describing languages, called Regular Expressions (RegEx): (0 ∪ 1)0* … (Gives a language) What does (0 ∪ 1)0* mean?  It's a language consisting of all strings starting with a 0 or 1 followed by any number of 0s.
  20. 20. RegEx – Cont. (0 ∪ 1)0* is: - (0 ∪ 1) = ({0}∪{1}) = {0,1} - 0* means {0}*, means a language consisting of of any number of 0s. The concatenation between (0 ∪ 1) and 0* is a shorthand for for (0 ∪ 1) ○ 0*   Applications of RegEx: many, Grep (text-search)in Unix and Linux, Perl, C++, text editors etc.... In C++, you can use regex r("(+|-)?[[:digit:]]+");, but why?
  21. 21. Regex- Example (0 ∪ 1)* = the language consisting of all possible strings of 0s and 1s. If ∑ ={0,1}, then ∑ is a shorthand for (0 ∪ 1).   Generally, if ∑ is any alphabet, the regex ∑ describes the language consisting of all strings of length 1 over the alphabet, and ∑* describes the language consisting of all strings over the alphabet. ∑*1 is the language that contains all strings that end with 1
  22. 22. Regex- Example-cont.  The language (0∑*) ∪ (∑*1) consists of all strings that either start with a 0 or end with a 1. * similar to mathematical precedence ( x before + for example), in regex, * is done first ,then concatenation, finally the union. Unless parentheses are used.
  23. 23. Formal Definition of regex  We say that R is a regex if R: 1. a for some a in the alphabet 2. ε, 3. ᶲ 4. (R1 ∪ R2) 5. (R1 ○ R2) 6. (R1*), where R1 is a regex