Intro to Theory of Computation

Transcription

1 Intro to Theory of Computation 1/19/2016 LECTURE 3 Last time: DFAs and NFAs Operations on languages Today: Nondeterminism Equivalence of NFAs and DFAs Closure properties of regular languages Sofya Raskhodnikova Sofya Raskhodnikova; based on slides by Nick Hopper L3.1

2 Nondeterminism Nondeterministic Finite Automaton (NFA) accepts a string w if there is a way to make it reach an accept state on input w. L3.2

3 I-clicker problem (frequency: AC) What is the language of this NFA? (0 k means 00 0 ) k ε 0 0 A. {0 k k is a multiple of 2}. ε 0 0 B. {0 k k is a multiple of 3}. C. {0 k k is a multiple of 6}. 0 D. {0 k k is a multiple of 2 or 3}. E. None of the above.

4 Formal Definition An NFA is a 5-tuple M = (Q, Σ,, q 0, F) Q is the set of states Σ is the alphabet : Q Σ ε P(Q) is the transition function q 0 Q is the start state F Q is the set of accept states P(Q) is the set of subsets of Q and Σ ε = Σ {ε} M accepts a string w if there is a path from q 0 to an accept state that w follows. 1/19/2016 Sofya Raskhodnikova; based on slides by Nick Hopper L3.4

5 Example 0,1 0,1 q 0 1 0,ε 1 q 1 q 2 q 3 N = (Q, Σ,, q 0, F) Q = {q 0, q 1, q 2, q 3 } Σ = {0,1} F = {q 3 } (q 0,0) = {q 0 } (q 0,1) = {q 0, q 1 } (q 1,ε) = {q 1,q 2 } (q 2,0) = L3.6

6 Nondeterminism Deterministic Computation accept or reject Nondeterministic Computation accept reject Ways to think about nondeterminism parallel computation tree of possible computations guessing and verifying the right choice L3.7

7 NFAs ARE SIMPLER THAN DFAs A DFA that recognizes the language {1}: 0 0,1 1 0,1 An NFA that recognizes the language {1}: 1 L3.8

8 A DFA recognizing {1} Theorem. Every DFA for language {1} must have at least 3 states. Proof: 1/19/2016 L3.9

9 Equivalence of NFAs & DFAs Theorem. Every NFA has an equivalent DFA. Corollary: A language is regular iff it is recognized by an NFA. L3.10

10 NFA to DFA Conversion Input: N = (Q, Σ,, q 0, F) Output: M = (Q, Σ,, q 0, F ) Intuition: Do the computation in parallel, maintaining the set of states where all threads are. reject Idea: accept Q = P(Q) L4.11

11 NFA to DFA Conversion Input: N = (Q, Σ,, q 0, F) Output: M = (Q, Σ,, q 0, F ) Q = P(Q) : Q Σ Q (R, ) = E( (r, ) ) r R q 0 = E({q 0 }) For R Q, let E(R) be the set of states reachable by ε-transitions from the states in R. for all R Q and Σ. F = { R Q R contains some accept state of N} L4.12

12 Example: NFA to DFA 1) a 1 b L3.13

13 Examples NFA to DFA 2) 1 0,1 ε L3.14

14 Regular Operations on languages Complement: A = { w w A } Union: A B = { w w A or w B } Intersection: A B = { w w A and w B } Reverse: A R = { w 1 w k w k w 1 A } Concatenation: A B = { vw v A and w B } Star: A* = { w 1 w k k 0 and each w i A } L3.15

15 Closure properties of the class of regular languages THEOREM. The class of regular languages is closed under all 6 operations. If A and B are regular, applying any of these operation yields a regular language. L3.16

16 I-clicker problem (frequency: AC) Let L be the set of words in English. Then L L R is A. The set of English words in alphabetical order, followed the same words in reverse alphabetical order. B. {w w is an English word or an English word written backwords}. C. {w w is an English word that is a palindrome}. D. None of the above.

17 Palindromes A palindrome is a word or a phrase that reads the same forward and backward. Examples mom madam Never odd or even. Stressed? No tips? Spit on desserts! L3.18

18 I-clicker problem (frequency: AC) Let L be the set of words in English. Then L L R is A. The set of English words in alphabetical order, followed the same words in reverse alphabetical order. B. {w w is an English word or an English word written backwords}. C. {w w is an English word that is a palindrome}. D. None of the above.