Download presentation

Presentation is loading. Please wait.

Published byJamil Gotts Modified over 2 years ago

1
Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer Science University of Virginia

2
**Midterm Take home? Vote now. Date:**

During the week before Spring break (out March 3, due March 6)? Don’t have to worry over the break After Spring break (4 days within March 16-23)? 1 more week of content but 2 more weeks to solve problems I will hold Skype/Google Hangout office hours over the break Think about it, vote on Thursday.

3
Pumping Lemma Goal: Give a sufficient condition for showing non-regularity Consider that L is an infinite regular language Let M be a DFA, L(M)=L Let’s say M has p states There must be some String w in L s.t. |w|>p By the pigeon-hole principle: Some state was visited at least twice! Taking that loop another time must give another string in the language. M x2 3 8 p 12 1 20

4
Pumping Lemma If L is a regular language then there is some number p (called pumping length) where if w is a string in L s.t. |w|>p then w can be divided into 3 pieces: w=xyz which satisfy: For each i≥0, L |y|>0 |xy|≤p Example: Consider: By condition 3 we know: Thus for i=2 we have more a’s than b’s, so L cannot be regular x M y 3 8 p ℕ 12 1 20 z

5
**Pumping Lemma For each i≥0, L |y|>0 |xy|≤p Show: is not regular**

Consider By condition 3 we know y , this leaves 4 cases: y Problem: in all of these I only change the first half! I needed to “remember” the first half of the string.

6
**Pumping Lemma For each i≥0, L |y|>0 |xy|≤p Show: is not regular**

Consider where q is a prime greater than p y may have only a’s, let y= where m≤p For i=p+1 we have Clearly, p(m+1) is not prime For let then So

7
**Pumping Lemma For each i≥0, L |y|>0 |xy|≤p Consider:**

This language is pumpable but not regular 3 cases: If then let x=ε, y=a If then let x=ε, y=c If then let x=ε, y=c Thus every string is pumpable Nonregular: Then use pumping Lemma!

8
**Myhill-Nerode Theorem**

Gives a necessary and sufficient condition for regularity! Idea: If two strings terminate in the same state in a DFA then their membership must be equivalent for any suffix. If and meet at state q then for any string , ends in the same state as We say and have no distinguishing extensions M q

9
**Myhill-Nerode Theorem**

Recall Equivalence Relation: A relation ~ is called an equivalence relation if: Reflexive: Symmetric: Transitive: The relation if and have no distinguishing extensions in language L forms an equivalence relation. Recall Equivalence Class: For equivalence relation ~, equivalence class Myhill Nerode Theorem: L is regular iff has a finite number of equivalence classes under relation

10
**Myhill-Nerode Theorem**

Myhill Nerode Theorem: L is regular iff has a finite number of equivalence classes under relation Given a finite set of equivalence classes, construct a DFA: Each equivalence class becomes a state: If character takes a string from equivalence class to then add a transition from the state for to the state for Accept states are those for equivalence classes of strings in the language.

11
**Myhill-Nerode Theorem**

Extra Credit: Use the Myhill-Nerode Theorem to show the following languages are non-regular: ℕ

12
Language Reversal Theorem: The regular languages are closed under reversal. Proof: Construction. Given a regular language L, show that the language is also regular. Let M be a DFA for L, construct M’ to be a NFA for L: M M’ ε q0 ε

13
**Half Let Show that HALF preserves regularity.**

e.g. if “RingoStarr” is in L, then “Ringo” is in HALF(L) Let M be a DFA for language L Intuition: follow the transitions of M for string v. “Check” that there is a path from v to an accept state that consumes |v| characters. How do we do this “check”? M

14
**Half Let Show that HALF preserves regularity.**

Intuition: follow the transitions of M for string v. “Check” that there is a path from v to an accept state that consumes |v| characters. How do we do this “check”? Use the machine for LR M M’ Σ ε Σ Σ ε Σ F={(q,q)} Σ

15
**Double Let Show that DOUBLE preserves regularity.**

e.g. if “BamBam” is in L, then “Bam” is in DOUBLE(L) Let M be a DFA for language L Intuition: Run L on w, in parallel non-deterministically “guess” the end state in machine M on w and check if starting from that guess puts the machine in an accept state M M “Guess”

16
Double Let Intuition: Run L on w, in parallel non-deterministically “guess” the end state in machine M on w and check if starting from that guess puts the machine in an accept state ε ε M ε M M Accept if: M ends in state M ends in an accept state M 1 M 2 M …

Similar presentations

OK

Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer.

Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on various types of web browsers and their features Ppt on product design Ppt on principles of management for class 12 Ppt on charles dickens a christmas carol Ppt on forward rate agreement example Ppt on bluetooth communication protocol Ppt on point contact diode Ppt on internet banking project Ppt on school management system software Viewer ppt online ticket