What did we talk about last time? Exam 3 Before review:

Transcription

1 Week 13 - Wednesday

2 What did we talk about last time? Exam 3 Before review: Graphing functions Rules for manipulating asymptotic bounds Computing bounds for running time functions

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4 Ten people are marooned on a deserted island They gather many coconuts and put them all in a community pile They are so tired that they decide to divide them into ten equal piles the next morning One castaway wakes up hungry and decides to take his share early After dividing up the coconuts, he finds he is one coconut short of ten equal piles He notices a monkey holding one coconut He tries to take the monkey's coconut so that the total is evenly divisible by 10 However, when he tries to take it, the monkey hits him on the head with it, killing him Later, another castaway wakes up hungry and also decides to take his share early On the way to the coconuts he finds the body of the first castaway and realizes that he is now be entitled to 1/9 of the total pile After dividing them up into nine piles he is again one coconut short of an even division and tries to take the monkey's (slightly) bloody coconut Again, the monkey hits the second man on the head and kills him Each of the remaining castaways goes through the same process, until the 10 th person to wake up realizes that the entire pile for himself What is the smallest number of coconuts in the original pile (ignoring the monkey's)?

5 Student Lecture

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7 Computer science grew out a lot of different pieces Mathematics Engineering Linguistics Describing an algorithm precisely requires that it be framed in terms of some formal language with exact rules

8 We say that a language is a set of strings A string is an ordered n-tuple of elements of an alphabet Σ or the empty string ε (which has no characters) An alphabet Σ is a finite set of characters

9 Let alphabet Σ = {a, b} Define a language L 1 over Σ to be the set of all strings that begin with the character a and have length at most three characters Write out L 1 A palindrome is a string which stays the same if the order of its characters is reversed Define a language L 2 over Σ to be the set of all palindromes made up of characters from Σ Write 10 strings in L 2

10 Let Σ be some alphabet For any nonnegative integer n, let Σ n be the set of all strings over Σ that have length n Σ + be the set of all strings over Σ that have length at least 1 Σ * be the set of all strings over Σ Σ * is called the Kleene closure of Σ and the * operator is often called the Kleene star

11 Let alphabet Σ = {x, y, z} Find Σ 0, Σ 1, and Σ 2 What is A = Σ 0 Σ 1? What is B = Σ 1 Σ 2? How would you describe these sets and set A B in words? Describe a systematic way of writing out Σ + How would you have to change your system to write out Σ *?

12 Let Σ be a finite alphabet Given strings x and y over Σ, the concatenation of x and y is the string made by writing x with y appended afterwards With languages L and L' over Σ, we can define the following new languages: Concatenation of L and L', written LL' LL' = { xy x L and y L' } Union of L and L', written L L' L L' = { x x L or x L' } Kleene closure of L, written L * L * = { x x is a concatenation of any finite number of strings in L }

13 Let alphabet Σ = {a, b} Let L 1 be the set of all strings consisting of an even number of a's (including the empty string) Let L 2 = {b, bb, bbb} Find L 1 L 2 L 1 L 2 (L 1 L 2 ) *

14 It's getting annoying trying to describe infinite languages using ellipses Notation called a regular expression can allow us to express languages precisely and compactly Given a finite alphabet Σ, we can define regular expressions recursively: I. Base: The empty set, the empty string ε, and any individual II. character in Σ is a regular expression Recursion: If r and s are regular expressions over Σ, then the following are too: a) Concatenation: (rs) b) Alternation: (r s) c) Kleene star: (r*) III. Restriction: Nothing else is a regular expression

15 For a finite alphabet Σ, the language L(r) defined by a regular expression r is as follows Base: L( ) =, L(ε) = {ε}, L(a) = {a} for every a Σ Recursion: If L(r) and L(r') are the languages defined by the regular expressions r and r' over Σ, then L(r r') = L(r)L(r') L(r r') = L(r) L(r') L(r * ) = (L(r)) *

16 Let Σ = {a, b, c} Let language L = a (b c)* (ab)* Write 5 strings in L Let language M = ab * (c ε) Write 5 strings in M

17 For the sake of consistency, regular expressions obey a particular order of precedence * is the highest precedence Concatenation is the next highest Alternation is the lowest Parentheses can be omitted if there is no ambiguity Write (a((bc)*)) with as few parentheses as possible Write a b* c, using parentheses to mark the precedence of each operation

18 As before, let Σ = {a, b} Can you describe (a b)* in another way? What about ( ε a* b* )*? Given that L = a*b(a b)*, write 5 strings that belong to L Let M = a* (ab)* Which of the following belong to M? a b aaaa abba ababab

19 Let Σ = {0, 1} Find regular expressions for the following languages: The language of all strings of 0's and 1's that have even length and in which the 0's and 1's alternate The language consisting of all strings of 0's and 1's with an even number of 1's The language consisting of all strings of 0's and 1's that do not contain two consecutive 1's The language that gives all binary numbers written in normal form (that is, without leading zeroes, and the empty string is not allowed)

20 Regular expressions are used in some programming languages (notably Perl) and in grep and other find and replace tools The notation is generally extended to make it a little easier, as in the following: [ A C] means any character in that range, [A C] means ( A B C ) [0 9] means ( ) [ABC] means (A B C ) ABC means the concatenation of A, B, and C A dot stands for any letter: A.C could match AxC, A&C, ABC ^ means NOT, thus [^D Z] means not the characters D through Z Repetitions: R? means 0 or 1 repetitions of R R* means 0 or more repetitions of R R+ means 1 or more repetitions of R Notations vary and have considerable complexity Use this notation to describe the regular expression for legal C++ identifiers

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22 Finite state automata Simplifying FSA's

23 Keep reading Chapter 12