1. Week 13 - Wednesday CS322

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

3. Questions?

4. Logical warmup • Ten people are marooned on a deserted island • During their first day 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 • That night 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 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 10th 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. Formal Languages

6. Formal languages • 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

7. Rules • 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

8. Examples • Let alphabet Σ = {a, b} • Define a language L1 over Σ to be the set of all strings that begin with the character a and have length at most three characters • Write out L1 • A palindrome is a string which stays the same if the order of its characters is reversed • Define a language L2 over Σ to be the set of all palindromes made up of characters from Σ • Write 10 strings in L2

9. Notation • Let Σ be some alphabet • For any nonnegative integer n, let • Σn 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

10. Examples • Let alphabet Σ = {x, y, z} • Find Σ0, Σ1,and Σ2 • What is A = Σ0Σ1? What is B = Σ1Σ2? 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*?

11. More notation • 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 | xis a concatenation of any finite number of strings in L }

12. Examples • Let alphabet Σ = {a, b} • Let L1 be the set of all strings consisting of an even number of a's (including the empty string) • Let L2 = {b, bb, bbb} • Find • L1L2 • L1L2 • (L1L2)*

13. Regular expressions • 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: • Base: The empty set, the empty string ε, and any individual character in Σ is a regular expression • Recursion: If r and s are regular expressions over Σ, then the following are too: • Concatenation: (rs) • Alternation: (r | s) • Kleene star: (r*) • Restriction: Nothing else is a regular expression

14. Languages defined by a regular expression • 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))*

15. Examples • 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

16. Order of precedence • 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

17. Equivalences • 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

18. Examples • 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)

19. Practical notation • 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 ( 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 ) • [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 1or more repetitions of R • Notations vary and have considerable complexity • Use this notation to describe the regular expression for legal C++ identifiers

20. Finite-State Automata

21. Finite-state automaton • A finite-state automaton is an idealized machine composed of five objects: • A finite set I, called the input alphabet, of input symbols • A set S of states the automaton can be in • A designated state s0 called the initial state • A designed set of states called the set of accepting states • A next-state functionN: S x I S that maps a current state with current input to the next state

22. Transition diagram • FSA's are often described with a state transition diagram • The starting state has an arrow • The accepting states are marked with circles • Each rule is represented by a labeled transition arrow • The following FSA represents a vending machine quarter 25¢ 75¢ half-dollar half-dollar quarter $1.25 0¢ quarter half-dollar quarter quarter quarter 50¢ $1 half-dollar half-dollar half-dollar

23. FSA example • Consider this FSA: • What are its states? • What are its input symbols? • What is the initial state of A? • What are the accepting states of A? • What is N(s1, 1)? • What's a verbal description for the strings accepted? 1 1 s0 s1 s2 0 0 0 1

24. Annotated next-state tables • Consider the same FSA: • We can also describe an FSA using an annotated next-state table • A next-state table gives shows what the transition is for each state for all possible input • An annotated next-state table also marks the initial state and accepting states • Find the annotated next-state table for this FSA 1 1 s0 s1 s2 0 0 0 1

25. Table to transition diagram • Consider the following annotated next-state table •  marks initial state •  marks accepting states): • Draw the corresponding transition state diagram

26. FSA example • Consider this FSA again: • Which state will be reached on the following inputs: • 01 • 0011 • 0101100 • 10101 • What's a verbal description for the strings accepted? 1 1 s0 s1 s2 0 0 0 1

27. Eventual-state function • Let A be a FSA with a set of states S, set of input symbols I, and next-state function N: XxI S • Let I* be the set of all strings over I • The eventual-state functionN*: S x I*  S is the following • N*(s,w) = the state that A goes to if the symbols of w are input to A in sequence, starting with A in state s • All of this is just a notational convenience so that we have a way of talking about the state that a string will transition an FSA to • We say that wis accepted by AiffN*(s0, w) is an accepting state of A • The language of A, L(A) = { w  I* | w is accepted by A }

28. Designing automata • Design a finite-state automaton that accepts the set of all strings of 0's and 1's such that the number of 1's in the string is divisible by 3 • Make a regular expression for this language • Design a finite-state automaton that accepts the set of all strings of 0's and 1's that contain exactly one 1 • Make a regular expression for this language

29. Upcoming

30. Next time… • More on finite state automata • Simplifying FSA's

31. Reminders • Read Chapter 12