CS322Week 13 - Wednesday

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

Questions?

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)?

Formal Languages

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

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

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

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 ΣΣ* is called the Kleene closure of Σ

and the * operator is often called the Kleene star

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*?

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 | x is a concatenation of any finite number of strings in L }

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)*

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:I. Base: The empty set, the empty string ε, and any

individual character in Σ is a regular expressionII. 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

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))*

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

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

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

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)

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 1 or more repetitions of R

Notations vary and have considerable complexity Use this notation to describe the regular expression for legal C++

identifiers

Finite-State Automata

Finite-state automaton

A finite-state automaton is an idealized machine composed of five objects:1. A finite set I, called the input alphabet, of

input symbols2. A set S of states the automaton can be in3. A designated state s0 called the initial state

4. A designed set of states called the set of accepting states

5. A next-state function N: S x I S that maps a current state with current input to the next state

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

0¢

25¢

75¢

50¢

$1

$1.25

half-dollar

half-dollar

half-dollar

quarter

quarterhalf-dollar

half-dollar

quarter quarterquarter

half-dollar

quarter

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?

s0 s1 s2

1

1

001

0

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

s0 s1 s2

1

1

001

0

Table to transition diagram Consider the following annotated next-state

table marks initial state marks accepting states):

Draw the corresponding transition state diagram

a b c U Z Y Y V V V V

Y Z V Y Z Z Z Z

FSA example

Consider this FSA again:

Which state will be reached on the following inputs:i. 01ii. 0011iii. 0101100iv. 10101

What's a verbal description for the strings accepted?

s0 s1 s2

1

1

001

0

Eventual-state function

Let A be a FSA with a set of states S, set of input symbols I, and next-state function N: X x I S

Let I* be the set of all strings over I The eventual-state function N*: 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 w is accepted by A iff N*(s0, w) is an accepting state of A

The language of A, L(A) = { w I* | w is accepted by A }

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

Upcoming

Next time…

More on finite state automata Simplifying FSA's

Reminders

Read Chapter 12