Week 13 - Friday

 What did we talk about last time?  Regular expressions

 You are paddling a canoe around a perfectly circular pond  Enjoying yourself immensely, you fail to notice that a goblin has crept up to the shore  You remember four things from your old lessons on goblin lore  Goblins can't swim  Goblins are always hungry for human flesh  Goblins can run four times as fast as people can paddle canoes  People can run faster than goblins  The goblin always assumes that you are making for the closest point on the shore and is always trying to cut you off  If you can get to the shore, you can escape, provided that the goblin isn't waiting for you (you need a little margin)  What's your escape strategy?

 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)

 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

Student Lecture

 A finite-state automaton is an idealized machine composed of five objects: 1. A finite set I, called the input alphabet, of input symbols 2. A set S of states the automaton can be in 3. A designated state s 0 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

 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 quarter half-dollar quarter half-dollar quarter

 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(s 1, 1)?  What's a verbal description for the strings accepted? s0s0 s0s0 s1s1 s1s1 s2s2 s2s

 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 s0s0 s0s0 s1s1 s1s1 s2s2 s2s

 Consider the following annotated next-state table   marks initial state   marks accepting states):  Draw the corresponding transition state diagram abc  UZYY  VVVV YZVY  ZZZZ

 Consider this FSA again:  Which state will be reached on the following inputs: i. 01 ii iii iv  What's a verbal description for the strings accepted? s0s0 s0s0 s1s1 s1s1 s2s2 s2s

 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 * (s 0, w) is an accepting state of A  The language of A, L(A) = { w  I * | w is accepted by A }

 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

 Kleene's Theorem shows that Finite-State Automata are equivalent to regular expressions  That is, for every finite-state automaton there is some equivalent regular expression, and vice versa  We won't prove it, but it should be intuitively clear because there are algorithms for building FSA's from regular expressions and vice versa

 Languages that can be expressed as an FSA or a regular expression are called regular  Some languages are not regular  For example, the language consisting of strings a k b k, meaning all strings that have a positive number of a's followed by the same number of b's, is not regular  Prove it (by contradiction)  Hint: We use the pigeonhole principle to show that more than one sequence of a p and a q must end up the in the same state

 List strings accepted by the FSA A  List strings accepted by the FSA B s0s0 s0s0 s1s1 s1s1 s2s2 s2s s3s3 s3s A s1s1 s1s s0s0 s0s0 B

 Two states of a finite-state automaton are *- equivalent if any string accepted by the automaton when it starts from one state is accepted when starting from the other  Given an automaton A with eventual-state function N *, we can formally say:  States s and t in A are*-equivalent iff N * (s,w) and N * (t,w) are both accepting states or both not  It turns out that *-equivalence defines an equivalence relation

 *-equivalence is hard to demonstrate directly  Instead, we'll focus on equivalence after k or fewer inputs  Given an automaton A with eventual-state function N *, we can formally say:  States s and t in A are k-equivalent iff N * (s,w) and N * (t,w) are both accepting states or both not, for all strings w of length k or less

 For k ≥ 0, k-equivalence is an equivalence relation  For k ≥ 0, the k-equivalence classes partition the set of all states of the automaton into a union of mutually disjoint subsets  For k ≥ 1, if two states are k-equivalent, they are also (k-1)-equivalent  For k ≥ 1, each k-equivalence class is a subset of a (k-1)-equivalence class  Any two states that are k-equivalent for all integers k ≥ 0 are *-equivalent

 Let A be an FSA with next-state function N  Given any states s and t in A: 1. s is 0-equivalent to t iff either s and t are both accepting states or they are both nonaccepting states 2. For every integer k ≥ 1, s is k-equivalent to t iff s and t are (k-1)-equivalent and for any input symbol m, N(s,m) and N(t,m) are also (k-1)-equivalent  These theorems essentially allow us to create a recursive definition for testing k-equivalence

 Find the 0-equivalence classes, the 1- equivalence classes, and the 2-equivalence classes for the following FSA: s1s1 s1s1 s2s2 s2s2 s3s3 s3s s4s4 s4s4 0 0 s0s0 s0s

 Keep finding k-equivalence classes for larger and larger values of k  If you ever find that the set of k-equivalence classes is equal to the set of (k+1)-equivalence classes, that is the set of *-equivalence classes  This is known as a fixed point in mathematics

 We can build a new FSA from the *-equivalence classes  Recall that [s] means the equivalence class of s  This FSA is called the quotient automaton A', and is defined from an FSA A with states S, input symbols I, and next-state function N as follows: 1. The set of states S' of A' is the set of *-equivalent classes of states of A 2. The set of input symbols I' of A' equals I 3. The initial state of A' is [s 0 ] where s o is the initial state of A 4. The accepting states of A' are the states of the form [s] where s is an accepting state of A 5. The next-state function N': S' x I  S' is: For all states [s] in S' and input symbols m, N'([s], m) = [N(s,m)]

 Let A be an FSA with states S, input symbols I, and next-state function N  To build A': 1. Find the set of 0-equivalence classes of S 2. For each integer k ≥ 1, find the k-equivalence classes of S until the k-equivalence classes are the same as the (k-1)-equivalence classes 3. Build a quotient automaton whose states are the equivalence classes given above with transition function N'([s],m) = [N(s,m)] for any input symbol m

 Find the quotient automaton for the following FSA s1s1 s1s1 s2s2 s2s2 s3s3 s3s s4s4 s4s4 0 0 s0s0 s0s

 Two automata A 1 and A 2 are equivalent iff L(A 1 ) = L(A 2 )  Proving the languages accepted by two automata can be difficult  However, the quotient automata for both A 1 and A 2 will be the same (except for labeling) if A 1 is equivalent to A 2

 Prove that the following two automata are equivalent by finding their quotient automata s0s0 s0s0 s2s2 s2s2 s1s1 s1s s3s3 s3s s0's0' s0's0' s1's1' s1's1' s2's2' s2's2' 0, s3's3' s3's3'

 Simplifying finite state automata  Context free languages  Push down automata

 Keep reading Chapter 12  Work on Assignment 10  Due Friday, April 22 before midnight