1. Week 14 - Friday

2.  What did we talk about last time?  Exam 3 post mortem  Finite state automata  Equivalence with regular expressions

4.  U2 has 17 minutes to cross a bridge for a concert  Plan a way to get them across in the darkness  They have one flashlight  A maximum of two people can cross the bridge at one time, and one of them must have the flashlight  The flashlight must be walked back and forth  Each band member walks at a different speed  Bono:1 minute to cross  The Edge:2 minutes to cross  Adam:5 minutes to cross  Larry:10 minutes to cross  A pair must walk together at the rate of the slower man's pace

5. Student Lecture

7.  List strings accepted by the FSA A  List strings accepted by the FSA B s0s0 s0s0 s1s1 s1s1 s2s2 s2s2 1 0 0 1 0 s3s3 s3s3 1 1 0 A s1s1 s1s1 1 00 1 s0s0 s0s0 B

8.  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

9.  *-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

10.  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

11.  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

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

13.  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

14.  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)]

15.  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

16.  Find the quotient automaton for the following FSA s1s1 s1s1 s2s2 s2s2 s3s3 s3s3 0 1 1 s4s4 s4s4 0 0 s0s0 s0s0 1 1 0 0 1

17.  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

18.  Prove that the following two automata are equivalent by finding their quotient automata s0s0 s0s0 s2s2 s2s2 s1s1 s1s1 0 1 1 s3s3 s3s3 0 0 1 1 0 s0's0' s0's0' s1's1' s1's1' s2's2' s2's2' 0, 1 0 1 s3's3' s3's3' 0 0 1 1

20.  A context free language is one that can be described by a context free grammar  Every regular language is context free, but there are context free languages that are not regular  Classic examples:  Strings of k 0's followed by k 1's  Palindromes made up of a's and b's  Legally nested parentheses  All of these involve counting arbitrary numbers of characters  Regular expressions can't count

21.  Instead of using regular expressions, a context free language is often described with a grammar  A grammar is a formal system of rewriting rules consisting of  Terminals: symbols of the alphabet  Non-terminals: symbols that produce other sequences of terminals and non-terminals  Grammars often start with the non-terminal starting symbol S  Any string that can be derived from S through some sequence of rule rewrites is a string in the language

22.  The following is a grammar that produces the language of strings of 1s and 0s that end in a 1 S  A1S  A1  A  1 | 0 | A1 | A0  This language is regular and is equivalent to (0 | 1)* 1  The following is a grammar that produces the language of a k b k where k ≥ 1 (which is not regular S  AS  A  A  ab | aAb

23.  Write a grammar that legal nesting of parentheses and braces in Java  Don't worry about the stuff that goes inside  Write a grammar for legal mathematical expressions in Java using variables and integers, +, -, *, /, and parentheses  Write a grammar that corresponds to the same language defined by the regular expression ab* (a | bb) (ba)*

24.  As you know, regular languages can be expressed by regular expressions and finite state automata  Thus, regular expressions are equal to FSAs in power  Context free languages can be expressed by context free grammars  There is also a class of automata called pushdown automata that correspond to context free languages

26.  To make the PDA for the language 0 k 1 k, k ≥ 1, we have the following:  In this case, the notation X/Y means that if X is on the top of the stack, replace it with Y  The top of the stack is Z  Notation is not well standardized for PDAs  It's awkward keeping track of the stack s1s1 s1s1 s2s2 s2s2 s0s0 s0s0 0: Z/AZ 0: A/AA 1: A/ε 1: A/A 1: Z/Z

28.  Noam Chomsky is a brilliant linguist who has recently focused mostly on political activism  Remember that a grammar consists of terminals (alphabet symbols), non-terminals, production rules, and a start symbol  He noted that grammars can be divided into four levels in terms of expressiveness:  Type-0 (Unrestricted grammars)  Type-1 (Context sensitive grammars)  Type-2 (Context free grammars)  Type-3 (Regular grammars)

29.  Each grammar has rules for what is a legal production rule  Let α, β, and γ be any combinations of terminals and non- terminals, where γ is non-empty  Unrestricted grammars  α  β (anything to anything)  Context-sensitive grammars  αAβ  αγβ (non-terminal in a particular context to anything)  Context-free grammars  A  γ (a single non-terminal to anything)  Regular grammars  A  a and A  aB (a single non-terminal to a single terminal or a terminal and a single non-terminal on the right side)

30.  Every kind of language has a particular kind of machine associated with it GrammarLanguagesAutomaton Production Rules Examples Type-0 Recursively enumerable Turing machine α  βα  β All languages, any computable functions Type-1 Context- sensitive Linear-bounded non- deterministic Turing machine αAβ  αγβ Most natural human languages Type-2Context-free Non-deterministic pushdown automaton A  γ Most programming languages Type-3RegularFinite state automaton A  a and A  aB Regular expressions

31.  Each set of languages in the hierarchy strictly contains the sets beneath it  Regular languages  Can be accepted by nondeterministic or deterministic finite automata (or a read-0nly Turing machine)  Are closed under union, intersection, complement, concatenation, and Kleene star  Context-free languages  Are defined by those languages accepted by nondeterministic pushdown automata  Are closed under union, concatenation, and Kleene star (but not under intersection or complement)  Deciding whether a string is in a context-free language can be determined in polynomial time  Deciding whether a language is empty is decidable  Context-sensitive languages  Are very rarely used  Are closed under union, intersection, complement, and Kleene star  Deciding whether a string is in a context-sensitive language is a PSPACE-complete problem Recursively enumerable Context- sensitive Context-free Regular

32.  A Turing machine is a mathematical model for computation  It consists of a head, an infinitely long tape, a set of possible states, and an alphabet of characters that can be written on the tape  A list of rules saying what it should write and should it move left or right given the current symbol and state 1011110000 A A

33.  You can specify a Turing machine with a table giving its behavior for a specific configuration  Turing's first example machine printed an infinite sequence of alternating 1s and 0s, separated by spaces: ConfigurationBehavior StateSymbolOperationResult State BBlankWrite 0, Move RightC CBlankMove RightE EBlankWrite 1, Move RightF FBlankMove RightB

34.  If an algorithm exists, a Turing machine can perform that algorithm  In essence, a Turing machine is the most powerful model we have of computation  Power, in this sense, means the ability to compute some function, not the speed associated with its computation  Do you own a Turing machine?

35.  Given a Turing machine and input x, does it reach the halt state?  As you know, there is no algorithm to determine this  If you had a Turing machine that could solve the halting problem, it would cause a logical contradiction

36.  Are two context-free languages the same?  Is the intersection of two context-free languages empty?  Is a context-free language equal to Σ *  Is a context-free language a subset of another context-free language?  Post correspondence problem:  You have lists a 1, a 2, … a n and b 1, b 2, … b n, where a i and b j are strings of some alphabet Σ with at least 2 symbols  Is there a value k ≥ 1 such that some sequence of k strings from the a list concatenated is equal to some sequence of k strings from the b list concatenated?  Is a given statement of first-order logic provable from a starting set of axioms?  Given a set of matrices, is there some sequence that they can be multiplied in (perhaps with repetitions) that will yield the zero matrix?

38.  Review first third of the course

39.  Review chapters 2 - 6  Finish Assignment 10  Due tonight before midnight