CS322

1. Week 14 - Friday CS322

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

3. Questions?

4. Logical warmup • 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. Simplifying FSA's Student Lecture

6. Simplifying FSA's

7. Comparison • List strings accepted by the FSA A • List strings accepted by the FSA B A B 0 0 1 s0 s1 0 0 1 s0 s1 1 1 1 s3 s2 1 0 0

8. *-equivalence • 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 iffN*(s,w) and N*(t,w) are both accepting states or both not • It turns out that *-equivalence defines an equivalence relation

9. k-equivalence • *-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 iffN*(s,w) and N*(t,w) are both accepting states or both not, for all strings w of length k or less

10. Facts about k-equivalence • 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. k-equivalence theorems • Let A be an FSA with next-state function N • Given any states s and t in A: • s is 0-equivalent to tiff either s and t are both accepting states or they are both nonaccepting states • For every integer k ≥ 1, s is k-equivalent to tiffs 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. k-equivalence examples • Find the 0-equivalence classes, the 1-equivalence classes, and the 2-equivalence classes for the following FSA: 1 1 0 s0 s1 s2 1 0 1 1 0 0 s4 s3 0

13. Finding the *-equivalence classes • 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. The quotient automaton • 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 automatonA', and is defined from an FSA A with states S, input symbols I, and next-state function N as follows: • The set of states S' of A' is the set of *-equivalent classes of states of A • The set of input symbols I' of A' equals I • The initial state of A' is [s0] where so is the initial state of A • The accepting states of A' are the states of the form [s] where s is an accepting state of A • 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. Constructing a quotient automaton • Let A be an FSA with states S, input symbols I, and next-state function N • To build A': • Find the set of 0-equivalence classes of S • 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 • 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. Quotient automaton example • Find the quotient automaton for the following FSA 1 1 0 s0 s1 s2 1 0 1 1 0 0 s4 s3 0

17. Equivalent automata • Two automata A1 and A2 are equivalent iffL(A1) = L(A2) • Proving the languages accepted by two automata can be difficult • However, the quotient automata for both A1 and A2 will be the same (except for labeling) if A1 is equivalent to A2

18. Proving equivalence • Prove that the following two automata are equivalent by finding their quotient automata s2 s1' 0, 1 0 1 1 0 0 1 1 s0 s0' s2' s1 0 0 0 1 1 1 s3 s3' 0

19. Context Free Languages

20. Context free languages • 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. Context free grammars • 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. Simple context free grammars • The following is a grammar that produces the language of strings of 1s and 0s that end in a 1 • 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 akbk where k ≥ 1 (which is not regular • SA • Aab| aAb

23. CFG examples • 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. Pushdown automata • 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

25. Pushdown automata • A pushdown automaton is an idealized machine with seven objects: • Q a finite set of states • Σ the finite input alphabet • Γ the finite stack alphabet • δ is a finite subset of Q x (Σ  ε) x Γ x Q x Γ*which gives a set of transition rules • q0is the start state • Z Γ is the initial stack symbol • F Q is the set of accepting states • All of this looks totally insane, but it's really just adding a stack to finite state automata

26. Pushdown automaton example • To make the PDA for the language 0k1k, k ≥ 1, we have the following: • In this case, the notation X/Ymeans 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 1: A/A 1: Z/Z s0 s1 s2 0: Z/AZ 0: A/AA 1: A/ε

27. Chomsky's Hierarchy of Grammars

28. Chomsky hierarchy • 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. Rules for grammars • Each grammar has rules for what is a legal production rule • Let α, β, and γbe 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. Languages broken down • Every kind of language has a particular kind of machine associated with it

31. Fun facts about formal languages • 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

32. Turing machine • 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 A

33. Turing machine example • 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:

34. Church-Turing thesis • 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. Halting problem • 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. Other undecidable problems • 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 a1, a2, … an and b1, b2, … bn, where ai and bj 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?

37. Upcoming

38. Next time… • Review first third of the course

39. Reminders • Review chapters 2 - 6 • Finish Assignment 10 • Due tonight before midnight