Week 14 - Friday

 What did we talk about last time?  Simplifying FSAs  Quotient automata

P RINCE His Royal Badness

 A sociologist named McSnurd gave a talk about a community that has clubs with the following attributes:  A person can belong to more than one club  Each club is named after a person  No two different clubs are named after the same person  Every person has a club named after him or her  A person may or may not belong to the club named after them ▪ If they do, that person is called sociable ▪ If they don't, that person is called unsociable  Perhaps surprisingly, there is a club that contains all of the unsociable people  Do you believe McSnurd's description?

 Another sociologist named McSnuff described a similar community:  As before, there is the same number of clubs as people, and each club is named after exactly one person  However, a person can be a member of a club openly or secretly ▪ If a person is not openly a member of the club named after them, they are suspicious ▪ If a person is secretly a member of the club named after them, they are a spy  Perhaps surprisingly, there is a club that contains all of the suspicious people  Do you believe McSnuff's description?  Do you know whether or not there is at least one spy in the community?

 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'

 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

 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

 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  ε | 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

 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)*

 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

 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

 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)

 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 1. Unrestricted grammars  α  β (anything to anything) 2. Context-sensitive grammars  αAβ  αγβ (non-terminal in a particular context to anything) 3. Context-free grammars  A  γ (a single non-terminal to anything) 4. 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)

 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

 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

 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 A

 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

 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?

 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

 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?

 Review first third of the course

 Review chapters  Finish Assignment 10  Due Monday before midnight