Finite State Machines Concepts DFA NFA

Finite State Machines Concepts DFA NFA

Graphs and More • Finites state machines are directed graphs.

Nodes-states, Edges-input • Each city represents a state you can be in. • Take and edge on some desire(input) • Imagine you are doing a food tour of the country, starting in Atlanta • You want to eat…Perogies, Philly Cheese Steak, Cuban

Edges as input • The food transitions are along the pathway. • Certain food tours are allowed, others just don’t work out! • Valid: Perogies, Philly Cheese Steak, Miami Cuban • Invalid: Perogies, Jambalaya, Cuban • Same food can be found in several places, and assume there is a finite set of foods – the alphabet • There are also a finite set of cities that you can be in, or go to – the states • Add restriction that for health reasons, the last food you eat has to be Cuban! ( ACCEPTING vs REJECT states) • What kind of questions can we answer!

FSM Input String {Yes, No} Finite State Machines (FSMs) • What are the problems we are trying to solve? • Acceptor – takes an input and tells you whether the input matches some specifications. (Are you in or are you out) • Transducers – working thru input data to modify the behavior of some process. • Key limitation - memory size is fixed • Independent of input size.

a a 0 1 2 b b a,b FSM Input String {Yes, No} The FSM model • Inputs: • Strings built from a fixed alphabet • Machine: A directed graph • Nodes: states of the machine • Edges: transitions from one state to another • Special states • Start • Finalor accepting

2 FSM Decider Example • Which strings of as and bs are accepted? • Input alphabet {a, b} • States {q0, q1, q2} • Start state q0 • Final states {q2} a a 0 1 b b a,b

The soda machine – concretely

Input alphabet $0.25 Return-coin States Ready-for-coins 25 50 Empty … The soda machine – abstractly

Making the DFA model precise • Input alphabet • SState set • q0  S Initial state • F  S Final states • :S   S Transition function M = (, S, q0, F, )

Building FSMs • An FSM is a directed graph • How large is the input alphabet? • How many states? • How fast must it run? • How to minimize space? • Representations • Matrix • Array of lists • Hashtable • Overlapping hashtable • Switch statement

Representing the machine • Matrix representation • Why? • Running Time?

Process machines • Coke machine • Speech recognition – add probability to the transitions • TCP protocol • Parsers (HTML)

TCP Protocol ( don’t worry …)

Deciders • “Are you IN or are you OUT” - Jerry Maguire

Simple DFA example • HTML Parser – Is it valid HTML? • Accepts strings that end in 1 , å= {0,1} • Create a machine that accepts string which contain the sub-string “001” • 0010, 1001 are in. • 11, 0000 are out • Working it out… Skip all 1’s, perk up when you see a 0. ( you may have just come upon the substring ). Skip back if you see a 1 too early. • 1. Seen no symbols of the pattern • 2. Just seen a 0 • 3. Just seen a 00 • 4. Just seen the entire pattern 001

Regular Expressions • Regular expression. • Way to describe sets of strings • Examples – • (“|” – OR) • (“*” – 0 or more, “+” – one or more) • (01)* • (a|b)*ab • this | that | theother • 00*11*22* =0+1+2+

Languages and FSMs • A language that can be DECIDED by an FSM is called regular. • Examples • (accepts only these and no other) • a(a|b)*b Easiest way to test whether a language is regular is to see if you can create a machine! L = {a(a|b)*b}

Problems types • What about… a(a|b)*b • What does this mean? • | - OR • * - 0 or more

Problems types • Ends with… (a+b)*ab • Begins with… ab(a+b)* • Contains substring ‘w1’ (a+b)*w1(a+b)* • Even / odd numbers of some character or both! • Leverage the power of the guess

Enhancing the model - NFA • Non-determinism • Allow the machine to be in more than one state at once • On a given input, you could follow multiple paths • If there EXISTS a pathway to an ACCEPT state to deal with the given input, ACCEPT • Examples: • 001 sub-string • a(a|b)*b

The Idea of Non-determinism • An NFA can be in more than one state at a time • 3 ways of looking at it • Coins/fingers on every state you can be at. • A tree of outcomes, put a lemming on each path, if any of them end on a final state, we ACCEPT • Oracle approach. Someone is telling you the right way to go

I like lemmings… • Lemming approach Deterministic Non-deterministic …

More examples • “[.?!][]\"')]*\$$\\|\t\\| \$[ \t\n]*” • [.?!][]\"')]*($|<tab>|)[<tab><C-j>]* • Emacs regexp: • Any of .? !followed by • Zero or more of ] “ ‘ )followed by • Any of end-of-line, tab, two spaces followed by • Zero or more of space, tab, newline

Big Question • Are there languages L that can be accepted by NFAs but not DFAs? • Are they more “powerful”?

Big Question • Are there languages L that can be accepted by NFAs but not DFAs? No!

What have we gained? • Can we solve harder problems? • We have assumed we can solve problems by only parsing the data once… • What about the ability to back-track? Turing Machines… ( If I only had a memory…)

What can’t you do • Counting (on an unbounded set) • L = {0n1n | n>0} • It needs to remember the number of 0’s it has seen. Since the number of 0’s is not limited we would have to track an infinite set of states. • Violates our memory condition… • We can do L = {0515}

What can’t you do • Counting (on an unbounded set) • L = {0n1n | n>0} • It needs to remember the number of 0’s it has seen. Since the number of 0’s is not limited we would have to track an infinite set of states. • Violates our memory condition… • We can do L = {0515} NEED MORE POWER

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Finite State Machines Concepts DFA NFA