Fall 2004COMP 3351 Finite Automata

Fall 2004COMP 3352 Finite Automaton Input String Output String Finite Automaton

Fall 2004COMP 3353 Finite Accepter Input “Accept” or “Reject” String Finite Automaton Output

Fall 2004COMP 3354 Transition Graph initial state final state “accept” state transition abba - Finite Accepter

Fall 2004COMP 3355 Initial Configuration Input String

Fall 2004COMP 3356 Reading the Input

Fall 2004COMP 3357

Fall 2004COMP 3358

Fall 2004COMP 3359

Fall 2004COMP Output: “accept” Input finished

Fall 2004COMP Rejection

Fall 2004COMP 33512

Fall 2004COMP 33513

Fall 2004COMP 33514

Fall 2004COMP Output: “reject” Input finished

Fall 2004COMP Another Rejection

Fall 2004COMP Output: “reject”

Fall 2004COMP Another Example

Fall 2004COMP 33519

Fall 2004COMP 33520

Fall 2004COMP 33521

Fall 2004COMP Output: “accept” Input finished

Fall 2004COMP Rejection

Fall 2004COMP 33524

Fall 2004COMP 33525

Fall 2004COMP 33526

Fall 2004COMP Output: “reject” Input finished

Fall 2004COMP Formalities Deterministic Finite Accepter (DFA) : (A finite) set of states : (A finite) set of input alphabet : transition function : initial state (an element of Q) : set of final states (a subset of Q )

Fall 2004COMP Input Alphabet

Fall 2004COMP Set of States

Fall 2004COMP Initial State

Fall 2004COMP Set of Final States

Fall 2004COMP Transition Function

Fall 2004COMP 33534

Fall 2004COMP 33535

Fall 2004COMP 33536

Fall 2004COMP Transition Function

Fall 2004COMP Extended Transition Function

Fall 2004COMP 33539

Fall 2004COMP 33540

Fall 2004COMP 33541

Fall 2004COMP Observation: There is a walk from to with label

Fall 2004COMP Example: There is a walk from to with label

Fall 2004COMP Recursive Definition

Fall 2004COMP 33545

Fall 2004COMP Languages Accepted by DFAs Take DFA Definition: The language contains all input strings accepted by = { strings that drive to a final state}

Fall 2004COMP Example accept

Fall 2004COMP Another Example accept

Fall 2004COMP Formally For a DFA Language accepted by :

Fall 2004COMP Observation Language rejected by :

Fall 2004COMP More Examples accept trap state

Fall 2004COMP = { all strings with prefix } accept

Fall 2004COMP = { all strings without substring }

Fall 2004COMP Regular Languages A language is regular if there is a DFA such that All regular languages form a language family

Fall 2004COMP { all strings with prefix } { all strings with suffix } { all strings without substring } Examples of regular languages: There exist automata that accept these Languages (see previous slides).

Fall 2004COMP Another Example The language is regular:

Fall 2004COMP There exist languages which are not Regular: There is no DFA that accepts such a language (we will prove this later!) Example: