Finite automate

Patterns for token digit  [0-9] digits  digit+ Letter  [A –Z a-z] If if then ten else else Relop <|<=|=|<>|>|>= Idletter(letter|digit)* Numdigits+ (.digits+)?(E(+|-)?digits+)? ws (blank|tab|newline)+

Transition Diagram A transition diagram is a stylized flowchart. Transition diagram is used to keep track of information about characters that are seen as the forward pointer scans the input. We do so by moving from position to position in the diagrams as characters are read.

Transition Diagram Start state Accept state Any state Have a collection of circles called states . Each state represents a condition that could occur during the process of scanning the input looking for a lexeme that matches one of several patterns . Start state Accept state Any state

Transition Diagram Edge : represent transitions from one state to another as caused by the input

Finite automata: why? Regular expressions = specification Finite automata = implementation

Definition : Finite Automaton

Finite Automata

State Diagram 1 q3 q1 q2 1 0, 1

Finite Automata Transition s1 a s2 Is read In state s1 on input a go to state s2 •If end of input and in accepting state => accept •Otherwise => reject

Data Representation

Exercise A finite automaton that accepts only “a” Language of FS is the set of accepted states stat input 1 a 2 Is accepted b ? a 1 2

Finite Automata Two types – both describe what are called regular languages Deterministic (DFA) – There is a fixed number of states and we can only be in one state at a time Nondeterministic (NFA) –There is a fixed number of states but we can be in multiple states at one time NFA and DFA stand for nondeterministic finite automaton and deterministic finite automaton, respectively.

NFA

Nondetermistic Finite Automata: NFA A nondeterministic finite automaton can be different from a deterministic one in that for any input symbol, nondeterministic one can transit to more than one states. epsilon transition

NFA 0,1 0,1 q3 1 q4 q1 q2 1 0,e

NFA

Test youself Construct an NFA to accept all strings terminating in 01 

Answer

Test yourself 2 Construct an NFA to accept those strings containing three consecutive zeroes

Answer 2

DFA

DFA Is special case of NFA where: There are no moves on input Ԑ For each state s and input symbol a , there exactly one edge out of a labeled a.

DFA Deterministic Finite Automata (DFA) One transition per input per state No Ԑ-moves Nondeterministic Finite Automata (NFA) Can have multiple transitions for one input in a given state Can have -moves

Test yourself (DFA1) 1- Construct a DFA to accept a string containing a zero followed by a one

Answer 1-

Test yourself (DFA2) 2-  Construct a DFA to accept a string containing two consecutive zeroes followed by two consecutive ones

answer 2-

Test yourself (DFA3) 3- Construct a DFA to accept a string containing even number of zeroes and any number of ones

Answer 3

Test yourself (DFA4) 4-  Construct a DFA to accept all strings which do not contain three consecutive zeroes  

Answer 4-

Construction of an NFA from regular Expression

Construction of an NFA from regular Expression Recursive construction Base cases follow base case definitions of regular expressions : -NFA that accepts the empty string – a single state that is the start and end state a: -NFA that accepts {a} – two-state machine (start and final state) with an a-transition Note: technically, we also need an -NFA for the empty language {} – easy

Construction of an NFA from regular Expression -NFA that accepts the empty string – a single state that is the start and end state a: -NFA that accepts {a} – two-state machine (start and final state) with an a-transition

Regular language Single final state Regular language NFA Single final state NFA

Example

Union NFA for Ԑ Ԑ Ԑ Ԑ

Example NFA for Ԑ Ԑ Ԑ Ԑ

Concatenation NFA for Ԑ Ԑ

Example NFA for Ԑ Ԑ

Star Operation Ԑ NFA for Ԑ Ԑ Ԑ Ԑ

Example NFA for Ԑ Ԑ Ԑ Ԑ

Test yourself Construct the NFA for r= (a|b)*abb

Steps 1,2

Step 3,4

Answer Construct the NFA for r= (a|b)*abb

3.4 3.6 3.7.4