[PDF] 1 Finite Automata and Regular Expressions PDF fare2.3.pdf

is a finite automaton recognizing A For example, justify why there would be a finite automaton recognizing the language represented by a ∪ (ab) ∗ Proof: We

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[PDF] Finite Automata

An FA accepts input string if final state is ac- cept state; otherwise it rejects Goddard 1: 4 Page 5 An Example FA A

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second section we define finite automata and provide many examples we prove another famous result in finite automata theory, the Kleene Theorem The

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1 Introducing Finite Automata 1 1 Problems and Computation Decision Problems Decision Problems Given input, decide “yes” or “no” • Examples: Is x an

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Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1 a finite set of states (often denoted Q) 2 a finite set Σ of symbols

[PDF] 1 Finite Automata and Regular Expressions

is a finite automaton recognizing A For example, justify why there would be a finite automaton recognizing the language represented by a ∪ (ab) ∗ Proof: We

1 Finite Automata and Regular Expressions

Motivation: Given a pattern (regular expression) for string searching, we might want to convert it into a deterministic finite automaton or nondeter- ministic finite automaton to make string searching more efficient; a determin- istic automaton only has to scan each input symbol once. Can this always be done? Theorem 1.1IfL1=L(M1)andL2=L(M2)for languagesLi⊆Σ∗then

1. there is an automatonMrecognizingL1∪L2

2. there is an automatonMrecognizingL1◦L2

3. there is an automaton recognizingL∗1

4. there is an automaton recognizingΣ∗-L1

5. there is an automaton recognizingL1∩L2

6. ifa∈Σthen there is an automaton recognizing{a}

7. there is an automaton recognizing∅

From all of these things it follows that ifAis a regular language then there is a finite automaton recognizingA. For example, justify why there would be a finite automaton recognizing the language represented bya∪(ab)∗. Proof:We will do the proof for nondeterministic automata since determin- istic and nondeterministic automata are of equivalent power.

1.1 Union

For union, supposeM1is (K1,Σ,∆1,s1,F1) andM2is (K2,Σ,∆2,s2,F2).

Then letMbe (K,Σ,∆,s,F) where

K=K1∪K2∪ {s}

F=F1∪F2

1∪∆2∪ {(s,e,s1),(s,e,s2)}

andsis a new state. ThenL(M) =L(M1)∪L(M2). Diagram: 1 M1 M2 MK1 K2 K1 K2s1 s2 s1 s2 se e Note thatϵarrows are convenient for this construction.

1.1.1 Example

p qa bRecognizes a*

Recognizes b*

2 p qa b

Recognizes a* U b*e

1.2 Concatenation

M1K1 s1 F2 K2 s2F1 K1 s1 F1 K1 s1 F1 K1 s1 F1 M2MK1 s1 F2 K2 s2 F1K1 s1 F1 K1 s1 F1 K1 s1 F1 ee e 3 The states inF1are no longer accepting states. ThenL(M) =L(M1)◦

L(M2).

1.2.1 Examplep

qa bRecognizes a*

Recognizes b*

paqbRecognizes a*b* e

1.3 Kleene star

M1K1 s1F1 K1 s1 F1 K1 s1 F1 K1 s1 F1 4 MsF K F ee ee

ThenL(M) =L(M1)∗.

1.3.1 Examplea,bRecognizes {a,b}a,bRecognizes {a,b}*e

e How would you modify this automaton to recognize{a,b}+? Another simple construction for Kleene star fails for this automaton:a b 5

1.4 Complementation

LetM1= (K,Σ,δ,s,F) be adeterministicfinite automaton. LetM be (K,Σ,δ,s,K-F). ThenL(M) = Σ∗-L(M1).

1.4.1 ExampleaabbM1

Recognizes strings with even number of a'saabbM

Recognizes strings with odd number of a's

Why does the automaton have to be deterministic for this to work? An example showing thatM1has to be deterministic for this construc- tion to work: 6 a a

1.5 Intersection

For this, note thatL1∩L2= Σ∗-((Σ∗-L1)∪(Σ∗-L2)).

1.6 Other operations

Parts 6 and 7 of the theorem are trivial. Ask students to do them. As a consequence of this theorem, if a languageLis regular, then there is a finite automatonMrecongizingL.

2 Example

We construct a nondeterministic finite automaton recognizingL((ab)∗∪a). a bquotesdbs_dbs3.pdfusesText_6

[PDF] [PDF] 1 Finite Automata and Regular Expressions

1 Finite Automata and Regular Expressions

1. there is an automatonMrecognizingL1∪L2

2. there is an automatonMrecognizingL1◦L2

3. there is an automaton recognizingL∗1

4. there is an automaton recognizingΣ∗-L1

5. there is an automaton recognizingL1∩L2

6. ifa∈Σthen there is an automaton recognizing{a}

7. there is an automaton recognizing∅

1.1 Union

Then letMbe (K,Σ,∆,s,F) where

K=K1∪K2∪ {s}

F=F1∪F2

1∪∆2∪ {(s,e,s1),(s,e,s2)}

1.1.1 Example

Recognizes b*

Recognizes a* U b*e

1.2 Concatenation

L(M2).

1.2.1 Examplep

Recognizes b*

1.3 Kleene star

ThenL(M) =L(M1)∗.

1.3.1 Examplea,bRecognizes {a,b}a,bRecognizes {a,b}*e

1.4 Complementation

1.4.1 ExampleaabbM1

Recognizes strings with even number of a'saabbM

Recognizes strings with odd number of a's

1.5 Intersection

1.6 Other operations

2 Example