Regular Languages

Prepared byManuel E. Bermúdez, Ph.D.

Associate ProfessorUniversity of Florida

Programming Language Translators

Regular Languages

We will study:

• Regular grammars• Relation to finite-state automata• Regular expressions• Equivalence among representations• Elimination on non-determinism• State minimization

Regular LanguagesDefinition: A grammar G = (Φ, Σ, P, S) is regular iff

either (but not both):Every production is of the form

A → or A → B (right linear)

Every production is of the form A → or A → B (left linear),where Σ*, and A, B Φ.

Regular LanguagesExamples:

G1: S → a R → abaU → bU → U Regular? Why?

→ bR U → b → S

G2: S → a R → Uaba → Ub U → b Regular? Why?

→ Rb → aS

Regular LanguagesLet’s devise a machine that accepts L(G1).

Observe thatS => a bU => bb …bR bS …

babaU1. Every sentential form (except sentences) has exactly one

nonterminal.2. The nonterminal occurs in the right-most position.3. Applicable productions depend only on that nonterminal.

=>=>

=>=>=>

Regular LanguagesEncode possible derivation sequences with a relation ⊢

on pairs of the form (q, ), whereq – current state – remaining string to accept

So, S → bU implies

(S, bβ) ⊢ (U, β)

State “sentential form ends in S”

“movesto”

state “sentential form ends in U”

Regular LanguagesDefine a graph, one node per nonterminal,

describing the possible actions on each sentential form. So,

S → bU implies ,

R → U implies ,

S → a implies .

S U

R U

S F

b

a

Regular LanguagesExample: S → a R → abaU U → b

→ bU → U →aS → bR

S

U

RF

aba

ε

b

b

a

b

a

Regular LanguagesIn general, Right-linear grammar → Transition diagram:

1. Nodes: Φ {f}, f Φ

2. if A → B

3. if A →

4.

A B

S

A Fα

α

Regular LanguagesExample: Is “babaa” in L(G)?

Node Input DerivationS babaa S =>U abaa bU =>S baa baS =>U aa babU =>S a babaS =>F babaa Yes.

Finite-State Automata

Definition: A (non-deterministic) finite-state automaton is a 5-tuple M = (Q, Σ, δ, s, F), where

Q is a finite set of states,

Σ is a finite set of transition symbols,

δ: Q x Σ {ε} → 2Q is a partial function called the transition function,

s Q is called the start state, and

F Q is the set of final states.

Finite-State AutomataAn FSA is the formal accepting mechanism of a regular

language. It requires that each transition be labeled by a string of length < 1.

The state diagram

is described by the FSA

S

U

RF

aba

ε

b

b

a

b

a

Finite-State Automata({S, R, U, F, X, Y}, {a, b}, δ, S, {F}), where

δ (S, a) = {F}δ (S, b) = {U, R} δ (R, ε) = {U}δ (R, a) = {X}δ (U, a) = {S}δ (U, b) = {F}δ (X, b) = {Y}δ (Y, a) = {U}

R X

X YY U

a

ba

Finite-State AutomataTWO “SYMPTOMS” OF NON-DETERMINISM:

Note: is not a problem

Fa

X

a

a X

ε

a1. 2.

Finite-State AutomataAdvantages of FSA’s:

Question: What language does the following grammar generate?

S → aA A → aB B → aC → ε → E → DC → bD D → bE E → bS

Difficult to see. Try FSA.

Finite-State Automata

Answer: L*, where L = {ab, aabb, aaabbb}

Summary: FSA’s are as powerful as right-linear regular grammars.

Are they more powerful? No. Can transform FSA → RGR.

Fε

S A

E D C

B

bεb

aa

ε

ba

Finite-State Automata

Transition Diagram (FSA) → Right-linear regular grammar

1. Φ = Q

2. A → aB if B δ (A, a)

3. A → a if f δ (A, a), and f F

4. Start symbol = Start state

Finite-State AutomataExample:

FSA:

RGR: A → aBB → bBD → cE → a → bD → c

→ bE → F F → dGG → H

→ εH → A

Conclusion: Right-linear regular grammars and FSA’s are equivalent.

A

G Fεε

ba

dH

B D E

bc

ε

Finite-State AutomataRelationship between Left-linear regular grammars and FSA’s:

Example: F → Sa U → Sb R → Sb → Ub → R S → Ua → Raba →

Derivations: Sbb ...F => Ub => Rb ... Rabab ... Sa => Uaa ... a

=>

=>

=>

=>

Finite-State AutomataSimilarities with right-linear grammars:1. Sentential forms have at most one nonterminal.2. Sentences have none.3. Applicable productions depend only on the one

nonterminal.

Differences with right-linear grammars:1. Nonterminals appears on left-most position.2. String generated right-to-left, versus left-to-right for right-linear grammars.

Finite-State AutomataLeft-linear Regular Grammar → FSA

1. if A → B.

2. if A → , S’ is a new start state.

3. F = {S}, S is the start symbol.

B A

S’ Aα

α

Finite-State AutomataExample:

F → Sa U → SB S → Ua → Ub → R → ε

R → Sb → Raba

S

U

RF

aba

ε

b

b

a

b

a

S’ε

Finite-State AutomataState Input DerivationS’ babaaa babaaaS babaaa Sbabaaa <=R abaaa Rabaaa <=U aa Uaa <=S a Sa <=F F

Finite-State Automata

FSA → Left-linear Regular Grammar:

1. A → B if

2. A → if

3. S’ → F if

B A

S Aα

α

F

New start symbol

Finite-State AutomataSummarizing:

RGR RGL

RE FSA

Note: Beware of attempts at direct conversion between left and right-linear grammars.

Done

Soon