Regular Languages

1
Regular Languages
Programming Language Concepts Lecture 4

• Prepared by
• Manuel E. Bermúdez, Ph.D.
• Associate Professor
• University of Florida

2
Regular Languages

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

3
Regular Languages

• Definition A grammar G (F, S, 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 ? ? S, and A, B ? F.

4
Regular Languages

• Examples
• 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

5
Regular Languages

• Lets devise a machine that accepts L(G1).
• Observe that
• S gt a
• bU gt bb
• bR bS
• babaU
• Every sentential form (except sentences) has
  exactly one nonterminal.
• The nonterminal occurs in the right-most
  position.
• Applicable productions depend only on that
  nonterminal.

gt
gt
gt
gt
gt
6
Regular Languages

• Encode possible derivation sequences with a
  relation ? on pairs of the form (q, ?), where
• q current state
• ? remaining string to accept
• So, S ? bU implies
• (S, bß) ? (U, ß)

State sentential form ends in S
moves to
state sentential form ends in U
7
Regular Languages

• Define a graph, one node per nonterminal,
  describing the possible actions on each
  sentential form. So,
• S ? bU implies ,
• R ? U implies ,
• S ? a implies .

b
S
U
?
R
U
a
S
F
8
Regular Languages

• Example S ? a R ? abaU U ? b
• ? bU ? U ?aS
• ? bR

S
b
a
R
b
e
a
F
aba
b
U
9
Regular Languages

• In general,
• Right-linear grammar ? Transition diagram
• Nodes F ? f, f ? F
• if A ? ? B
• if A ? ?

a
A
B
a
A
F
S
10
Regular Languages

• Example Is babaa in L(G)?
• Node Input Derivation
• S babaa S gt
• U abaa bU gt
• S baa baS gt
• U aa babU gt
• S a babaS gt
• F babaa Yes.

11
Finite-State Automata

• Definition A (non-deterministic) finite-state
  automaton is a 5-tuple M (Q, S, d, s, F), where
• Q is a finite set of states,
• S is a finite set of transition symbols,
• d Q x S ? e ? 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.

12
Finite-State Automata

• An FSA is the formal accepting mechanism of a
  regular language. It requires that each
  transition be labeled by a string of length lt 1.
• The state diagram
• is
  described by the FSA

S
b
a
R
b
e
a
F
aba
b
U
13
Finite-State Automata

• (S, R, U, F, X, Y, a, b, d, S, F), where
• d (S, a) F
• d (S, b) U, R
• d (R, e) U
• d (R, a) X
• d (U, a) S
• d (U, b) F
• d (X, b) Y
• d (Y, a) U

a
R
X
b
X
Y
a
Y
U
14
Finite-State Automata

• TWO SYMPTOMS OF NON-DETERMINISM
• Note is not a problem

a
e
X
1.
2.
a
X
a
a
F
15
Finite-State Automata

• Advantages of FSAs
• Question What language does the following
  grammar generate?
• S ? aA A ? aB B ? aC
• ? e ? E ? D
• C ? bD D ? bE E ? bS
• Difficult to see. Try FSA.

16
Finite-State Automata
e
a
a
F
S
A
B
e
e
a
b
b
b
E
D
C

• Answer L, where L ab, aabb, aaabbb
• Summary FSAs are as powerful as right-linear
  regular grammars.
• Are they more powerful? No. Can transform
• FSA ? RGR.

17
Finite-State Automata

• Transition Diagram (FSA) ? Right-linear regular
  grammar
• F Q
• A ? aB if B ? d (A, a)
• A ? a if f ? d (A, a), and f ? F
• Start symbol Start state

18
Finite-State Automata

• Example
• FSA
• RGR A ? aB B ? bB D ? cE
• ? a ? bD ? c
• ? b
• E ? F F ? dG G ? H
• ? e
• H ? A
• Conclusion Right-linear regular grammars and
  FSAs are equivalent.

b
b
a
c
A
B
D
E
e
e
e
d
G
F
H
19
Finite-State Automata

• Relationship between Left-linear regular grammars
  and FSAs
• Example F ? Sa U ? Sb R ? Sb
• ? Ub ? R S ? Ua
• ? Raba ?
• Derivations Sbb ...
• F gt Ub gt Rb ...
• Rabab ...
• Sa gt Uaa ...
• a

gt
gt
gt
gt
20
Finite-State Automata

• Similarities with right-linear grammars
• Sentential forms have at most one nonterminal.
• Sentences have none.
• Applicable productions depend only on the one
  nonterminal.
• Differences with right-linear grammars
• Nonterminals appears on left-most position.
• String generated right-to-left, versus
• left-to-right for right-linear grammars.

21
Finite-State Automata

• Left-linear Regular Grammar ? FSA
• if A ? B ?.
• if A ? a, S is a new start state.
• F S, S is the start symbol.

a
B
A
a
S
A
22
Finite-State Automata

• Example
• F ? Sa U ? SB S ? Ua
• ? Ub ? R ? e
• R ? Sb ? Raba

S
S
b
a
R
b
e
a
F
aba
b
U
23
Finite-State Automata

• State Input Derivation
• S babaaa babaaa
• S babaaa Sbabaaa lt
• R abaaa Rabaaa lt
• U aa Uaa lt
• S a Sa lt
• F F

24
Finite-State Automata

• FSA ? Left-linear Regular Grammar
• A ? B? if
• A ? ? if
• S ? F if

a
B
A
a
S
A
F
New start symbol
25
Finite-State Automata

• Summarizing
• RGR RGL
• RE FSA
• Note Beware of attempts at diirect conversion
  between left and right-linear grammars.

Done
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