Lexical Analysis

Challenge the future
Delft
University of
Technology
Course IN4303
Compiler Construction
Eduardo Souza, Guido Wachsmuth, Eelco Visser
Lexical Analysis

today’s lecture
Lexical Analysis
Overview
2

today’s lecture
Lexical Analysis
Overview
Lexical analysis
2

today’s lecture
Lexical Analysis
Overview
Lexical analysis
Regular languages
• regular grammars
• regular expressions
• finite state automata
2

today’s lecture
Lexical Analysis
Overview
Lexical analysis
Regular languages
Equivalence of formalisms
• constructive approach
2

today’s lecture
Lexical Analysis
Overview
Lexical analysis
Regular languages
Equivalence of formalisms
• constructive approach
Tool generation
2

Lexical Analysis
Regular Grammars
I
3

formal languages
Lexical Analysis 4
Recap: A Theory of Language

formal languages
Lexical Analysis
vocabulary Σ
finite, nonempty set of elements (words, letters)
alphabet
4

formal languages
Lexical Analysis
vocabulary Σ
alphabet
string over Σ
finite sequence of elements chosen from Σ
word, sentence, utterance
4

formal languages
Lexical Analysis
vocabulary Σ
alphabet
string over Σ
finite sequence of elements chosen from Σ
word, sentence, utterance
formal language λ
set of strings over a vocabulary Σ
λ ⊆ Σ*
4

formal grammars
Lexical Analysis 5

formal grammars
Lexical Analysis
formal grammar G = (N, Σ, P, S)
nonterminal symbols N
terminal symbols Σ
production rules P ⊆ (N∪Σ)*
N (N∪Σ)*
× (N∪Σ)*
start symbol S∈N
5

formal grammars
Lexical Analysis
terminal symbols Σ
N (N∪Σ)*
× (N∪Σ)*
start symbol S∈N
5
nonterminal symbol

formal grammars
Lexical Analysis
terminal symbols Σ
N (N∪Σ)*
× (N∪Σ)*
start symbol S∈N
5
context

formal grammars
Lexical Analysis
terminal symbols Σ
N (N∪Σ)*
× (N∪Σ)*
start symbol S∈N
5
replacement

formal grammars
Lexical Analysis
terminal symbols Σ
N (N∪Σ)*
× (N∪Σ)*
start symbol S∈N
grammar classes
type-0, unrestricted
type-1, context-sensitive: (a A c, a b c)
type-2, context-free: P ⊆ N × (N∪Σ)*
type-3, regular: (A, x) or (A, xB)
5

right regular grammar
Lexical Analysis 6
Num → “0” Num
Num → “1” Num
Num → “2” Num
Num → “3” Num
Num → “4” Num
Num → “5” Num
Num → “6” Num
Num → “7” Num
Num → “8” Num
Num → “9” Num
Num → “0”
Num → “1”
Num → “2”
Num → “3”
Num → “4”
Num → “5”
Num → “6”
Num → “7”
Num → “8”
Num → “9”
Decimal Numbers

Lexical Analysis 7
Id → “a” R
…
Id → “z” R
R → “a” R
…
R → “z” R
R → “0” R
…
R → “9” R
Id → “a”
…
Id → “z”
R → “a”
…
R → “z”
R → “0”
…
R → “9”
Identifiers

formal languages
Lexical Analysis 8

formal languages
Lexical Analysis
8

formal languages
Lexical Analysis
derivation relation G ⊆ (N∪Σ)*
× (N∪Σ)*
w G w’
∃(p, q)∈P: ∃u,v∈(N∪Σ)*
:
w=u p v ∧ w’=u q v
8

formal languages
Lexical Analysis
× (N∪Σ)*
w G w’
∃(p, q)∈P: ∃u,v∈(N∪Σ)*
:
formal language L(G) ⊆ Σ*
L(G) = {w∈Σ*
| S G
*
w}
8

formal languages
Lexical Analysis
× (N∪Σ)*
w G w’
∃(p, q)∈P: ∃u,v∈(N∪Σ)*
:
formal language L(G) ⊆ Σ*
L(G) = {w∈Σ*
| S G
*
w}
classes of formal languages
8

Lexical Analysis 9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
S → aA
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
S → aA
A → aB
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
S → aA
A → aB
B → aC
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
S → aA
A → aB
B → aC
C → aD
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
G:
S → a
S → aA
A → aB
B → aC
C → aD
D → aS
9
Example
Is G regular? 
L(G) = ?

Lexical Analysis
Regular Expressions
II
10

overview
Lexical Analysis
Recap: Regular Expressions
basics
• symbol from an alphabet
• ε
combinators
• alternation: E1 | E2
• concatenation: E1 E2
• repetition: E*
• optional: E? = E | ε
• one or more: E+ = E E*
11

regular expressions
Lexical Analysis 12
Num: (0|1|2|3|4|5|6|7|8|9)+
Id: (a|…|z)(a|…|z|0|…|9)*
Decimal Numbers & Identifiers

formal languages
Lexical Analysis
Regular Expressions
basics
• L(a) = {“a”}
• L(ε) = {“”}
combinators
• L(E1 | E2) = L(E1) ∪ L(E2)
• L(E1 E2) = L(E1) · L(E2)
• L(E*) = L(E)*
13

regular expressions
Lexical Analysis 14
Num: (0|1|2|3|4|5|6|7|8|9)+
A valid Num is any word that consists of a sequence of
one or more digits.
Id: (a|…|z)(a|…|z|0|…|9)*
A valid Id is any word that starts with a lowercase letter
followed by sequence of zero or more lowercase letters or
digits.

Lexical Analysis 15
Example
Gr:

Lexical Analysis 15
Example
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*

Lexical Analysis 15
Example
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*
L(Gr) = ?

Lexical Analysis 16
Example
Gr:

Lexical Analysis 16
Example
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*

Lexical Analysis 16
Example
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*
L(Gr) = The set of words that start with
zero or more capital letters, followed by
zero or more decimal digits.

Lexical Analysis
Finite Automata
III
17

formal definition
Lexical Analysis 18
Finite Automata

formal definition
Lexical Analysis
finite automaton M = (Q, Σ, T, q0, F)
states Q
input symbols Σ
transition function T
start state q0∈Q
final states F⊆Q
18
Finite Automata

formal definition
Lexical Analysis
states Q
input symbols Σ
start state q0∈Q
final states F⊆Q
transition function
nondeterministic FA T : Q × Σ → P(Q)
NFA with ε-moves T : Q × (Σ ∪ {ε}) → P(Q)
deterministic FA T : Q × Σ → Q
18
Finite Automata

finite automata
Lexical Analysis
19
1 2
0-9
0-9
1 2
a-z a-z
0-9

formal languages
Lexical Analysis 20
Nondeterministic Finite Automata

formal languages
Lexical Analysis
20

formal languages
Lexical Analysis
transition function T : Q × Σ → P(Q)
T({q1,..., qn}, x) := T(q1, x) ∪ … ∪ T(qn, x)
T*({q1,..., qn}, ε) := {q1,..., qn}
T*({q1,..., qn}, xw) := T*(T({q1,..., qn}, x), w)
20

formal languages
Lexical Analysis
transition function T : Q × Σ → P(Q)
T({q1,..., qn}, x) := T(q1, x) ∪ … ∪ T(qn, x)
T*({q1,..., qn}, ε) := {q1,..., qn}
T*({q1,..., qn}, xw) := T*(T({q1,..., qn}, x), w)
formal language L(M) ⊆ Σ*
L(M) = {w∈Σ*
| T*
({q0}, w) ∩ F ≠ ∅}
20

formal languages
Lexical Analysis 21
3 4
f
1 2
a-z a-z
0-9
i

formal languages
Lexical Analysis 22
Deterministic Finite Automata

formal languages
Lexical Analysis
22

formal languages
Lexical Analysis
transition function T : Q × Σ → Q
T*(q, ε) := q
T*(q, xw) := T*(T(q, x), w)
22

formal languages
Lexical Analysis
transition function T : Q × Σ → Q
T*(q, ε) := q
T*(q, xw) := T*(T(q, x), w)
formal language L(M) ⊆ Σ*
L(M) = {w∈Σ*
| T*
(q0, w) ∈ F}
22

formal languages
Lexical Analysis 23
1 2
0-9
0-9
1 2
a-z a-z
0-9

Lexical Analysis
Equivalence
IV
24

formalisms
Lexical Analysis
Regular Languages
25
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

formalisms
Lexical Analysis
Regular Languages
26
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

formalisms
Lexical Analysis
Regular Languages
27
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

Lexical Analysis 28
NFA construction

Lexical Analysis
28
NFA construction

Lexical Analysis
finite automaton M = (N ∪ {f}, Σ, T, S, F)
28
NFA construction

Lexical Analysis
(X, aY)∈P : (X, a, Y)∈T
(X, a)∈P : (X, a, f)∈T
28
NFA construction

Lexical Analysis
(X, aY)∈P : (X, a, Y)∈T
(X, a)∈P : (X, a, f)∈T
final states F
(S, ε)∈P : F = {S, f}
else: F = {f}
28
NFA construction

Example
Lexical Analysis 29
Num → “0” Num
Num → “1” Num
Num → “2” Num
Num → “3” Num
Num → “4” Num
Num → “5” Num
Num → “6” Num
Num → “7” Num
Num → “8” Num
Num → “9” Num
Num → “0”
Num → “1”
Num → “2”
Num → “3”
Num → “4”
Num → “5”
Num → “6”
Num → “7”
Num → “8”
Num → “9”
NFA construction

Example
Lexical Analysis 29
Num → “0” Num
Num → “1” Num
Num → “2” Num
Num → “3” Num
Num → “4” Num
Num → “5” Num
Num → “6” Num
Num → “7” Num
Num → “8” Num
Num → “9” Num
Num → “0”
Num → “1”
Num → “2”
Num → “3”
Num → “4”
Num → “5”
Num → “6”
Num → “7”
Num → “8”
Num → “9”
NFA construction
N
f

Example
Lexical Analysis 29
Num → “0” Num
Num → “1” Num
Num → “2” Num
Num → “3” Num
Num → “4” Num
Num → “5” Num
Num → “6” Num
Num → “7” Num
Num → “8” Num
Num → “9” Num
Num → “0”
Num → “1”
Num → “2”
Num → “3”
Num → “4”
Num → “5”
Num → “6”
Num → “7”
Num → “8”
Num → “9”
NFA construction
N
f
0-9

Example
Lexical Analysis 29
Num → “0” Num
Num → “1” Num
Num → “2” Num
Num → “3” Num
Num → “4” Num
Num → “5” Num
Num → “6” Num
Num → “7” Num
Num → “8” Num
Num → “9” Num
Num → “0”
Num → “1”
Num → “2”
Num → “3”
Num → “4”
Num → “5”
Num → “6”
Num → “7”
Num → “8”
Num → “9”
NFA construction
N
f
0-9
0-9

Example
Lexical Analysis
(X, aY)∈P : (X, a, Y)∈T
(X, a)∈P : (X, a, f)∈T
final states F
(S, ε)∈P : F = {S, f}
else: F = {f}
30
NFA construction
G:
S → a
S → aA
A → aB
B → aC
C → aD
D → aS

formalisms
Lexical Analysis
Regular Languages
31
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

regular expressions
Lexical Analysis
NFA construction
32
a
a
(x) εε
x
x*
ε
x
ε

regular expressions
Lexical Analysis
NFA construction
33
x|y
x
xy
y
ε
ε
ε
ε
εε
x y
ε

regular expressions
Lexical Analysis
NFA construction
34
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*

formalisms
Lexical Analysis
Regular Languages
35
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

ε elimination
Lexical Analysis
additional final states
• states with ε-moves into final states
• become final states themselves
additional transitions
• ε-move from source to target state
• transitions from target state
• add these transitions to the source state
36
NFA construction

ε elimination
Lexical Analysis
NFA construction
additional final states
• states with ε-moves into final states
• become final states themselves
additional transitions
• ε-move from source to target state
• transitions from target state
• add these transitions to the source state
37
Gr:
S → ('A'= | … | 'Z')*(0 | … | 9)*

formalisms
Lexical Analysis
Regular Languages
38
right regular
grammars
left regular
grammars
regular
expressions
NFAs DFAs
NFAs with
ε-moves

eliminating nondeterminism
Lexical Analysis
nondeterministic finite automaton M = (Q, Σ, T, q0, F)
deterministic finite automaton M’ = (P(Q), Σ, T’, {q0}, F’)
transition function T’
• T’({q1,..., qn}, x) = T({q1,..., qn}, x) = T(q1, x) ∪ … ∪ T(qn, x)
final states F’ = {S⎮S⊆Q, S∩F ≠ ∅}
• all states that include a final state of the original NFA
39
Powerset construction

example
Lexical Analysis
40
3 4
f
1 2
a-z a-z
0-9
i

example
Lexical Analysis
41
3 4
f
1 2
a-z a-z
0-9
i
1 2
a-h
j-z
a-z
0-9
24
f
i
23
a-e,g-z
0-9
a-z
0-9

lessons learned
Lexical Analysis
Summary
43

lessons learned
Lexical Analysis
Summary
What are the formalisms to describe regular languages?
43

lessons learned
Lexical Analysis
Summary
Why are these formalisms equivalent?
• constructive proofs
43

lessons learned
Lexical Analysis
Summary
Why are these formalisms equivalent?
• constructive proofs
How can we generate compiler tools from that?
• implement DFAs
43

learn more
Lexical Analysis
Literature
44

learn more
Lexical Analysis
Literature
Formal languages
Noam Chomsky: Three models for the description of language. 1956
J. E. Hopcroft, R. Motwani, J. D. Ullman: Introduction to Automata
Theory, Languages, and Computation. 2006
44

learn more
Lexical Analysis
Literature
Formal languages
Noam Chomsky: Three models for the description of language. 1956
J. E. Hopcroft, R. Motwani, J. D. Ullman: Introduction to Automata
Theory, Languages, and Computation. 2006
Lexical analysis
Andrew W. Appel, Jens Palsberg: Modern Compiler Implementation in
Java, 2nd edition. 2002
44

Lexical Analysis
Copyrights
45

copyrights
Lexical Analysis
Pictures
Slide 1:
Book Scanner by Ben Woosley, some rights reserved
Slides 4, 5, 8:
Noam Chomsky by Fellowsisters, some rights reserved
Slide 6, 7, 11, 15:
Tiger by Bernard Landgraf, some rights reserved
Slide 18:
Coffee Primo by Dominica Williamson, some rights reserved
Slide 32:
Pine Creek by Nicholas, some rights reserved
47

Lexical Analysis

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Lexical Analysis