Ch02

1. 제 2 장
형식 언어
컴파일러 입문

2. 목 차
2.1 언어(Language)
2.2 문법(Grammar)
2.3 문법의 분류
Inroduction to FL theory
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꼭 기억해야 할 세 가지 개념
1. 언어의 정의
2. 문법의 정의 및 개념
3. 인식기의 의미

3. Language
 Basic definitions
(1) alphabet
 a finite set of symbols.
 ex) T1 = {ㄱ,ㄴ,ㄷ,...,ㅎ,ㅏ,ㅑ, … ,ㅡ,ㅣ}
T2 = {A,B,C, … ,Z, a, b, c, …, z}
T3 = {main, int, char, …, while}
(2) string(or sentence, word)
 a sequence of symbols from some alphabet T.
(3) length
 the number of symbols in the string.
 denoted by |ω|
Inroduction to FL theory
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4. (4) empty string
 a string consisting of no symbols.
 denoted by ε or λ.
(5) T*
denotes the set of all strings of symbols over the
alphabet T, including the empty string.
T+
= T*
- {ε}
T*
: T star
T+
: T dagger
(6) Language is any set of strings over an alphabet.(Text p.40)
(or A Language L over the alphabet T is a subset of T*
.)
L ⊆ T*
Inroduction to FL theory
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5.  Two problems
(1) How do we represent a language ?
If the language is finite, the answer is easy.
If the language is infinite, we are faced with the problem
of finding a finite representation for the language.
 Set description
 Grammar : Generating Scheme
 Recognizer : Recognition Scheme
(2) Does there exist a finite representation for every language ?
No !
 This is not always possible.
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L1 = {a, ab, ba, aba}
L2 = T* = {, a, b, aa, ab, ….}
L3 = {anbn | n1}
L4 = {wwR | wT*}

6.  More definitions
(1) concatenation
 u = a1a2a3...an, v = b1b2b3...bm , u • v = a1a2a3...anb1b2b3...bm
 u • v를 보통 uv로 표기.
 uε= u = εu
 ∀u,v ∈ T*
, uv ∈ T*
.
 |uv| = |u| + |v|
(2) an
represents n a's.
a0
= ε
(3) the reversal of a string ω, denoted ωR
is the string ω
written in reverse order:
i.e., if ω = a1a2...an then ωR
= anan-1...a1.
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(ωR)R=ω

7. (4) language product
LL' = {xy| x ∈ L and y ∈ L'}
(5) The powers of a language L are defined recursively by:
L0
= {ε}
Ln
= LLn-1
for n  1.
(6) L*
: reflexive transitive closure
= L0
∪ L1
∪ L2
∪ ...∪ Ln
∪… =
(7) L+
: transitive closure
= L1
∪ L2
∪... ∪ Ln
∪ ...
= L*
- L0


0i
i
L
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L = {a, ab, ba, aba}
L0 = {}
L2 = {aa, aab, aba, aaba, …}

8. Grammar
 Language
 문장(sentence)들을 원소로 갖는 집합
 언어를 어떻게 표현할 것인가 ?
 Grammar
 terminal : 정의된 언어의 알파벳
 nonterminal :
 스트링을 생성하는 데 사용되는 중간 과정의 심볼
 언어의 구조를 정의하는데 사용
 grammar symbol (V)
Inroduction to FL theory
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9. Inroduction to FL theory
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 G = (VN, VT, P, S)
 VN : a finite set of nonterminal symbols
 VT : a finite set of terminal symbols
VN ∩ VT =  , VN∪ VT = V
 P : a finite set of production rules
α → β, α∈ V+
, β∈ V*
lhs rhs
 S : start symbol(sentence symbol)

10.  [예] G = ( {S, A}, {a, b}, P, S ) Text p.47 [예 8]
 P : S → aAS S → a
A → SbA A → ba A → SS
⇒ S → aAS | a
A → SbA | ba | SS
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11.  Derivation
1. ⇒ : “directly produce” or “directly derive”
if α → β∈ P and  , δ∈ V*
then
 αδ ⇒  βδ
2. ⇒ : Suppose α1,α2,...,αn ∈ V*
and α1 ⇒α2 ⇒… ⇒αn,
then α1 ⇒ αn
(zero or more derivations)
3. ⇒ : one or more derivations.
cf) → : production rule에서 사용.
“may be replaced by”
⇒ : derivation할 때 사용한다.
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*
+
*

12. Ex. P : S → aA | bB | ε
A → bS
B → aS
S ⇒ abba 유도 과정
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S ⇒ aA (생성규칙 S → aA)
⇒ abS (생성규칙 A → bS)
⇒ abbB (생성규칙 S → bB)
⇒ abbaS (생성규칙 B → aS)
⇒ abba (생성규칙 S → )

13.  L(G) : Language generated by grammar G
L(G) = {ω | S ⇒ ω, ω ∈ VT
*
}
☞ ω is a sentential form of G if S ⇒ ω and ω ∈ V*
.
ω is a sentence of G if S ⇒ ω and ω ∈ VT
*
.
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*
*
*
S ⇒ aA ( 생성규칙 S → aA를 적용 )
⇒ abS ( 생성규칙 A → bS를 적용 )
⇒ abbB ( 생성규칙 S → bB를 적용 )
⇒ abbaS ( 생성규칙 B → aS를 적용 )
⇒ abba ( 생성규칙 S → 를 적용 )
S ⇒ abba
sentential form : aA, abS, abbB, abbaS, abba
sentence : abba

14.  G1 = ( {S}, {a}, P, S ) 을 이용하여 L(G1)
P : S → a | aS
L (G1) = { an
| n  1 }
 Language design
Grammar Language
generation
design
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15.  주어진 문법으로부터 생성되는 언어 발견 과정
 1. start symbol로부터 길이가 짧은 순으로 생성규칙 적용
 2. 생성된 문장들의 형태를 고려하여 일정한 규칙 발견
 G = ( {A, B, C}, {a, b, c}, P, A)
P : A → abc A → aBbc
Bb → bB Bc → Cbcc
bC → Cb aC → aaB
aC → aa
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L(G) = { anbncn | n  1 }

16. (===>) ex1) S → 0S1 | 01
ex2) S → aSb | c
ex3) A → aB
B → bB | b
ex4) O → a
O → aE
E → aO
ex5) A → abc A → aBbc
Bb → bB Bc → Cbcc
bC → Cb aC → aaB
aC → aa
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17.  문법 기술 방법
 embedded rule : S → aSb
 right-recursive rule : A → aA
 left-recursive rule : A → Aa
 L = { an
| n  0 } 일 때 문법 : A → aA | ε
 L = { an
| n  1 } 일 때 문법 : A → aA | a
 Embedded production A → aAb
ex1) L1 = { an
bn
| n  0 }
ex2) L2 = { 0i
1j
| i  j, i,j  1 }
ex3) Constructs of Conventional PL
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18. ex) C 언어의 정수 선언 부분 :
 정수선언 부분은 여러 개의 정수선언으로 구성되며 하나의 선
언은 int a,a,a;와 같은 형태를 갖는다. 여기서 a는 임의의
identifier를 나타낸다.
 그리고 ; 으로 각각의 선언을 구분한다. 예를 들어, int i,j; int sum;
과 같다.
※ 문법을 고안할 때, nonterminal의 이름은 구문 구조를
대변할 수 있는 명칭으로 쓰는 것이 바람직하다.
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19.  In order to prove that a grammar generates a
language L
i) Every sentence generated by the grammar is in L.
ii) Every string in L can be generated by the grammar.
교과서 55쪽
[예16]
proof) (=>) Every sentence derivable from S is balanced.
(<=) Every balanced string is derivable from S.
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G = ( { S }, { ( , ) }, {S → (S)S |ε}, S )
⇔ All strings of balanced parentheses.

20. (=>) Every sentence derivable from S is balanced.
(i.e., S ⇒ ω, ω: balanced)
By induction on the number of steps in a derivation.
i) n = 1 일 때, S ⇒ ε, ε is surely balanced.
ii) Suppose that all derivations of fewer than n steps
produce balanced sentences.
iii) Consider a leftmost derivation of exactly n steps.
S ⇒ (S)S ⇒ (x)S ⇒ (x)y
By the hypothesis x, y : balanced.
Thus (x)y balanced.
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21. (<=) Every balanced string is derivable from S.
By induction on the length of a string.
i) |ω| = 0, S ⇒ ε
(the empty string is derivable from S.)
ii) Assume that every balanced string of length less than 2n is derived from S.
iii) Consider a balanced string ω of length 2n.
Let (x) : shortest prefix of ω being balanced.
Thus ω = (x)y, where x, y : balanced.
Since |x|, |y | < 2n, they are derivable from S by inductive hypothesis.
Thus S ⇒ (S)S ⇒ (x)S ⇒ (x)y = ω
Therefore, (x)y is also derivable from S.
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22. Chomsky Hierarchy
 Noam Chomsky
 According to the form of the productions.
α → β ∈ P
 Type 0 : No restrictions(unrestricted grammar)
 Type 1 : Context-sensitive grammar(CSG).
 → β, |  |  | β|
 Type 2 : Context-free grammar(CFG).
A → , where A : nonterminal,  ∈ V*
.
 Type 3 : Regular grammar(RG).
A → tB or A → t, (right-linear)
A → Bt or A → t, (left-linear)
where, A, B : nonterminal, t ∈ VT
*.
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23.  REL (Recursively Enumerable Language)
CSL (Context Sensitive Language)
CFL (Context Free Language)
RL (Regular Language)
 Examples of Formal Language
 simple matching language : Lm = {an
bn
| n ≥ 0} CFL
 double matching language : Ldm = {an
bn
cn
| n ≥ 0} CSL
 mirror image language : Lmi = {ωωR
| ω ∈ VT
*
} CFL
 palindrome language : Lr = {ω | ω = ωR
} CFL
 parenthesis language : Lp = {ω | ω: balanced parenthesis} CFL
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24.  The Chomsky Hierarchy of Languages
unrestricted language
context-sensitive language
context-free language
regular language
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25.  Languages & Recognizers
Grammar Language Recognizer
type 0
(unrestricted)
recursively
enumerable set
Turing
machine
type 1
(context-sensitive)
context-sensitive
language
Linear Bounded
Automata
type 2
(context-free)
context-free
language
Pushdown
Automata
type 3
(regular)
regular
language
Finite Automata
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