컴파일러 입문 - :: 동국대학교 OCW · Inroduction to FL theory Page 3 Language Basic definitions

http://plac.dongguk.ac.kr

제 2 장

형식 언어

컴파일러 입문


Inroduction to FL theory Page 2

목 차

2.1 언어(Language)

2.2 문법(Grammar)

2.3 문법의 분류



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}

T3 = {auto, break, case, … , 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|

꼭 기억해야 할 세 가지 개념

1. 언어의 정의

2. 문법의 정의 및 개념

3. 인식기의 의미



(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*



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.



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. (ωR)R=ω

Text p. 41



(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



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)



Text p. 43 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)



[예] G = ( {S, A}, {a, b}, P, S ) Text p.43 [예 8]

P : S → aAS S → a

A → SbA A → ba A → SS ⇒ S → aAS | a A → SbA | ba | SS



Derivation 1. ⇒ : "directly produce" or "directly derive" if α → β∈ P and γ , δ∈ V* then γ αδ ⇒ γ 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.

*

+

*



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

*.

P : S → aA | bB | ε A → bS B → aS S ⇒ abba 유도 과정

S ⇒ aA (생성규칙 S → aA) ⇒ abS (생성규칙 A → bS) ⇒ abbB (생성규칙 S → bB) ⇒ abbaS (생성규칙 B → aS) ⇒ abba (생성규칙 S → ε)

*

*

*

*



G1 = ( {S}, {a}, P, S ) 을 이용하여 L(G1)

P : S → a | aS L (G1) = { an | n ≥ 1 } Language design

Grammar Languagegeneration

design

Text p. 46



G = ( {A, B, C}, {a, b, c}, P, A)

P : A → abc A → aBbc Bb → bB Bc → Cbcc bC → Cb aC → aaB aC → aa

L(G) = { anbncn | n ≥ 1 }



(===>) ex1) S → 0S1 | 01 ex2) S → aSb | c ex3) A → aB B → bB | b ex4) A → abc A → aBbc Bb → bB Bc → Cbcc bC → Cb aC → aaB aC → aa



Grammar Design

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 = { anbn | n ≥ 0 }

ex2) L2 = { 0i1j | i ≠ j, i,j ≥ 1 }

ex3) Constructs of Conventional PL



1) 파스칼 언어의 상수 정의 부분 :

상수정의 부분은 CONST라는 예약어로 시작하며 하나의 상수 정

의는 a=b의 형태를 갖는다. 여기서, a는 identifier를 b는 상수를

나타내는 terminal 심벌이다. 상수정의부분은 선택적이며 각각의

상수정의는 ; 으로 구분한다.

다음은 상수정의 부분의 예이다. CONST ON = TRUE; OFF = FALSE; EPSILON = 1.0E-10;



2) C 언어의 정수 선언 부분 :

정수선언 부분은 여러 개의 정수선언으로 구성되며 하나의 선언

은 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.



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 51쪽 [예16]

proof) (=>) Every sentence derivable from S is balanced. (<=) Every balanced string is derivable from S.

G = ( { S }, { ( , ) }, {S → (S)S |ε}, S ) ⇔ All strings of balanced parentheses.



(=>) 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.

* *

*



(<=) 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.

* *



Noam Chomsky According to the form of the productions.

α → β ∈ P Type 0 : No restrictions(unrestricted grammar) Type 1 : Context-sensitive grammar(CSG).

α → β, | α | ≤ | β| Type 2 : Context-free grammar(CFG).

A → α, where A : nonterminal, α ∈ 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

*.

Chomsky Hierarchy



REL (Recursively Enumerable Language)

CSL (Context Sensitive Language)

CFL (Context Free Language)

RL (Regular Language)

Examples of Formal Language

simple matching language : Lm = {anbn | n ≥ 0} CFL double matching language : Ldm = {anbncn | n ≥ 1} CSL mirror image language : Lmi = {ωωR | ω ∈ VT

*} CFL palindrome language : Lr = {ω | ω = ωR } CFL parenthesis language : Lp = {ω | ω: balanced parenthesis} CFL



The Chomsky Hierarchy of Languages

unrestricted language

context-sensitive language

context-free language

regular language



Languages & Recognizers

Grammar Language Recognizer

type 0(unrestricted)

type 1(context-sensitive)

type 2(context-free)

type 3(regular)

recursively enumerable set

context-sensitive language

context-free language

regular language

Turing Machine

Linear Bounded Automata

Pushdown Automata

Finite Automata

Documents

컴파일러 입문 - :: 동국대학교 OCW · Inroduction to FL theory Page 3 Language Basic definitions