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2.1 Language

꼭 기억해야 할 세 가지 개념 1. 언어의 정의 2. 문법의 정의 및 개념 3. 인식기의 의미. 2.1 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}

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2.1 Language

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  1. 꼭 기억해야 할 세 가지 개념 1. 언어의 정의 2. 문법의 정의 및 개념 3. 인식기의 의미 2.1 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|

  2. (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*

  3. (ωR)R=ω • 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) anrepresents n a's. a0 = ε (3) the reversalof a string ω, denoted ωR is the string ω written in reverse order: i.e., if ω = a1a2...an then ωR = anan-1...a1.

  4. (4) language product LL' = {xy| x ∈ L and y ∈ L'} (5) The powersof a language L are defined recursively by: L0 = {ε} Ln = LLn-1 for n  1. (6) L* : reflexivetransitiveclosure = L0 ∪ L1 ∪ L2 ∪ ...∪ Ln ∪… = (7) L+ : transitive closure = L1 ∪ L2 ∪... ∪ Ln ∪ ... = L* - L0

  5. 2.2 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)

  6. 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)

  7. [예] G = ( {S, A}, {a, b}, P, S )Text p.47 [예제 2.8] • P : S → aAS S → a A → SbA A → ba A → SS ⇒ S → aAS | a A → SbA | ba | SS

  8. Inroduction to FL theory * * + • 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.

  9. * * * * • 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 → )

  10. G1 = ( {S}, {a}, P, S )을 이용하여 L(G1) P : S → a | aS L (G1) = { an | n  1 } • Language design

  11. 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 }

  12. (===>) 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

  13. 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

  14. 1) 파스칼 언어의 상수 정의 부분: • 상수정의 부분은 CONST라는 예약어로 시작하며 하나의 상수 정의는 a=b의 형태를 갖는다. 여기서, a는 identifier를 b는 상수를 나타내는 terminal심벌이다. 상수정의부분은 선택적이며 각각의 상수정의는; 으로 구분한다. • 다음은 상수정의 부분의 예이다. CONST ON = TRUE; OFF = FALSE; EPSILON = 1.0E-10;

  15. 2) C 언어의 정수 선언 부분: • 정수선언 부분은 여러 개의 정수선언으로 구성되며 하나의 선언은 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.

  16. 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 [예제 2.16] 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.

  17. * * * (=>) 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 = 1일 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.

  18. * * (<=) 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.

  19. 2.3 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*.

  20. 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

  21. The Chomsky Hierarchy of Languages

  22. Languages & Recognizers

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