Regular Language and Regular Grammar Objectives • Regular Expression and Regular Language • Regular Expression vs Regular Language • Regular Grammar Regular Expression Alphabet Σ 1. ∅, λ, a∈Σ are regular expressions (known as primitive regular expressions). 2. If r 1 and r 2 are regular expressions, so are r 1 + r 2 , r 1 . r 2 , r 1 * , and (r 1 ). Operator Precedence • parentheses • star-closure (*) • concatenation (.) • union (+) Languages Associated with Regular Expressions • Each regular expression stands for a set of strings of symbols in Σ  each regular expression represents a language, called regular language • r L(r) Example 3.1 • L(a) = {a} • L((a + b.c)* ) = {λ, a, bc, aa, abc, bca, bcbc, aaa, aabc, } • L(a + b +)  syntax error Regular Languages 1. L(∅) = {} 2. L(λ) = {λ} 3. L(a) = {a} 4. L(r 1 + r 2 ) = L(r 1 )∪L(r 2 ) 5. L(r 1 . r 2 ) = L(r 1 )L(r 2 ) 6. L(r 1 * ) = (L(r 1 )) * 7. L((r 1 )) = L(r 1 ) Example 3.2 L(a * . (a + b)) = L(a * ) L(a + b) = (L(a)) * (L(a)∪L(b)) = {λ, a, aa, aaa, }.{a, b} = {a, aa, aaa, , b, ab, aab, } Example 3.3 r = (a + b) * (a + bb) L(r) = ? Example 3.3 r = (a + b) * (a + bb) L(r) = {w| w ends with a or bb} [...]... no pair of consecutive zeros} r = (1 + 01)* (0 + λ) Equivalent Regular Expression r1 and r2 are equivalent iff L(r1) = L(r2) Example 3.8 r1 = a (b + c) r2 = a b + a c L(r1) = L(r2) = {ab, ac} Regular Expressions and Languages Given a regular expression r, there exists an NFA that accepts L(r)  L(r) is a regular language Primitive NFAs q0 q0 q0 q1 λ a NFA that accepts ∅ q1 NFA that accepts {λ} q1... NFAs (cont’d) λ M(r1) λ M(r2) λ Primitive NFAs (cont’d) NFA that accepts L(r1*) λ λ M(r1) λ λ Example 3.9 r = (a + bb)*(ba* + λ) λ a λ λ λ b λ b λ λ λ λ λ λ λ λ b λ λ λ a λ λ λ λ Languages and Regular Expressions Given a regular language L, there exists a regular expression r such that L = L(r) Generalized Transition Graph Generalized transition graph is a transition graph whose edges are labelled with... Expression r1 and r2 are equivalent iff L(r1) = L(r2) • Example 3.14 r1 = a (b + c) r2 = a b + a c L(r1) = L(r2) = {ab, ac} Grammar Recalled • Formal grammar: G = (V, T, S, P) V: finite set of variables T: finite set of terminal symbols S∈V: start variable P: finite set of productions Right-linear Grammar • Right-linear grammar: G = (V, T, S, P) Productions are of the form: A → xB A→x A, B ∈ V and x ∈... Right-linear grammar: G = (V, T, S, P) Productions are of the form: A → xB A→x A, B ∈ V and x ∈ T* Left-linear Grammar • Left-linear grammar: G = (V, T, S, P) Productions are of the form: A → Bx A→x A, B ∈ V and x ∈ T* . Regular Language and Regular Grammar Objectives • Regular Expression and Regular Language • Regular Expression vs Regular Language • Regular Grammar Regular Expression Alphabet. expressions). 2. If r 1 and r 2 are regular expressions, so are r 1 + r 2 , r 1 . r 2 , r 1 * , and (r 1 ). Operator Precedence • parentheses • star-closure (*) • concatenation (.) • union (+) Languages. Regular Expressions • Each regular expression stands for a set of strings of symbols in Σ  each regular expression represents a language, called regular language • r L(r) Example 3.1 • L(a) = {a} • L((a