5/10/16. Grammar. Automata and Languages. Today s Topics. Grammars Definition A grammar G is defined as G = (V, T, P, S) where:

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1 Grammar Automata and Languages Grammar Prof. Mohamed Hamada oftware Engineering Lab. The University of Aizu Japan Regular Grammar Context-free Grammar Context-sensitive Grammar Left-linear Grammar right-linear Grammar 2 Today s Topics Right-Linear Left-Linear Regular Derivation: Leftmost & Rightmost A grammar G is defined as G = (V, T, P, where: V : Finite set of variables/non-terminals (We use capital letters A,B,C, for variables) T : Alphabet/Finite set of terminals (We use small letters b,c, for terminals) P : Finite set of rules/productions : tart symbol V V T = φ Rule : α β + α (V T), β (V T) * Derivation Tree 3 Each grammar G defines a language L(G), which is the set of strings in T* (=Σ*) that G can generate from. It is all about the production rules. 4 1

2 Given a grammar G = (V, T, P, For a string w=uxv we can apply the production rule xày to w so we get a string z=uyv. In this case we write w z, which reads w drives z. If w 1 w 2 w n, we say that w 1 drives w n and we write w 1 * w n 5 Example Given a grammar G = (V, T,P, V={A, B, C} T={ b, x} = A And P is: AaBxàaBAaBb CaBxàaBAaCb ABCàλ 6 Let G=(V, Τ, P, be a grammar. w ( V T)* is a sentential form, if * G w w T* is a sentence, if The language of G, L(G) = { w * T* w G } * G w ome Remarks The language L(G) = { w T* : * w } contains only strings of terminals, not variables. Notation: We summarize several rules for one variable: A B A 01 by A B 01 AA A AA 7 8 2

3 Example Given the grammar: G = {{ },{ b},{ λ, ab}, Right-Linear A Grammar G= (V, T,P, is called rightlinear grammar if every production is of the form A à xb, or A à x where A,B V, x T* The language generated by this grammar is: n n L( G) = { a b n 0} 9 Example: The grammar x à 0x 1y y à 0x 1z z à 0x 1z λ Is a right-linear grammar. 10 Left-Linear Regular A Grammar G= (V, T,P, is called leftlinear grammar if every production is of the form A à Bx, or A à x where A,B V, x T* Example: The grammar x à x0 y1 y à x0 z1 z à x0 z1 λ Is a left-linear grammar. 11 A Grammar G= (V, T,P, is called regular grammar if its is left- or right-linear Example: The grammar x à x0 y1 y à x0 z1 z à x0 z1 λ Is a left-linear grammar, hence is Regular Grammar Example: The grammar x à 0x 1y y à 0x 1z z à 0x 1z λ Is a right-linear grammar, Hence is Regular Grammar 12 3

4 Example 1 Write a grammar that generate the language: L = { w { b}* length( w) is EVEN} Example 2 E λ aae abe bae bbe E λ ao bo O ae be Write a grammar that generate the language: L = { w { b}* w has EVEN number of b's} E λ ae bo O ao be Example 3 Write a grammar that generate the language: L = { w { b, c}* w does not contain abc} reset b reset c reset a seena λ seena a seena c reset b seenab λ seenab a seena 13 b reset λ 14 CFG = (V, T, P, 15 V : Finite set of variables/non-terminals T : Alphabet/Finite set of terminals P : Finite set of rules/productions : tart symbol V V T = φ Rule : A ω A V ω (V T) * 16 4

5 Context-freeness: An A-rule can be applied whenever A occurs in a string, irrespective of the context (that is, nonterminals and terminals around A). Derivation One-step Derivation uav A ω uωv w is derivable from v in CFG, if there is a finite sequence of rule applications such that: v w... w = n w 1 17 In this case we can write this derivation as v * w 18 Derivation The derivation as v * w is called: Leftmost derivation: if in every step the leftmost variable is selected for reduction Rightmost derivation: if in every step the rightmost variable is selected for reduction Example 1 Let G = ({}, {b},,p) with for P: a and bb, and λ. ome derivations from this grammar: aa aaaa aabbaa aabbaa bb baab baab, and so on. In general. ww R for w {b}*. 19 L(G)={ww R : w {b}*} 20 5

6 Example 2 G = ({, A, B},{ b}, { AB, A aa λ, B Bb λ}, L( G) = L( a * b*) Leftmost Derivation : AB aab ab abb ab Rightmost Derivation : AB ABb Ab aab ab 21 Example 3 Take the CFG 0 1 ( ( ( ( (, which generates all proper Boolean formulas that use 0, 1,,,, ( and ). Then (0) ((0) (1)) can be derived in the following ways [leftmost] ( ( (0) ( (0) (( () (0) ((0) () (0) ((0) (1)) [rightmost] ( ( ( (( () ( (( (1)) ( ((0) (1)) (0) ((0) (1)) [something else] ( ( (0) ( (0) (( () (0) (( (1)) (0) ((0) (0)) 22 Example 4 Consider the CFG: G = {{ },{ b},{ λ, ab}, Derivation of aabb is ab aabb aabb Example 5 Consider the CFG G: L( B) = { b aa aba B bb b m n L( = { a b m > 0} m a n n > 0 m > 0} 23 L(G)= L( 24 6

7 Example 6 Consider the CFG G 1 : The language generated by G 1 is: aa B B bb λ n m n L( G1) = { a b a n 0 m 0} Consider the CFG G 2 : The language generated by G 2 is: abc λ n n L( G2 ) = {( ab) c n 0} 25 Example 7 G 1 : Consider the CFGs G 1 and G 2 : AB A aa a B bb λ n L( = { a b L( = L( a + m m 0 n > 0} * b ) G 2 : The language generated by G 1 and G 2 is: L(G 1 ) = L(G 2 ) = L( a ab B bb λ 26 Example 8 Write a CFG to generate the language: AbAbA A aa λ a * ba * ba * a B B ba A aa bc C ac λ Left to right generation of string. Exercise 1 WRITE A CFG FOR THE EMPTY ET G = { {}, Σ,, }

8 Exercise 2 What is the CFG ({},{(,)}, P, that produces the language of correct parentheses like (), (()), or ()(())? ( λ Example Consider the CFG G=({,Z},{0,1}, P, with P: 01 0Z1 Z 0Z λ What is the language generated by G? Answer: L(G) = {0 i 1 j i j } pecifically, yields the 0 j+k 1 j according to: 01 0 j 1 j 0 j Z1 j 0 j 0Z1 j 0 j+k Z1 j 0 j+k ε1 j = 0 j+k 1 j Exercise Can you make Context Free for the following? a) { 0 n 1 n : n 0} b) { 0 n 1 m : n,m 0} c) Arithmetic b,c formulas like a+b c+a (without ()) Answers: a) 01 λ b) 0 R and R 1R λ c) a b c + 31 Derivation Tree For a CFG G=(V,T,,P) a derivation tree has the following properties: 1) The root is labeled 2) Each leaf is from T {λ} 3) Each interior node is from V 4) If node has label A V and ( its children a 1 a n (from L to R), then P must have the rule ) ( ) A a 1 a n (with a j V T {λ}) 5) A leaf labeled λ is a single child (has no siblings). 0 ( 0 ) ( 1 ) For partial derivation trees we have: 2a) Each leaf is from V T {λ} Example 32 8

9 Derivation Tree: Example Take the CFG 0 1 ( ( ( ( (, which generates all proper Boolean formulas that use 0, 1,,,, ( and ). The derivation * (0) ((0) (1)) can be expressed by the following derivation tree: 0 ( ) ( ) ( ) ( ) Derivation Tree: Notes Application of a production rule A x is represented by node A with children x. (Note that the tree is ordered: the ordering of the nodes matters.) The root has variable. The yield of is expressed by the leaves of the tree. 0 ( ) ( ) ( ) ( ) Derivation Tree: Notes Looking at a tree you see the derivation without the unnecessary information about its order. Theorem: Let G be a CFG. We have w L(G) if and only if there exists a derivation tree of G with yield w. Also, y is a sentential form of G if and only if there exists a partial derivation tree for G. Remember: the root always has to be. Example 1 Consider the CFG G: G = {{ },{ b},{ λ, ab}, The derivation of aabb is: ab aabb aabb Derivation tree is a b a λ b

10 Example 2 A 0A1 00A11 00B11 00#11 A A A B 0 0 # Example 3 <EXPR> <EXPR> + <EXPR> <EXPR> <EXPR> * <EXPR> <EXPR> ( <EXPR> ) <EXPR> a Build a parse tree for a + a * a <EXPR> <EXPR> <EXPR> <EXPR> <EXPR> a + a * a 38 Exercise WRITE A CFG FOR EVEN-LENGTH PALINDROME over Σ={b,c,d}? aa for all a Σ λ 39 10