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Context-Free Languages & Grammars (Cfls & Cfgs)

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Context-Free Languages & Grammars (Cfls & Cfgs) Context-Free Languages & Grammars (CFLs & CFGs) Reading: Chapter 5 1 Not all languages are regular n So what happens to the languages which are not regular? n Can we still come up with a language recognizer? n i.e., something that will accept (or reject) strings that belong (or do not belong) to the language? 2 Context-Free Languages n A language class larger than the class of regular languages n Supports natural, recursive notation called “context- free grammar” n Applications: Context- n Parse trees, compilers Regular free n XML (FA/RE) (PDA/CFG) 3 An Example n A palindrome is a word that reads identical from both ends n E.g., madam, redivider, malayalam, 010010010 n Let L = { w | w is a binary palindrome} n Is L regular? n No. n Proof: N N (assuming N to be the p/l constant) n Let w=0 10 k n By Pumping lemma, w can be rewritten as xyz, such that xy z is also L (for any k≥0) n But |xy|≤N and y≠e + n ==> y=0 k n ==> xy z will NOT be in L for k=0 n ==> Contradiction 4 But the language of palindromes… is a CFL, because it supports recursive substitution (in the form of a CFG) n This is because we can construct a “grammar” like this: Same as: 1. A ==> e A => 0A0 | 1A1 | 0 | 1 | e Terminal 2. A ==> 0 3. A ==> 1 Variable or non-terminal Productions 4. A ==> 0A0 5. A ==> 1A1 How does this grammar work? 5 How does the CFG for palindromes work? An input string belongs to the language (i.e., accepted) iff it can be generated by the CFG G: n Example: w=01110 A => 0A0 | 1A1 | 0 | 1 | e n G can generate w as follows: Generating a string from a grammar: 1. A => 0A0 1. Pick and choose a sequence 2. => 01A10 of productions that would allow us to generate the 3. => 01110 string. 2. At every step, substitute one variable with one of its productions. 6 Context-Free Grammar: Definition n A context-free grammar G=(V,T,P,S), where: n V: set of variables or non-terminals n T: set of terminals (= alphabet U {e}) n P: set of productions, each of which is of the form V ==> a1 | a2 | … n Where each ai is an arbitrary string of variables and terminals n S ==> start variable CFG for the language of binary palindromes: G=({A},{0,1},P,A) P: A ==> 0 A 0 | 1 A 1 | 0 | 1 | e 7 More examples n Parenthesis matching in code n Syntax checking n In scenarios where there is a general need for: n Matching a symbol with another symbol, or n Matching a count of one symbol with that of another symbol, or n Recursively substituting one symbol with a string of other symbols 8 Example #2 n Language of balanced paranthesis e.g., ()(((())))((()))…. n CFG? G: S => (S) | SS | e How would you “interpret” the string “(((()))()())” using this grammar? Sà SS à (S) à (SS)à((S)SS)à (((S))(S)(S))à (((()))()()) 9 Example #3 m n n A grammar for L = {0 1 | m≥n} n CFG? G: S => 0S1 | A A => 0A | e How would you interpret the string “00000111” using this grammar? Sà0S1à 0 0S1 1à 0 00S11 1à 0 000A11 1à 0 0000A11 1à 000001111 10 Example #4 A program containing if-then(-else) statements if Condition then Statement else Statement (Or) if Condition then Statement CFG? 11 More examples n n L1 = {0 | n≥0 } n n L2 = {0 | n≥1 } i j k n L3={0 1 2 | i=j or j=k, where i,j,k≥0} i j k n L4={0 1 2 | i=j or i=k, where i,j,k≥1} 12 Applications of CFLs & CFGs n Compilers use parsers for syntactic checking n Parsers can be expressed as CFGs 1. Balancing paranthesis: n B ==> BB | (B) | Statement n Statement ==> … 2. If-then-else: n S ==> SS | if Condition then Statement else Statement | if Condition then Statement | Statement n Condition ==> … n Statement ==> … 3. C paranthesis matching { … } 4. Pascal begin-end matching 5. YACC (Yet Another Compiler-Compiler) 13 More applications n Markup languages n Nested Tag Matching n HTML n <html> …<p> … <a href=…> … </a> </p> … </html> n XML n <PC> … <MODEL> … </MODEL> .. <RAM> … </RAM> … </PC> 14 Tag-Markup Languages Roll ==> <ROLL> Class Students </ROLL> Class ==> <CLASS> Text </CLASS> Text ==> Char Text | Char Char ==> a | b | … | z | A | B | .. | Z Students ==> Student Students | e Student ==> <STUD> Text </STUD> Here, the left hand side of each production denotes one non-terminals (e.g., “Roll”, “Class”, etc.) Those symbols on the right hand side for which no productions (i.e., substitutions) are defined are terminals (e.g., ‘a’, ‘b’, ‘|’, ‘<‘, ‘>’, “ROLL”, etc.) 15 Structure of a production head derivation body A =======> a1 | a2 | … | ak The above is same as: 1. A ==> a1 2. A ==> a2 3. A ==> a3 … K. A ==> ak 16 CFG conventions n Terminal symbols <== a, b, c… n Non-terminal symbols <== A,B,C, … n Terminal or non-terminal symbols <== X,Y,Z n Terminal strings <== w, x, y, z n Arbitrary strings of terminals and non- terminals <== a, b, g, .. 17 Syntactic Expressions in Programming Languages result = a*b + score + 10 * distance + c terminals variables Operators are also terminals Regular languages have only terminals n Reg expression = [a-z][a-z0-1]* n If we allow only letters a & b, and 0 & 1 for constants (for simplification) n Regular expression = (a+b)(a+b+0+1)* 18 String membership How to say if a string belong to the language defined by a CFG? 1. Derivation n Head to body Both are equivalent forms 2. Recursive inference n Body to head G: Example: A => 0A0 | 1A1 | 0 | 1 | e n w = 01110 A => 0A0 n Is w a palindrome? => 01A10 => 01110 19 Simple Expressions… n We can write a CFG for accepting simple expressions n G = (V,T,P,S) n V = {E,F} n T = {0,1,a,b,+,*,(,)} n S = {E} n P: n E ==> E+E | E*E | (E) | F n F ==> aF | bF | 0F | 1F | a | b | 0 | 1 20 Generalization of derivation n Derivation is head ==> body n A==>X (A derives X in a single step) n A ==>*G X (A derives X in a multiple steps) n Transitivity: IF A ==>*GB, and B ==>*GC, THEN A ==>*G C 21 Context-Free Language n The language of a CFG, G=(V,T,P,S), denoted by L(G), is the set of terminal strings that have a derivation from the start variable S. n L(G) = { w in T* | S ==>*G w } 22 Left-most & Right-most G: Derivation Styles E => E+E | E*E | (E) | F F => aF | bF | 0f | 1f | e * Derive the string a*(ab+10) from G: E = =>G a*(ab+10) nE nE n==> E * E n==> E * E Left-most n==> F * E n==> E * (E) Right-most n==> aF * E n==> E * (E + E) derivation: derivation: n==> a * E n==> E * (E + F) n==> a * (E) n==> E * (E + 1F) Always Always substitute n==> a * (E + E) n==> E * (E + 10F) n==> a * (F + E) n==> E * (E + 10) substitute leftmost n==> a * (aF + E) n==> E * (F + 10) rightmost variable n==> a * (abF + E) n==> E * (aF + 10) variable n==> a * (ab + E) n==> E * (abF + 0) n==> a * (ab + F) n==> E * (ab + 10) n==> a * (ab + 1F) n==> F * (ab + 10) n==> a * (ab + 10F) n==> aF * (ab + 10) n==> a * (ab + 10) n==> a * (ab + 10) 23 Leftmost vs. Rightmost derivations Q1) For every leftmost derivation, there is a rightmost derivation, and vice versa. True or False? True - will use parse trees to prove this Q2) Does every word generated by a CFG have a leftmost and a rightmost derivation? Yes – easy to prove (reverse direction) Q3) Could there be words which have more than one leftmost (or rightmost) derivation? Yes – depending on the grammar 24 How to prove that your CFGs are correct? (using induction) 25 Gpal: CFG & CFL A => 0A0 | 1A1 | 0 | 1 | e n Theorem: A string w in (0+1)* is in L(Gpal), if and only if, w is a palindrome. n Proof: n Use induction n on string length for the IF part n On length of derivation for the ONLY IF part 26 Parse trees 27 Parse Trees n Each CfG can be represented using a parse tree: n Each internal node is labeled by a variable in V n Each leaf is terminal symbol n for a production, A==>X1X2…Xk, then any internal node labeled A has k children which are labeled from X1,X2,…Xk from left to right Parse tree for production and all other subsequent productions: A ==> X1..Xi..Xk A X1 … Xi … Xk 28 Examples E A E + E 0 A 0 f f 1 A 1 a 1 e Derivation Recursiveinference Parse tree for 0110 Parse tree for a + 1 G: G: E => E+E | E*E | (E) | F A => 0A0 | 1A1 | 0 | 1 | e F => aF | bF | 0f | 1f | 0 | 1 | a | b 29 Parse Trees, Derivations, and Recursive Inferences Production: A ==> X ..X ..X A 1 i k X1 … Xi … Xk Derivation Recursive inference Left-most Parse tree derivation Derivation Right-most derivation Recursive inference 30 Interchangeability of different CFG representations n Parse tree ==> left-most derivation n DFS left to right n Parse tree ==> right-most derivation n DFS right to left n ==> left-most derivation == right-most derivation n Derivation ==> Recursive inference n Reverse the order of productions n Recursive inference ==> Parse trees n bottom-up traversal of parse tree 31 Connection between CFLs and RLs 32 What kind of grammars result for regular languages? CFLs & Regular Languages n A CFG is said to be right-linear if all the productions are one of the following two forms: A ==> wB (or) A ==> w Where: • A & B are variables, • w is a string of terminals n Theorem 1: Every right-linear CFG generates a regular language n Theorem 2: Every regular language has a right-linear grammar n Theorem 3: Left-linear CFGs also represent RLs 33 Some Examples 0 1 0,1 0 1 ØA => 01B | C 1 0 B => 11B | 0C | 1A A B C 1 0 A B 1 C C => 1A | 0 | 1 0 Right linear CFG? Right linear CFG? Finite Automaton? 34 Ambiguity in CFGs and CFLs 35 Ambiguity in CFGs n A CFG is said to be ambiguous if there exists a string which has more than one left-most derivation Example: S ==> AS | e LM derivation #1: LM derivation #2: S => AS S => AS A ==> A1 | 0A1 | 01 => 0A1S => A1S =>0A11S => 0A11S => 00111S => 00111S Input string: 00111 => 00111 => 00111 Can be derived in two ways 36 Why does ambiguity matter? Values are E ==> E + E | E * E | (E) | a | b | c | 0 | 1 different !!! string = a * b + c E • LM derivation #1: •E => E + E => E * E + E E + E (a*b)+c ==>* a * b + c E * E c a b E • LM derivation #2 •E => E * E => a * E => E * E a*(b+c) a * E + E ==>* a * b + c a E + E The calculated value depends on which b c of the two parse trees is actually used.
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