Ambiguity of Grammar in Context Free Grammar - Review Sheet | CMSC 330, Study notes of Programming Languages

Download Ambiguity of Grammar in Context Free Grammar - Review Sheet | CMSC 330 and more Study notes Programming Languages in PDF only on Docsity! 1 CMSC 330: Organization of Programming Languages Context-Free Grammars CMSC 330 2 Review • Why should we study CFGs? • What are the four parts of a CFG? • How do we tell if a string is accepted by a CFG? • What’s a parse tree? CMSC 330 3 Review A sentential form is a string of terminals and non- terminals produced from the start symbol Inductively: – The start symbol – If
A is a sentential form for a grammar, where (
and   (N|)*), and A   is a production, then
 is a sentential form for the grammar • In this case, we say that
A derives
 in one step, which is written as
A 
 CMSC 330 4 Leftmost and Rightmost Derivation • Example: S  a | SbS String: aba Leftmost Derivation Rightmost Derivation S  SbS  abS  aba S  SbS  Sba  aba At every step, apply production At every step, apply production to leftmost non-terminal to rightmost non-terminal • Both derivations happen to have the same parse tree • A parse tree has a unique leftmost and a unique rightmost derivation • Not every string has a unique parse tree • Parse trees don’t show the order productions are applied CMSC 330 5 Another Example (cont’d) S  a | SbS • Is ababa in this language? A leftmost derivation S  SbS  abS  abSbS  ababS  ababa Another leftmost derivation S  SbS  SbSbS  abSbS  ababS  ababa CMSC 330 6 Ambiguity • A string is ambiguous for a grammar if it has more than one parse tree – Equivalent to more than one leftmost (or more than one rightmost) derivation • A grammar is ambiguous if it generates an ambiguous string – It’s can be hard to see this with manual inspection • Exercise: can you create an unambiguous grammar for S  a | SbS ? 2 CMSC 330 7 Are these Grammars Ambiguous? (1) S  aS | T T  bT | U U  cU |  (2) S  T | T T  Tx | Tx | x | x (3) S  SS | () | (S) CMSC 330 8 Ambiguity of Grammar (Example 3) • 2 different parse trees for the same string: ()()() • 2 distinct leftmost derivations : S ⇒ SS ⇒ SSS ⇒()SS ⇒()()S ⇒()()() S ⇒ SS ⇒ ()S ⇒()SS ⇒()()S ⇒()()() • We need unambiguous grammars to manage programming language semantics CMSC 330 9 More on Leftmost/Rightmost Derivations • Is the following derivation leftmost or rightmost? S  aS  aT  aU  acU  ac – There’s at most one non-terminal in each sentential form, so there's no choice between left or right non- terminals to expand • How about the following derivation? – S  SbS  SbSbS  SbabS  ababS  ababa CMSC 330 10 Tips for Designing Grammars 1. Use recursive productions to generate an arbitrary number of symbols A  xA |  Zero or more x’s A  yA | y One or more y’s 2. Use separate non-terminals to generate disjoint parts of a language, and then combine in a production G = S  AB A  aA |  B  bB |  L(G) = a*b* CMSC 330 11 Tips for Designing Grammars (cont’d) 3. To generate languages with matching, balanced, or related numbers of symbols, write productions which generate strings from the middle {anbn | n 0} (not a regular language!) S  aSb |  Example: S  aSb  aaSbb  aabb {anb2n | n 0} S  aSbb |  CMSC 330 12 Tips for Designing Grammars (cont’d) {anbm | m 2n, n 0} S  aSbb | B |  B  bB | b The following grammar also works: S  aSbb | B B  bB |  How about the following? S  aSbb | bS |