Context free grammars

Context free grammars

• 1.
  Costas Busch -LSU 1 Context-Free Languages
• 2.
  Costas Busch -LSU 2 Regular Languages }0:{ nba nn }{ R ww **ba *)( ba  Context-Free Languages
• 3.
  Costas Busch -LSU 3 Context-Free Languages Pushdown Automata Context-Free Grammars stack automaton
• 4.
  Costas Busch -LSU 4 Context-Free Grammars
• 5.
  Costas Busch -LSU 5 Grammars Grammars express languages Example: the English language grammar verbpredicate nounarticlephrasenoun predicatephrasenounsentence    _ _
• 6.
  Costas Busch -LSU 6 sleepsverb runsverb dognoun catnoun thearticle aarticle      
• 7.
  Costas Busch -LSU 7 Derivation of string “the dog sleeps”: sleepsdogthe verbdogthe verbnounthe verbnounarticle verbphrasenoun predicatephrasenounsentence       _ _
• 8.
  Costas Busch -LSU 8 Derivation of string “a cat runs”: runscata verbcata verbnouna verbnounarticle verbphrasenoun predicatephrasenounsentence       _ _
• 9.
  Costas Busch -LSU 9 Language of the grammar: L = { “a cat runs”, “a cat sleeps”, “the cat runs”, “the cat sleeps”, “a dog runs”, “a dog sleeps”, “the dog runs”, “the dog sleeps” }
• 10.
  Costas Busch -LSU 10 catnoun  Variables Sequence of Terminals (symbols) Productions predicatephrasenounsentence _ Sequence of Variables
• 11.
  Costas Busch -LSU 11 Another Example   S aSbSGrammar: Variable Sequence of terminals and variables The right side may be 
• 12.
  Costas Busch -LSU 12 Grammar: Derivation of string :   S aSbS abaSbS  ab aSbS  S
• 13.
  Costas Busch -LSU 13 Grammar: Derivation of string : aabbaaSbbaSbS  aSbS  S aabb   S aSbS
• 14.
  Costas Busch -LSU 14 aaabbbaaaSbbbaaSbbaSbS  aaaabbbbaaaaSbbbb aaaSbbbaaSbbaSbS   Other derivations:   S aSbSGrammar:
• 15.
  Costas Busch -LSU 15   S aSbS }0:{  nbaL nn Grammar: Language of the grammar 
• 16.
  Costas Busch -LSU 16 We write: Instead of: aaabbbS *  aaabbbaaaSbbbaaSbbaSbS  for zero or more derivation steps A Convenient Notation
• 17.
  Costas Busch -LSU 17 nww * 1  nwwww  321 in zero or more derivation steps In general we write: If: ww * Trivially:
• 18.
  Costas Busch -LSU 18   S aSbS aaabbbS abS S * * *    Example Grammar Possible Derivations baaaaaSbbbbaaSbbS  
• 19.
  Costas Busch -LSU 19   S aSbS |aSbS  thearticle aarticle   theaarticle | Another convenient notation:
• 20.
  Costas Busch -LSU 20 Formal Definitions  PSTVG ,,, Set of variables Set of terminal symbols Start variable Set of productions Grammar:
• 21.
  Costas Busch -LSU 21 Context-Free Grammar: All productions in are of the form sA  String of variables and terminals ),,,( PSTVG  P Variable
• 22.
  Costas Busch -LSU 22 |aSbS   PSTVG ,,, }{SV  },{ baT  },{  SaSbSP variables terminals productions start variable Example of Context-Free Grammar
• 23.
  Costas Busch -LSU 23 For a grammar with start variableG S String of terminals or *},:{)( * TwwSwGL   Language of a Grammar:
• 24.
  Costas Busch -LSU 24 context-free grammar : }0:{)(  nbaGL nn nn baS   G Example: Since, there is derivation for any 0n |aSbS 
• 25.
  Costas Busch -LSU 25 A language is context-free if there is a context-free grammar with L G )(GLL  Context-Free Language definition:
• 26.
  Costas Busch -LSU 26 since context-free grammar : }0:{  nbaL nn G Example: is a context-free language generates LGL )( |aSbS 
• 27.
  Costas Busch -LSU 27 ||bSbaSaS  abbaabSbaaSaS  Context-free grammar :G Example derivations: abaabaabaSabaabSbaaSaS  )(GL }*},{:{ bawwwR  Palindromes of even length Another Example
• 28.
  Costas Busch -LSU 28 || SSaSbS  ababSaSbSSSS  Context-free grammar :G Example derivations: abababaSbabSaSbSSSS  }prefixanyin )()(and ),()(:{ v vnvn wnwnw ba ba  )(GL () ((( ))) (( )) Describes matched parentheses: )b(, a Another Example
• 29.
  Costas Busch -LSU 29 Derivation Order and Derivation Trees
• 30.
  Costas Busch -LSU 30 Derivation Order Consider the following example grammar with 5 productions: ABS .1   A aaAA .3 .2   B BbB .5 .4
• 31.
  Costas Busch -LSU 31 aabaaBbaaBaaABABS 54321  Leftmost derivation order of string aab : At each step, we substitute the leftmost variable ABS .1   A aaAA .3 .2   B BbB .5 .4 Derivation of aab
• 32.
  Costas Busch -LSU 32 aabaaAbAbABbABS 32541  Rightmost derivation order of string aab: At each step, we substitute the rightmost variable ABS .1   A aaAA .3 .2   B BbB .5 .4 Derivation of aab
• 33.
  Costas Busch -LSU 33 aabaaAbAbABbABS 32541  Rightmost derivation of :aab aabaaBbaaBaaABABS 54321  Leftmost derivation of :aab ABS .1   A aaAA .3 .2   B BbB .5 .4
• 34.
  Costas Busch -LSU 34 Derivation Trees Consider the same example grammar: aabaaBbaaABbaaABABS  And a derivation of :aab ABS  |aaAA  |BbB 
• 35.
  Costas Busch -LSU 35 ABS  S BA ABS  |aaAA  |BbB  yield AB
• 36.
  Costas Busch -LSU 36 aaABABS  a a A S BA ABS  |aaAA  |BbB  yield aaAB
• 37.
  Costas Busch -LSU 37 aaABbaaABABS  S BA a a A B b ABS  |aaAA  |BbB  yield aaABb
• 38.
  Costas Busch -LSU 38 aaBbaaABbaaABABS  S BA a a A B b  ABS  |aaAA  |BbB  yield aaBbBbaa 
• 39.
  Costas Busch -LSU 39 aabaaBbaaABbaaABABS  yield aabbaa  S BA a a A B b   Derivation Tree ABS  |aaAA  |BbB  (parse tree)
• 40.
  Costas Busch -LSU 40 aabaaBbaaBaaABABS  aabaaAbAbABbABS  TRY IT OUT! Sometimes, derivation order doesn’t matter Leftmost derivation: Rightmost derivation:
• 41.
  Costas Busch -LSU 41 aabaaBbaaBaaABABS  aabaaAbAbABbABS  S BA a a A B b   Give same derivation tree Sometimes, derivation order doesn’t matter Leftmost derivation: Rightmost derivation:
• 42.
  Costas Busch -LSU 42 Ambiguity
• 43.
  Costas Busch -LSU 43 Grammar for mathematical expressions ))(()( aaaaaaa  Example strings: Denotes any number aEEEEEE |)(|| 
• 44.
  Costas Busch -LSU 44 A leftmost derivation for aaa  aaaEaa EEaEaEEE *  aEEEEEE |)(||  PARSE TREE ???
• 45.
  Costas Busch -LSU 45 A leftmost derivation for aaa  E EE EE  a a a  aaaEaa EEaEaEEE *  aEEEEEE |)(|| 
• 46.
  Costas Busch -LSU 46 aaaEaa EEaEEEEEE   Another leftmost derivation for aEEEEEE |)(||  aaa  PARSE TREE ???
• 47.
  Costas Busch -LSU 47 E EE  a a  EE a aaaEaa EEaEEEEEE   Another leftmost derivation for aEEEEEE |)(||  aaa 
• 48.
  Costas Busch -LSU 48 aaa  E EE  a a  EE a E EE EE  a a a  Two derivation trees for aEEEEEE |)(|| 
• 49.
  Costas Busch -LSU 49 E EE   EE E EE EE  2 2 2 2 2 2 222  aaa take 2a
• 50.
  Costas Busch -LSU 50 E EE   EE E EE EE   6222  2 2 2 2 2 2 8222  4 2 2 2 6 2 2 24 8 Good Tree Bad Tree Compute expression result using the tree
• 51.
  Costas Busch -LSU 51 Two different derivation trees may cause problems in applications which use the derivation trees: • Evaluating expressions • In general, in compilers for programming languages
• 52.
  Costas Busch -LSU 52 Ambiguous Grammar: A context-free grammar is ambiguous if there is a string which has: two different derivation trees or two leftmost derivations G )(GLw (Two different derivation trees give two different leftmost derivations and vice-versa)
• 53.
  Costas Busch -LSU 53 E EE  a a  EE a E EE EE  a a a  string aaa  has two derivation trees aEEEEEE |)(||  this grammar is ambiguous since Example:
• 54.
  Costas Busch -LSU 54 string aaa  has two leftmost derivations aaaEaa EEaEEEEEE   aaaEaa EEaEaEEE *  aEEEEEE |)(||  this grammar is ambiguous also because
• 55.
  Costas Busch -LSU 55 IF_STMT if EXPR then STMT | if EXPR then STMT else STMT Another ambiguous grammar: Variables Terminals Very common piece of grammar in programming languages
• 56.
  Costas Busch -LSU 56 If expr1 then if expr2 then stmt1 else stmt2 TWO Parse Trees or Derivation Trees ???
• 57.
  Costas Busch -LSU 57 If expr1 then if expr2 then stmt1 else stmt2 IF_STMT expr1 then elseif expr2 then STMT stmt1 if IF_STMT expr1 then else if expr2 then STMT stmt2if stmt1 stmt2 Two derivation trees
• 58.
  Costas Busch -LSU 58 In general, ambiguity is bad and we want to remove it Sometimes it is possible to find a non-ambiguous grammar for a language But, in general ιt is difficult to achieve this
• 59.
  Costas Busch -LSU 59 aE EE EEE EEE     )( aEF FFTT TTEE |)( | |    Ambiguous Grammar Non-Ambiguous Grammar Equivalent generates the same language A successful example:
• 60.
  Costas Busch -LSU 60 aaaFaaFFa FTaTaTFTTTEE   E E T T  F F a T F a a aEF FFTT TTEE |)( | |    Unique derivation tree for aaa 
• 61.
  Costas Busch -LSU 61 }{}{ mmnmnn cbacbaL  0, mn An un-successful example: every grammar that generates this language is ambiguous L is inherently ambiguous:
• 62.
  Costas Busch -LSU 62 }{}{ mmnmnn cbacbaL  | |11 aAbA AcSS   | |22 bBcB BaSS  21 | SSS  Example (ambiguous) grammar for :L
• 63.
  Costas Busch -LSU 63 The string Lcba nnn  has always two different derivation trees (for any grammar) S 1S S 2S 1S c 2Sa For example