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 0n |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 2a
- 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