1 / 18

7 . Properties of Context-Free Languages

7 . Properties of Context-Free Languages. CIS 5513 - Automata and Formal Languages – Pei Wang. Chomsky normal form. A CFL can be generated by many CFGs

rbeach
Download Presentation

7 . Properties of Context-Free Languages

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 7. Properties of Context-Free Languages CIS 5513 - Automata and Formal Languages – Pei Wang

  2. Chomsky normal form A CFL can be generated by many CFGs Every CFL  {ɛ} can be generated by a CFG in Chomsky normal form (CNF), where each rule is in the form of A → BC or A → a, i.e., every variable becomes either two variables or one terminal Every CFG can be converted into CNF in several steps

  3. Removing ɛ-productions A symbol A is nullable if A * ɛ, i.e., there is a production A →ɛ, or A → B1B2 … Bk where B1B2 … Bk are all nullable If A is nullable, then B → CAD should produce a variant B → CD, and A cannot derive ɛ anymore in B → CAD All the ɛ-productions can be eliminated by treating all the variables the above way

  4. Removing ɛ-productions: example S → AB A → aAA | ɛ B → bBB | ɛ S, A, and B are all nullable. New grammar: S → AB | A | B A → aAA | aA | a B → bBB | bB | b

  5. Removing unit productions A unit production has the form A → B, and (A, B) is a unit pair if A * B A unit pair can be removed by expanding the involved variables all the way until the result is not a unit production If there is a cycle of expansion like A → B → C → → A then all the variables involved can be merged

  6. Removing unit productions: example I → a | b | Ia | Ib | I0 | I1 F → I | (E) T → F | T * F E → T | E + T changes to I → a | b | Ia | Ib | I0 | I1 F → a | b | Ia | Ib | I0 | I1 | (E) T → a | b | Ia | Ib | I0 | I1 | (E) | T * F E → a | b | Ia | Ib | I0 | I1 | (E) | T * F | E + T

  7. Removing useless symbols A symbol X is useful if it is both reachable and generating, i.e., S * αXβ * w Removing a useless symbol in a grammar will not change the language it generates • Eliminate nongenerating symbols and all productions involving such symbols • Eliminate unreachable symbols The order of the above steps matters

  8. Useless symbols: example Given CFG: S → AB | a A → b B is not generating, so the grammar is S → a A → b Now A is not reachable, so the grammar is S → a

  9. CFG to Chomsky normal form Convert a CFG into CNF (not unique): • Eliminate ɛ-productions • Eliminate unit productions • Eliminate useless symbols • Change non-CNF productions into CNF productions, i.e., A → BCD becomes A → BE, E → CD A → Fg becomes A → FG, G → g

  10. Decision properties of CFL’s [Complexity-related topics will not be covered] Whether a CFL is empty can be decided by checking whether the start symbol of its grammar is generating Whether a string belongs to a CFL can be decided using dynamic programming to incrementally build up the string

  11. Testing membership in a CFL The CYK algorithm: use a CFG in CNF to incrementally find all variables that generate the substrings The triangular table is filled bottom-up, where Xij comes fromXikX(k+1)j for all possible k values, according to the grammar

  12. Membership decision for CFL

  13. Greibach normal form Every nonempty CFL without ɛ can be generated from a grammar each of whose production rule has the form A → aα where a is a terminal, and α is a string of zero or more variables This form can be obtained from PDA with a single state and accept by empty stack

  14. Pumping lemma for CFL A sufficiently long string must be derived by using the same variable repeatedly in a path of the parse tree

  15. Pumping lemma for CFL (cont) A part of the parse tree can be repeated: S * uAy A * vAx A * w

  16. Languages that are not CFL The pumping lemma can be used to show that some languages are not CFL: • L = {0m1m2m | m 1} : for the n in pumping lemma, pick the word z = 0n1n2n = uvwxy, since there are n 1’s in the middle, vwx cannot contains both 0 and 2, so repeat it will produce a word not in the language • To prove L = {ww} is not CFL, pump the word 0n1n0n1n , then discuss the cases

  17. Closure properties of CFL CFLs are closed under the operation of • Substitution (replace a terminal by a CFL) • Union • Concatenation • Closure (* and +) • Reversal • Homomorphism • Inverse homomorphism

  18. Closure properties of CFL (cont.) CFL’s are not closed under complement, intersection, and difference Example: {0n1n2i | n 1, i 1} and {0i1n2n | n 1, i 1} are both CFL’s, but their intersection is not Example: {0,1}*  {ww} is CFL, but {ww} is not The intersection or difference of a CFL and a regular language is a CFL

More Related